Frequently Asked Questions in Mathematics

The Sci.Math FAQ Team. Editor: Alex López-Ortiz

e-mail: alopez-o@neumann.UWaterloo.ca

Contents

1 Introduction 4 1.1 Why a list of Frequently Asked Questions? . . . . . . . . . . . . . . . . . . . 4 1.2 Frequently Asked Questions in Mathematics? . . . . . . . . . . . . . . . . . 5

2 Fundamentals 6

2.1 Algebraic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Monoids and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.2 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.3 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.4 Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 What are numbers? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Construction of the Number System . . . . . . . . . . . . . . . . . . 11 2.2.3 Construction of N . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.4 Construction of Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.5 Construction of Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.6 Construction of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2.7 Construction of C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.8 Rounding things up . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.9 What’s next? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Number Theory 16

3.1 Fermat’s Last Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.1 History of Fermat’s Last Theorem . . . . . . . . . . . . . . . . . . . . 16 3.1.2 What is the current status of FLT? . . . . . . . . . . . . . . . . . . . 16 3.1.3 Related Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.4 Did Fermat prove this theorem? . . . . . . . . . . . . . . . . . . . . . 18

3.2 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.1 Largest known Mersenne prime . . . . . . . . . . . . . . . . . . . . . 20 3.2.2 Largest known prime . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.3 Largest known twin primes . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.4 Largest Fermat number with known factorization . . . . . . . . . . . 21 3.2.5 Algorithms to factor integer numbers . . . . . . . . . . . . . . . . . . 21 3.2.6 Primality Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2.7 List of record numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.8 What is the current status on Mersenne primes? . . . . . . . . . . . . 23 3.2.9 Formulae to compute prime numbers . . . . . . . . . . . . . . . . . . 25

4 Special Numbers and Functions 27 4.1 How to compute digits of π? . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Euler’s formula: e^{iπ} = −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3 What is 0^{0} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.3.1 Why is 0.9999 … = 1? . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.4 Name for f(x)^{f(x)} = x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.5 Some Famous Mathematical Constants . . . . . . . . . . . . . . . . . . . . . 37

5 Human Interest 38 5.1 Indiana bill sets the value of π to 3 . . . . . . . . . . . . . . . . . . . . . . . 38 5.2 Fields Medal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2.1 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.2.2 Table of Awardees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Erdos Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.4 Why is there no Nobel in mathematics? . . . . . . . . . . . . . . . . . . . . 49 5.5 International Mathematics Olympiad and Other Competitions . . . . . . . . 51 5.6 Who is N. Bourbaki? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Mathematical Trivia 52 6.1 Names of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7 Famous Problems in Mathematics 56 7.1 The Four Colour Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.2 The Trisection of an Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.3 Which are the 23 Hilbert Problems? . . . . . . . . . . . . . . . . . . . . . . 58 7.4 Unsolved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.4.1 Does there exist a number that is perfect and odd? . . . . . . . . . . 58 7.4.2 Collatz Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.4.3 Goldbach’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.4.4 Twin primes conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 59

8 Mathematical Games 60 8.1 The Monty Hall problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 8.2 Master Mind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 9 Axiom of Choice and Continuum Hypothesis 62

9.1 The Axiom of Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

9.1.1 Relevance of the Axiom of Choice . . . . . . . . . . . . . . . . . . . . 62 9.2 Cutting a sphere into pieces of larger volume . . . . . . . . . . . . . . . . . . 66 9.3 The Continuum Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

10 Formulas of General Interest 73 10.1 How to determine the day of the week, given the month, day and year …. 73 10.2 Symbolic Computation Packages . . . . . . . . . . . . . . . . . . . . . . . . . 76 10.3 Formula for the Surface Area of a sphere in Euclidean N-Space . . . . . . . 78 10.4 Formula to compute compound interest. . . . . . . . . . . . . . . . . . . . . 79

11 References, General Bibliography and Textbooks 83

12 The Sci.Math FAQ Team 86 12.1 Copyright Notice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Chapter 1

Introduction

1.1 Why a list of Frequently Asked Questions?

The Net, as users call the Internet, and specially newsgroups, (i.e. Usenet) created a demand of knowledge without parallel since the invention of the printing press. Surprisingly, the type of knowledge demanded from and by the Usenet community had, in most cases, little in common –both in structure and content– with that of printed in current publications. This defined Usenet as more of an alternative to books rather than a replacement thereof^{1}

In the Net, questions posed are, more often than not, at the level of an amateur practitioner –even in cases where the question was posed by a professional in the field. Similarly, the quality of the answers varies greatly, ranging from the incorrect or disrespectful, to summaries of the state of the art in the topic in question.

Other characteristics of communication on the Net are simply inherited from restrictions of the medium. The unit of knowledge is a screenful worth of text (a scrit, from screen and bit). Articles exceeding that limit are usually disregarded.

The lack of memory of the medium generates a repetition of topics, much to the chagrin of old time citizens of the Net. Frequently asked questions lists palliate some of these deficiencies by providing a record of relevant information while at the same time never being outdated.

Thus, typically a list of frequently asked questions is “posted” at least once a month, and updated at least as frequently. And, in what must be a first for an information based product, FAQ lists “expire” on a given date, very much like any other perishable item.

1It could be argued that books fulfill their mandate and purpose to everybody’s satisfaction. Thus, even though the Net could, in principle, replace the need for books, people choose not to do so. Instead it’s domain is defined, by its very nature to be disjoint from books.

1.2 Frequently Asked Questions in Mathematics?

If I had to describe the contents of the FAQ in Mathematics in a single sentence, I would call it mathematical gossip or perhaps non-trivial mathematical trivia.

The FAQ list is a compilation of knowledge of interest to most professional and amateur mathematicians, ranging from advanced topics such as Wiles’ proposed proof to Fermat’s Last Theorem to the list of Fields Medal winners.

Fundamentals

2.1 Algebraic structures

We will attempt to give a brief explanation of the following concepts:

• N is a monoid

• Z is an integral domain

• Q is a field

• in the field R the order is complete

• the field C is algebraically complete

If you have been asked by a child to give them arithmetic problems, so they could show off their newly learned skills in addition and subtraction I’m sure that after a few problems such as: 2+3, 9−5, 10+2 and 6−4, you tried tossing them something a little more difficult: 4 − 7 only to be told “That’s not allowed.”

What you may not have realized is that you and the child did not just have different objects in mind (negative numbers) but entirely different algebraic systems. In other words a set of objects (they could be natural numbers, integers or reals) and a set of operations, or rules regarding how the numbers can be combined.

We will take a very informal tour of some algebraic systems, but before we define some of the terms, let us build a structure which will have some necessary properties for examples and counterexamples that will help us clarify some of the definitions.

We know that any number that is divided by six will either leave a remainder, or will be divided exactly (which is after all the remainder 0). Let us write any number by the remainder n it leaves after division by six, denoting it as [n]. This means that, 7, 55 and 1 will all be written [1], which we call the class to which they all belong: i.e. 7 ∈ [1], 55 ∈ [1], or, a bit more technically, they are all equivalent to 1 modulo 6. The complete set of class will contain six elements, and this is called partitioning numbers into equivalent classes because it separates (or partitions) all of our numbers into these classes, and any one number in a class is equivalent to any other in the same class.

One interesting thing we can do with these classes is to try to add or to multiply them. What can [1] + [3] mean? We can, rather naively try out what they mean in “normal” arithmetic: [1] + [3] = [1 + 3] = [4]. So far so good, let us try a second example 25 ∈ [1] and 45 ∈ [3], their sum is 70 which certainly belongs to [4]. Here we see what we meant above by equivalence, 25 is equivalent to 1 as far as this addition is concerned. Of course this is just one example, but fortunately it can be proven that the sum of two classes is always the class of the sums.

Now this is the kind of thing we all do when we add hours for example, 7 (o’ clock) plus 6 hours is 1 (o’ clock), and all we are really doing is adding hours (modulo 12).

The neat part comes with multiplication, as we will see later on. But for now just remember, it can be proven that something like [4] × [5] = [2] will work: the product of two classes is the class of the product.

Now for some of the necessary terminology.

2.1.1 Monoids and Groups

We need to define a group.

Let us take a set of objects and a rule (called a binary operation) which allows us to combine any two elements of this set. Addition is an example from math, or ANDing in some computer language.

The set must be closed under the operation. That means that when two elements are combined the result must also be in the set. For example the set containing even numbers will always give us an even number when two elements are added together. But if we restrict ourselves to odd numbers, their sum is not an odd number and so we know right off the bat that the set of odd numbers and addition cannot constitute a group. Some books will consider closure in the definition of binary operation, and others add it as one of the requirements for a group along with the ones that follow below.

The set and the operation is called a group if the binary operation satisfies the following criteria:

• the operation is associative, which means it doesn’t matter how you group the elements you are operating on, for example in our set of remainders: [1] + ([3] + [4]) = ([1] + [3]) + [4]

• there is an identity element, meaning: one of the elements combined with the others in the set doesn’t change them in the least. For example the zero in addition, or the one in multiplication.

• every element has an inverse with respect to that operation. If you combine an element and its inverse you get the identity (of that operation) back.

(Be careful with this last one, −3 is the inverse of 3 in addition, since they give us 0 when added, but 1/3 is the inverse of 3 with respect to multiplication, since 3 × 1/3 = 1 the identity under multiplication.)

So we can see that the set of natural numbers N(with the operation of addition) is not even a group, since there is no inverse for 5, for example. (In other words there is no natural number which added to 5 will give us zero.) And so the third rule for our operation is violated. But it still has some structure, even if it is not as rich as the ones we’ll see later on.

Sets with an associative operation (the first condition above) are called semigroups, and if they also have an identity element (the second condition) then they are called monoids.

Our set of natural numbers under addition is then an example of a monoid, a structure that is not quite a group because it is missing the requirement that every element have an inverse under the operation (Which is why in elementary school 4 − 7 is not allowed.)

What about the set of integers, is it a group?

By itself this question is nonsensical. Why? Well, we have not mentioned under what operation. OK, let us say: the set of integers with addition.

Now, addition is associative, the zero does not change any number when added to it, and for every number n we can add −n and get zero. So it’s a group all right.

In fact it is a special kind of group. When we can perform the operation on the two elements in any order (e.g a + b = b + a) then the group is called commutative, or Abelian in honor of Abel. Not every operation is commutative, for example three minus two is certainly not the same as two minus three. Our set of integers under addition is then an Abelian group.

2.1.2 Rings

If we take an Abelian group (remember: a set with a binary operation) and we define a second operation on it we get a bit more of a structure than we had with just a group.

If the second operation is associative, and it is distributive over the first then we have a ring. Note that the second operation may not have an identity element, nor do we need to find an inverse for every element with respect to this second operation. As for what distributive means, intuitively it is what we do in math when perform the following change: a × (b + c) = (a × b) + (a × c).

If the second operation is also commutative then we have what is called a commutative ring. The set of integers (with addition and multiplication) is a commutative ring (with even an identity - called unit element - for multiplication).

Now let us go back to our set of remainders. What happens if we multiply [5] × [1]? We see that we get [5], in fact we can see a number of things according to our definitions above, [5] is its own inverse, and [1] is the multiplicative element. We can also show easily enough (by creating a complete multiplication table) that it is commutative. But notice that if we take [3] and [2], neither of which are equal to the class that the zero belongs to [0], and we multiply them, we get [3] × [2] = [0]. This bring us to the next definition. In a commutative ring, let us take an element which is not equal to zero and call it a. If we can find a non-zero element, say b that combined with a equals zero ( a × b = 0) then a is called a zero divisor.

A commutative ring is called an integral domain if it has no zero divisors. Well the set Z with addition and multiplication fullfills all the necessary requirements, and so it is an integral domain. Notice that our set of remainders is not an integral domain, but we can build a similar set with remainders of division by five, for example, and voilà, we have an integral domain.

Let us take, for example, the set Q of rational numbers with addition and multiplication - I’ll leave out the proof that it is a ring, but I think you should be able to verify it easily enough with the above definitions. But to give you a head start, notice the addition of rationals follow all the requirements for an abelian group. If we remove the zero we will have another abelian group, and that implies that we have something more than a ring, in fact, as we will see in the next section.

2.1.3 Fields

Now we can make one step further. If the elements of a ring, excluding the zero, form an abelian group (with the second operation) then it is a field. For example, write the multiplication table of the remainders of division by 5, and you will see that it satisfies all the requirements for a group: (You will probably have noticed that the group does not contain the number five itself since [5] = [0].)

1 2 3 4
1 1 2 3 4
2 2 4 1 3
3 3 1 4 2
4 4 3 2 1

(Why isn’t the set of divisors of six - excluding the zero and under multiplication - a group? That’s easy enough, since we have excluded the zero we do not have the result of [2] × [3] in our set, so it isn’t closed.)

2.1.4 Ordering

Given a ring, we can say that it is ordered when you have a special subset of that ring behaves in a very special way. If any two elements of that special subset are added or multiplied their sum and their product are again in the special subset. Take the negative numbers in R, can they be that special subset? Well the sum seems to be allright, it is also a negative number. But things don’t work with the product: it is positive. What about the positive numbers? Yep, and in fact we call that special subset, the set of positive elements. Now, we gave the definition for an ordered ring, we can also define an ordered field the same way.

But what does a complete ordered field mean? Well the definition looks rather nasty: it is complete if every non-empty subset which posesses an upper bound has a least upper bound.

Let’s translate some of that, trying to lose as little information on the way. A bound is something that guarantees that all of the elements of your set are on one side of it (reasonably enough). For example, certainly all negative reals are less than 100, so 100 is a bound (it is in fact an upper bound ’cause all negatives are “below” it). But there are lots of other bounds, 1, 5, 26 will all do nicely. The question now is, of all of these (upper bounds) which is the smallest, that is which one is “the border” so to speak? Does it always exists?

Let’s take the rationals, and look at the following numbers:

1.4, 1.41, 1.414, 1.4142, 1.41421, …

Now each of these is a rational number (it can be written as a fraction), and they are getting closer and closer to a number we’ve probably seen before (just take out your calculator and find the square root of two). So we can write the shorthand for this series as \sqrt{2}. Certainly we can find an upper bound for this series, 3 will do nicely, but so can 1.5, or 1.42. But what is the smallest. Well there isn’t any. Not among the rational at least, because no matter what fraction you give I can give you one closer to the square root of two. What about the square root of two itself? Well it’s not a rational number (I’ll skip the proof, but it is really rather easy) so you can’t use it. If you want another series which is really neat look at the section on “Euler’s formula” in the FAQ.

And that is where the reals come in. Any set or reals that is bounded you can certainly find the smallest of these bounds. (By the way this “least upper bound” is abbreviated “l.u.b.”, or “sup” for supremum.) We can also turn things around and talk of lower bounds, and of the largest of these etc. but most of that will be just a mirror image of what we have dealt with so far.

So that should be it. And for years that did seem to be it, we seemed to have all the numbers we’d ever care to have.

There was just one small stick in the works, but most people just sort of pretended not to notice, and that was that not all polynomials had solutions. One simple polynomial of this kind is x^{2} + 1 = 0. It’s so simple, yet there’s no self respecting number that would solve this polynomial. There were these funny answers which seemed like they should be solutions but no one could make any sense out of them, so they were considered imaginary solutions. Which was really too bad because they were given the name of imaginary numbers and now that the name stuck we realize that they are numbers just as good as any of the ones we have been using for centuries. And in fact that takes us to the last great pinnacle in this short excursion. The field of complex numbers.

We can define an algebraically closed field as a field where every nonconstant polynomial (i.e. one with an x in it from high school days) has a zero in the field. Whew! This in short means that as long as the polynomial is not a constant number (which is no fun anyways) but something which looks like it wants a solution, like 5x^{3} − 2x^{2} + 6 = 0 it will always have one, if you are working with complex numbers and not just reals.

There is another definition which is probably just as good, but may or may not be easier: A field is algebraically closed if every polynomial splits into linear factors. Linear factors are briefly factors not containing x to any power of two or higher, in other words in the form: ax + b. For example x^{2} + x − 6 can be factored as (x + 3)(x − 2), but if we are in the field of reals we cannot factor x^{2} + 1, but we can in the field of complex numbers: x^{2} + 1 = (x − i)(x + i), where, you may recall, i^{2} = −1.

2.2 What are numbers?

2.2.1 Introduction

Informally:

• N = {0, 1, …} or N = {1, 2, …} Wether 0 is in N depends on where you live and what is your field of interest. At the informal level it is a religious topic.

• Z = {…, −1, 0, 1, …}

• Q = {p/q|p, q ∈ Z and q ≠ 0}

• R = {d_{0}.d_{1}d_{2} … |d_{0} ∈ Z and 0 ≤ d_{i} ≤ 9 for i > 0}

• C = {a + b · i|a, b ∈ R and i^{2} = −1}

2.2.2 Construction of the Number System

Formally (following the mainstream in math) the numbers are constructed from scratch out of the axioms of Zermelo Fraenkel set theory (a.k.a. ZF set theory) [Enderton77, Henle86, Hrbacek84]. The only things that can be derived from the axioms are sets with the empty set at the bottom of the hierarchy. This will mean that any number is a set (it is the only thing you can derive from the axioms). It doesn’t mean that you always have to use set notation when you use numbers: just introduce the numerals as an abbreviation of the formal counterparts.

The construction starts with N and algebraically speaking, N with its operations and order is quite a weak structure. In the following constructions the structures will be strengthen one step at the time: Z will be an integral domain, Q will be a field, for the field R the order will be made complete, and field C will be made algebraically complete.

Before we start, first some notational stuff:

• a pair (a, b) = {{a}, {a, b}},

• an equivalence class [a] = {b|a ≡ b},

• the successor of a is s(a) = a ∪ {a}.

Although the previous notations and the constructions that follow are the de facto standard ones, there are different definitions possible.

2.2.3 Construction of N

• {} ∈ N

• if a ∈ N then s(a) ∈ N

• N is the smallest possible set such that the preceding rules hold.

Informally n = {0, …, n − 1} (thus 0 = {}, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}). We will refer to the elements of N by giving them a subscript _{n}. The relation <_{n} on N is defined as: a_{n} <_{n} b_{n} iff a_{n} ∈ b_{n}. We can define +_{n} as follows:

• a_{n} +_{n} 0_{n} = a_{n}

• a_{n} +_{n} s(b_{n}) = s(a_{n} +_{n} b_{n})

Define ∗_{n} as:

• a_{n} ∗_{n} 0_{n} = 0_{n}

• a_{n} ∗_{n} s(b_{n}) = (a_{n} ∗_{n} b_{n}) +_{n} a_{n}

2.2.4 Construction of Z

We define an equivalence relation on N × N: (a_{n}, b_{n}) ≡_{z} (c_{n}, d_{n}) iff a_{n} +_{n} d_{n} = c_{n} +_{n} b_{n}. Note that ≡_{z} “simulates” a subtraction in N. Z = {[(a_{n}, b_{n})]_{z}|a_{n}, b_{n} ∈ N}. We will refer to the elements of Z by giving them a subscript _{z}. The elements of N can be embedded as follows: embed_{n} : N → Z such that embed_{n}(a_{n}) = [(a_{n}, 0_{n})]_{z}. Furthermore we can define:

• [(a_{n}, b_{n})]_{z} <_{z} [(c_{n}, d_{n})]_{z} iff a_{n} +_{n} d_{n} <_{n} c_{n} +_{n} b_{n}

• [(a_{n}, b_{n})]_{z} +_{z} [(c_{n}, d_{n})]_{z} = [(a_{n} +_{n} c_{n}, b_{n} +_{n} d_{n})]_{z}

• [(a_{n}, b_{n})]_{z} ∗_{z} [(c_{n}, d_{n})]_{z} = [((a_{n} ∗_{n} c_{n}) +_{n} (b_{n} ∗_{n} d_{n}), (a_{n} ∗_{n} d_{n}) +_{n} (c_{n} ∗_{n} b_{n}))]_{z}

2.2.5 Construction of Q

We define an equivalence relation on Z × (Z\{0_{z}}): (a_{z}, b_{z}) ≡_{q} (c_{z}, d_{z}) iff a_{z} ∗_{z} d_{z} = c_{z} ∗_{z} b_{z}. Note that ≡_{q} “simulates” a division in Z. Q = {[(a_{z}, b_{z})]_{q}|a_{z} ∈ Z and b_{z} ∈ Z\{0_{z}}}. We will refer to the elements of Q by giving them a subscript _{q}. The elements of Z can be embedded as follows: embed_{z} : Z → Q such that embed_{z}(a_{z}) = [(a_{z}, 1_{z})]_{q}. Furthermore we can define:

• [(a_{z}, b_{z})]_{q} <_{q} [(c_{z}, d_{z})]_{q} iff a_{z} ∗_{z} d_{z} <_{z} c_{z} ∗_{z} b_{z} when 0_{z} <_{z} b_{z} and 0_{z} <_{z} d_{z}

• [(a_{z}, b_{z})]_{q} +_{q} [(c_{z}, d_{z})]_{q} = [((a_{z} ∗_{z} d_{z}) +_{z} (c_{z} ∗_{z} b_{z}), b_{z} ∗_{z} d_{z})]_{q}

• [(a_{z}, b_{z})]_{q} ∗_{q} [(c_{z}, d_{z})]_{q} = [(a_{z} ∗_{z} c_{z}, b_{z} ∗_{z} d_{z})]_{q}

2.2.6 Construction of R

The construction of R is different (and more awkward to understand) because we must ensure that the cardinality of R is greater than that of Q. Set c is a Dedekind cut iff

• {} ⊂ c ⊂ Q (strict inclusions!)

• c is closed downward:

if a_{q} ∈ c and b_{q} <_{q} a_{q} then b_{q} ∈ c

• c has no largest element:

there isn’t an element a_{q} ∈ c such that b_{q} <_{q} a_{q} for all b_{q} ≠ a_{q} ∈ c

You can think of a cut as taking a pair of scissors and cutting Q in two parts such that one part contains all the small numbers and the other part contains all large numbers. If the part with the small numbers was cut in such a way that it doesn’t have a largest element, it is called a Dedekind cut. R = {c|c is a Dedekind cut}. We will refer to the elements of R by giving them a subscript _{r}. The elements of Q can be embedded as follows: embed_{q} : Q → R such that embed_{q}(a_{q}) = {b_{q}|b_{q} <_{q} a_{q}}. Furthermore we can define:

• a_{r} <_{r} b_{r} iff a_{r} ⊂ b_{r} (strict inclusion!)

• a_{r} +_{r} b_{r} = {c_{q} +_{q} d_{q}|c_{q} ∈ a_{r} and d_{q} ∈ b_{r}}

• −_{r}a_{r} = {b_{q}| there exists an c_{q} ∈ Q such that b_{q} <_{q} c_{q} and (−1)_{q} ∗_{q} c_{q} ∉ a_{r}}

• |a_{r}|_{r} = a_{r} ∪ −_{r}a_{r}

• ∗_{r} is defined as:

– if not a_{r} <_{r} 0_{r} and not b_{r} <_{r} 0_{r}
then a_{r} ∗_{r} b_{r} = 0_{r} ∪ {c_{q} ∗_{q} d_{q}|c_{q} ∈ a_{r} and d_{q} ∈ b_{r}}
– if a_{r} <_{r} 0_{r} and b_{r} <_{r} 0_{r} then a_{r} ∗_{r} b_{r} = |a_{r}|_{r} ∗_{r} |b_{r}|_{r}
– otherwise a_{r} ∗_{r} b_{r} = −_{r}(|a_{r}|_{r} ∗_{r} |b_{r}|_{r})

There exists an alternative definition of R using Cauchy sequences: a Cauchy sequence is a s : N → Q such that s(i_{n}) +_{q} ((−1)_{q} ∗_{q} s(j_{n})) can be made arbitrary near to 0_{q} for all sufficiently large i_{n} and j_{n}. We will define an equivalence relation ≡_{r} on the set of Cauchy sequences as: r ≡_{r} s iff r(m_{n}) +_{q} ((−1)_{q} ∗_{q} s(m_{n})) can be made arbitrary close to 0_{q} for all sufficiently large m_{n}. R = {[s]_{r}|s is a Cauchy sequence}. Note that this definition is close to “decimal” expansions.

2.2.7 Construction of C

C = R×R. We will refer to the elements of C by giving them a subscript_{c}. The elements of R can be embedded as follows: embed_{r} : R → C such that embed_{r}(a_{r}) = (a_{r}, 0_{r}). Furthermore we can define:

• (a_{r}, b_{r}) +_{c} (c_{r}, d_{r}) = (a_{r} +_{r} c_{r}, b_{r} +_{r} d_{r})

• (a_{r}, b_{r}) ∗_{c} (c_{r}, d_{r}) = ((a_{r} ∗_{r} c_{r}) +_{r} −_{r}(b_{r} ∗ d_{r}), (a_{r} ∗_{r} d_{r}) +_{r} (b_{r} ∗_{r} c_{r}))

There exists an elegant alternative definition using ideals. To be a bit sloppy: C = R[x]/ < (x∗_{r}x)+_{r}1_{r} >, i.e. C is the resulting quotient ring of factoring ideal < (x∗_{r}x)+_{r}1_{r} > out of the ring R[x] of polynomials over R. The sloppy part is that we need to define concepts like quotient ring, ideal, and ring of polynomials. Note that this definition is close to working with i^{2} = −1: (x ∗_{r} x) +_{r} 1_{r} = 0_{r} can be rewritten as (x ∗_{r} x) = (−1)_{r}.

2.2.8 Rounding things up

At this moment we don’t have that N is a subset of Z, Z of Q, etc. But we can get the inclusions if we look at the embedded copies of N, Z, etc. Let

• N′ = ran embed_{r} ∘ embed_{q} ∘ embed_{z} ∘ embed_{n}

• Z′ = ran embed_{r} ∘ embed_{q} ∘ embed_{z}

• Q′ = ran embed_{r} ∘ embed_{q}

• R′ = ran embed_{r}

For these sets we have N′ ⊆ Z′ ⊆ Q′ ⊆ R′ ⊆ C. Furthermore these sets have all the properties that the “informal” numbers have.

2.2.9 What’s next?

Well, for some of the more alien parts of math we can extend this standard number system with some exotic types of numbers. To name a few:

• Cardinals and ordinals Both are numbers in ZF set theory [Enderton77, Henle86, Hrbacek84] and so they are sets as well. Cardinals are numbers that represent the sizes of sets, and ordinals are numbers that represent well ordered sets. Finite cardinals and ordinals are the same as the natural numbers. Cardinals, ordinals, and their arithmetic get interesting and “tricky” in the case of infinite sets.

• Hyperreals These numbers are constructed by means of ultrafilters [Henle86] and they are used in non-standard analysis. With hyperreals you can treat numbers like Leibnitz and Newton did by using infinitesimals.

• Quaternions and octonions Normally these are constructed by algebraic means (like the alternative C definition that uses ideals) [Shapiro75, Dixon94]. Quaternions are used to model rotations in 3 dimensions. Octonions, a.k.a. Cayley numbers, are just esoteric artifacts :-). Well, if you know where they are used for, feel free to contribute to the FAQ.

• Miscellaneous Just to name some others: algebraic numbers [Shapiro75], p-adic numbers [Shapiro75], and surreal numbers (a.k.a. Conway numbers) [Conway76].

Cardinals and ordinals are commonly used in math. Most mortals won’t encounter (let alone use) hyperreals, quaternions, and octonions.

References

J.H. Conway. On Numbers and Games, L.M.S. Monographs, vol. 6. Academic Press, 1976.

H.B. Enderton. Elements of Set Theory. Academic Press, 1977.

G.M. Dixon. Division Algebras; Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics. Kluwer Academic, 1994.

J.M. Henle. An Outline of Set Theory. Springer Verlag, 1986.

K. Hrbacek and T. Jech. Introduction to Set Theory. M. Dekker Inc., 1984.

L. Shapiro. Introduction to Abstract Algebra. McGraw-Hill, 1975.

This subsection of the FAQ is Copyright (c) 1994, 1995 Hans de Vreught. Send comments and or corrections relating to this part to J.P.M.deVreught@cs.tudelft.nl

Number Theory

3.1 Fermat’s Last Theorem

3.1.1 History of Fermat’s Last Theorem

Pierre de Fermat (1601-1665) was a lawyer and amateur mathematician. In about 1637, he annotated his copy (now lost) of Bachet’s translation of Diophantus’ Arithmetika with the following statement:

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

In English, and using modern terminology, the paragraph above reads as:

There are no positive integers such that x^{n} + y^{n} = z^{n} for n > 2. I’ve found a remarkable proof of this fact, but there is not enough space in the margin [of the book] to write it.

Fermat never published a proof of this statement. It became to be known as Fermat’s Last Theorem (FLT) not because it was his last piece of work, but because it is the last remaining statement in the post-humous list of Fermat’s works that needed to be proven or independently verified. All others have either been shown to be true or disproven long ago.

3.1.2 What is the current status of FLT?

Theorem 1 (Fermat’s Last Theorem) There are no positive integers x, y, z, and n > 2 such that x^{n} + y^{n} = z^{n}.

Andrew Wiles, a researcher at Princeton, claims to have found a proof. The proof was presented in Cambridge, UK during a three day seminar to an audience which included some of the leading experts in the field. The proof was found to be wanting. In summer 1994, Prof. Wiles acknowledged that a gap existed. On October 25th, 1994, Prof. Andrew Wiles released two preprints, Modular elliptic curves and Fermat’s Last Theorem, by Andrew Wiles, and Ring theoretic properties of certain Hecke algebras, by Richard Taylor and Andrew Wiles. The first one (long) announces a proof of, among other things, Fermat’s Last Theorem, relying on the second one (short) for one crucial step.

The argument described by Wiles in his Cambridge lectures had a serious gap, namely the construction of an Euler system. After trying unsuccessfully to repair that construction, Wiles went back to a different approach he had tried earlier but abandoned in favor of the Euler system idea. He was able to complete his proof, under the hypothesis that certain Hecke algebras are local complete intersections. This and the rest of the ideas described in Wiles’ Cambridge lectures are written up in the first manuscript. Jointly, Taylor and Wiles establish the necessary property of the Hecke algebras in the second paper.

The new approach turns out to be significantly simpler and shorter than the original one, because of the removal of the Euler system. (In fact, after seeing these manuscripts Faltings has apparently come up with a further significant simplification of that part of the argument.)

The papers were published in the May 1995 issue of Annals of Mathematics. For single copies of the issues send e-mail to jlorder@jhunix.hcf.jhu.edu for further directions.

In summary:

Both manuscripts have been published. Thousands of people have a read them. About a hundred understand it very well. Faltings has simplified the argument already. Diamond has generalized it. People can read it. The immensely complicated geometry has mostly been replaced by simpler algebra. The proof is now generally accepted. There was a gap in this second proof as well, but it has been filled since October 1994.

3.1.3 Related Conjectures

A related conjecture from Euler

x^{n} + y^{n} + z^{n} = c^{n} has no solution if n is ≥ 4

Noam Elkies gave a counterexample, namely 2682440^{4}+15365639^{4}+18796760^{4} = 20615673^{4}. Subsequently, Roger Frye found the absolutely smallest solution by (more or less) brute force: it is 95800^{4}+217519^{4}+414560^{4} = 422481^{4}. ”Several years”, Math. Comp. 51 (1988) 825-835.

This synopsis is quite brief. A full survey would run too many pages.

References

[1] J.P.Butler, R.E.Crandall,& R.W.Sompolski, Irregular Primes to One Million. Math. Comp., 59 (October 1992) pp. 717-722.

Fermat’s Last Theorem, A Genetic Introduction to Algebraic Number Theory. H.M. Edwards. Springer Verlag, New York, 1977.

Thirteen Lectures on Fermat’s Last Theorem. P. Ribenboim. Springer Verlag, New York, 1979.

Number Theory Related to Fermat’s Last Theorem. Neal Koblitz, editor. Birkhäuser Boston, Inc., 1982, ISBN 3-7643-3104-6

3.1.4 Did Fermat prove this theorem?

No he did not. Fermat claimed to have found a proof of the theorem at an early stage in his career. Much later he spent time and effort proving the cases n = 4 and n = 5. Had he had a proof to his theorem earlier, there would have been no need for him to study specific cases.

Fermat may have had one of the following “proofs” in mind when he wrote his famous comment.

• Fermat discovered and applied the method of infinite descent, which, in particular can be used to prove FLT for n = 4. This method can actually be used to prove a stronger statement than FLT for n = 4, viz, x^{4} + y^{4} = z^{2} has no non-trivial integer solutions. It is possible and even likely that he had an incorrect proof of FLT using this method when he wrote the famous “theorem”.

• He had a wrong proof in mind. The following proof, proposed first by Lame’ was thought to be correct, until Liouville pointed out the flaw, and by Kummer which latter became and expert in the field. It is based on the incorrect assumption that prime decomposition is unique in all domains.

The incorrect proof goes something like this:

We only need to consider prime exponents (this is true). So consider x^{p} + y^{p} = z^{p}. Let r be a primitive p-th root of unity (complex number)

Then the equation is the same as:

(x + y)(x + ry)(x + r^{2}y)...(x + r^{p−1}y) = z^{p}

Now consider the ring of the form:

a_{1} + a_{2}r + a_{3}r^{2} + ... + a_{p−1}r^{p−1}

where each a_{i} is an integer

Now if this ring is a unique factorization ring (UFR), then it is true that each of the above factors is relatively prime.

From this it can be proven that each factor is a pth power from which FLT follows. This is usually done by considering two cases, the first where p divides none of x, y, z; the second where p divides some of x, y, z. For the first case, if x + yr = u ∗ t^{p}, where u is a unit in Z[r] and t is in Z[r], it follows that x = y(modp). Writing the original equation as x^{p} + (−z)^{p} = (−y)^{p}, it follows in a similar fashion that x = −z(modp). Thus 2 ∗ x^{p} = x^{p} + y^{p} = z^{p} = −x^{p}(modp) which implies 3 ∗ x^{p} = 0(modp) and from there p divides one of x or 3|x. But p > 3 and p does not divides x; contradiction. The second case is harder. The problem is that the above ring is not an UFR in general.

Another argument for the belief that Fermat had no proof —and, furthermore, that he knew that he had no proof— is that the only place he ever mentioned the result was in that marginal comment in Bachet’s Diophantus. If he really thought he had a proof, he would have announced the result publicly, or challenged some English mathematician to prove it. It is likely that he found the flaw in his own proof before he had a chance to announce the result, and never bothered to erase the marginal comment because it never occurred to him that anyone would see it there.

Some other famous mathematicians have speculated on this question. Andre Weil, writes:

Only on one ill-fated occasion did Fermat ever mention a curve of higher genus x^{n} + y^{n} = z^{n}, and then hardly remains any doubt that this was due to some misapprehension on his part […] for a brief moment perhaps […] he must have deluded himself into thinking he had the principle of a general proof.

Winfried Scharlau and Hans Opolka report:

Whether Fermat knew a proof or not has been the subject of many speculations. The truth seems obvious … [Fermat’s marginal note] was made at the time of his first letters concerning number theory [1637]… as far as we know he never repeated his general remark, but repeatedly made the statement for the cases n = 3 and 4 and posed these cases as problems to his correspondents […] he formulated the case n = 3 in a letter to Carcavi in 1659 […] All these facts indicate that Fermat quickly became aware of the incompleteness of the [general] “proof” of 1637. Of course, there was no reason for a public retraction of his privately made conjecture.

However it is important to keep in mind that Fermat’s “proof” predates the Publish or Perish period of scientific research in which we are still living.

References

From Fermat to Minkowski: lectures on the theory of numbers and its historical development. Winfried Scharlau, Hans Opolka. New York, Springer, 1985.

Basic Number Theory. Andre Weil. Berlin, Springer, 1967

3.2 Prime Numbers

3.2.1 Largest known Mersenne prime

Mersenne primes are primes of the form 2^{p} − 1. For 2^{p} − 1 to be prime we must have that p is prime.

2^{2976221} − 1 is prime. It was discovered in 1997.

3.2.2 Largest known prime

The largest known prime is the Mersenne prime described above. The largest known non-Mersenne prime, is 391581 ∗ 2^{216193} − 1, discovered by Brown, Noll, Parady, Smith, Smith, and Zarantonello.

Throughout history, the largest known prime has almost always been a Mersenne prime; the period between Brown et al’s discovery in August 1989 and Slowinski & Gage’s in March 1992 is one of the few exceptions.

You can help find more primes. For more information see: The Great Internet Mersenne Prime Search home page on http://www.mersenne.org

References

Brown, Noll, Parady, Smith, Smith, and Zarantonello. Letter to the editor. American Mathematical Monthly, vol. 97, 1990, p. 214.

3.2.3 Largest known twin primes

The two largest known twin primes are 242206083 ∗ 2^{3}8880 ± 1. with 11713 digits, found by Indlekofer and Ja’rai in November, 1995. They are also the first known gigantic twin primes (primes with at least 10,000 digits).

Recent record holders are:

• 190116 ∗ 3003 ∗ 10^{5120} ± 1, with 5129 digits, by Harvey Dubner.

• 697053813 ∗ 2^{16352} ± 1, with 4932 digits, found by Indlekofer and Ja’rai in 1994.

• 1691232 ∗ 1001 ∗ 10^{4020} ± 1 with 4030 digits, found by H. Dubner.

• 4650828 ∗ 1001 ∗ 10^{3429} ± 1. Found by H. Dubner as well.

The two largest Sophie Germain primes (i.e. p and 2p + 1 are both primes) are p = 2687145 ∗ 3003 ∗ 10^{5072} − 1 and q = 2p + 1, found by Harvey Dubner, in October 3, 1995.

References

B. K. Parady and J. F. Smith and S. E. Zarantonello, Smith, Noll and Brown. Largest known twin primes. Mathematics of Computation, vol.55, 1990, pp. 381-382.

3.2.4 Largest Fermat number with known factorization

F_{11} = (2^{2^{11}}) + 1 which was factored by Brent & Morain in 1988. F_{9} = (2^{2^{9}}) + 1 = 2^{512} + 1 was factored by A.K. Lenstra, H.W. Lenstra Jr., M.S. Manasse & J.M. Pollard in 1990. F_{10} was factored by Richard Brent who found a 40-digit factor of 2^{1024} + 1 on October 20, 1995. The cofactor is a 252 digit number, which is not so easy to factor. Luckily, this number was also prime.

3.2.5 Algorithms to factor integer numbers

There are several known algorithms that have subexponential estimated running time, to mention just a few:

• Continued fraction algorithm.

• Quadratic sieve algorithm.

• Class Group method.

• Elliptic curve algorithm.

• Number field sieve.

• Dixon’s random squares algorithm.

• Valle’s two-thirds algorithm.

• Seysen’s class group algorithm.

References

A.K. Lenstra, H.W. Lenstra Jr. Algorithms in Number Theory. J. van Leeuwen (ed.), Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity Elsevier, pp. 673-715, 1990.

3.2.6 Primality Testing

The problem of primality testing and factorization are two distinct problems. If we concentrate on primality testing, we never need to know the actual factors. The only question to be answered is ”is the number in question prime or composite.”

Wilson’s Theorem: The integer p is prime if and only if (p−1)! is congruent to −1(modp) Since there is no known method for rapidly computing (N − 1)!(modN) in, say, log N

steps, so Wilson’s characterization of primes is of no practical value to the testing of the primality of N.

There are many different primality tests and they can be classified in at least three different ways:

1. Tests for numbers of special forms versus Tests for generic numbers

2. Tests with full justification versus Tests with justification based on conjectures

3. Deterministic tests versus Probabilistic or Monte Carlo tests

Miller’s Test In 1976, G. L. Miller proposed a primality test, which was justified using a generalized

form of Riemann’s hypothesis.

The APR Test The primality test devised by L. M. Adleman, C. Pomerance and R. S. Rumely (1983),

also known as the APR test, represents a breakthrough because:

1. It is applicable to arbitrary natural numbers N, without requiring the knowledge of factors of N − 1 or N + 1.

2. The running time t(N) is almost polynomial.

3. The test is justified rigorously, and for the first time ever in this domain, it is necessary to appeal to deep results in the theory of algebraic numbers; it involves calculations with roots of unity and the general reciprocity law for the power residue symbol.

The running time of the APR is at the present the world record for a deterministic primality test.

Soon afterwards, H. Cohen & A. K. Lenstra (1984) modified the APR test, making it more flexible, using Gauss sums in the proof (instead of the reciprocity law), and having the new test programmed for practical applications. It was the first primality test in existence that can routinely handle numbers of up 100 decimal digits, and it does so in about 45 seconds.

Monte Carlo methods Ribenboim mentions three Monte Carlo tests, due to R. Baillie & Wagstaff, Jr. (1980),

R. Solovay & V. Strassen (1977), and M. O. Rabin (1976, 1980).

Elliptic Curves Primality Proving, ECPP ECPP stands for ”Elliptic Curves and Primality Proving”. The algorithm is described

in:

A. O. L. Atkin and F. Morain
"Elliptic curves and primality proving"
To appear.

It is a deterministic algorithm that gives a certificate of primality for numbers that can be as large as 10**1000 (one thousand).

References

[1] A. O. L. Atkin and F. Morain
"Elliptic curves and primality proving"
To appear in Math. Comp.
%  Lieven Marchand <mal@bewoner.dma.be>
[2] F. Morain
"Courbes elliptiques et tests de primalite'"
The`se, Universite' de Lyon I, 1990.
Available at:
http://lix.polytechnique.fr/~morain/english-index.html

This subsection is copyright (C) 1995. Harry J. Smith, HJSmith@ix.netcom.com.

3.2.7 List of record numbers

Chris Caldwell (caldwell@utm.edu) maintains a list called ”The Largest Known Primes.” Some of the ways to get this list are:

web:     http://www.utm.edu/research/primes/largest.html
gopher:  unix1.utm.edu, directory 1/user/Public_FTP/pub/math/primes
ftp:     math.utm.edu, directory /pub/math/primes

Finger primes@math.utm.edu for a few record primes and the current ways to get the lists. He would like to know of any new titanic primes (over 1000 digits) so that he can add them to his list.

3.2.8 What is the current status on Mersenne primes?

The following Mersenne primes are known.

Number  p  Year Discoverer
1-4  2,3,5,7 pre-1500
5  13  1461 Anonymous
6-7  17,19  1588 Cataldi
8  31  1750 Euler
9  61  1883 I.M. Pervushin
10  89  1911 Powers
11  107  1914 Powers
12  127  1876 Lucas
13-14  521,607  1952 Robinson
15-17 1279,2203,2281  1952 R. M. Robinson
18  3217  1957 Riesel
19-20  4253,4423  1961 Hurwitz & Selfridge
21-23 9689,9941,11213  1963 Gillies
24  19937  1971 Tuckerman
25  21701  1978 Noll & Nickel
26  23209  1979 Noll
27  44497  1979 Slowinski & Nelson
28  86243  1982 Slowinski
29  110503  1988 Colquitt & Welsh
30  132049  1983 Slowinski
31  216091  1985 Slowinski
32  756839  1992 Slowinski & Gage
33  859433  1994 Slowinski & Gage
34  1257787  1996 Slowinski & Gage
35  1398269  1996 Armengaud, Woltman, et. al.
36???  2976221  1996 Spence, Woltman, et. al.

The way to determine if 2^{p} − 1 is prime is to use the Lucas-Lehmer test:

Lucas_Lehmer_Test(p):
u := 4
for i from 3 to p do
u := u^2-2 mod 2^p-1
od
if u == 0 then
2^p-1 is prime
else
2^p-1 is composite
fi

All exponents less than 1,481,800 have now been tested at least once.

References

An introduction to the theory of numbers. G.H. Hardy, E.M. Wright. Fifth edition, 1979, Oxford.

3.2.9 Formulae to compute prime numbers

There is no polynomial which gives all the prime numbers. This is a simple exercise to prove. There is no non-constant polynomial that only takes on prime values. The proof is simple enough that an high school student could probably discover it. See, for example, Ribenboim’s book The Book of Prime Number Records.

Note, however, by the work of Jones, Sato, Wada, and Wiens, there is a polynomial in 26 variables such that the set of primes coincides with the set of positive values taken by this polynomial. See Ribenboim, pp. 147-150.

But most people would object to the term “formula” restricted to mean polynomial. Can we not use summation signs, factorial, and the floor function in our “formula”? If so, then indeed, there are formulas for the prime numbers. Some of them are listed below.

A reasonable interpretation of the word “formula” is simply “Turing machine that halts on all inputs”. Under this interpretation, there certainly are halting Turing machines which compute the n-th prime number. However, nobody knows how to compute the n-th prime in time polynomial in log n. That’s still an open question.

Herb Wilf has addressed the question, “What is a formula?” in his article, “What is an answer?” which appeared in the American Mathematical Monthly, 89 (1982), 289-292. He draws a distinction between “formula” and “good formula”. Anyone who claims “there is no formula for the prime numbers” should read this article.

Here are just a few articles that discuss “formulas” for primes. Almost all of these do not require computation of the primes ahead of time. Most of them rely on standard mathematical functions such as summation, factorial, greatest integer function, etc.

References

C. Isenkrahe. Math. Annalen 53 (1900), 42-44.

W. H. Mills. Bulletin of the American Mathematical Society 53 (1947), 604.

L. Moser. Mathematics Magazine 23 (1950), 163-164.

E. M. Wright. American Mathematical Monthly 58 (1951), 616-618. (Correction, 59 (1952), 99.)

E. M. Wright. Journal of the London Mathematical Society 29 (1954), 63-71.

B. R. Srinivasan. Journal of the Indian Mathematical Society 25 (1961), 33-39.

C. P. Willans. Mathematics Gazette 48 (1964), 413-415. V. C. Harris. Nordisk Mat. Tidskr. 17 (1969), 82. U. Dudley. American Mathematical Monthly 76 (1969), 23-28. C. Vanden Eynden. American Mathematical Monthly 79 (1972), 625. S. W. Golomb. American Mathematical Monthly 81 (1974), 752-754. Algorithmic Number Theory. J.O. Shallit, E. Bach. (to be published, MIT Press). A Course in Computational Algebraic Number Theory. Henri Cohen. Springer-Verlag, Graduate Texts in Math, 1993.

Special Numbers and Functions

4.1 How to compute digits of π?

Symbolic Computation software such as Maple or Mathematica can compute 10,000 digits of π in a blink, and another 20,000-1,000,000 digits overnight (range depends on hardware platform).

It is possible to retrieve 1.25+ million digits of π via anonymous ftp from the site wuarchive.wustl.edu, in the files pi.doc.Z and pi.dat.Z which reside in subdirectory doc/misc/pi. New York’s Chudnovsky brothers have computed 2 billion digits of π on a homebrew computer.

The current record is held by Yasumasa Kanada and Daisuke Takahashi from the University of Tokyo with 51 billion digits of π (51,539,600,000 decimal digits to be precise).

Nick Johnson-Hill has an interesting page of Φπ trivia at: http://www.users.globalnet.co.uk/ nickjh/Pi.htm The new record for the number of digits of π is 4.29496 billion decimal digits of pi were

calculated and verified by 28th August ’95.

Related documents are available with anonymous ftp to www.cc.u-tokyo.ac.jp.

 ftp://www.cc.u-tokyo.ac.jp/

This computations were made by Yasumasa Kanada, at the University of Tokyo.

There are essentially 3 different methods to calculate π to many decimals.

1. One of the oldest is to use the power series expansion of atan(x) = x−x^{3}/3+x^{5}/5−… together with formulas like π = 16 ∗ atan(1/5) − 4 ∗ atan(1/239). This gives about 1.4 decimals per term.

2. A second is to use formulas coming from Arithmetic-Geometric mean computations. A beautiful compendium of such formulas is given in the book π and the AGM, (see references). They have the advantage of converging quadratically, i.e. you double the number of decimals per iteration. For instance, to obtain 1 000 000 decimals, around 20 iterations are sufficient. The disadvantage is that you need FFT type multiplication to get a reasonable speed, and this is not so easy to program.

3. A third one comes from the theory of complex multiplication of elliptic curves, and was discovered by S. Ramanujan. This gives a number of beautiful formulas, but the most useful was missed by Ramanujan and discovered by the Chudnovsky’s. It is the following (slightly modified for ease of programming):

Set k_{1} = 545140134; k_{2} = 13591409; k_{3} = 640320; k_{4} = 100100025; k_{5} = 327843840; k_{6} = 53360;

Then π = \frac{k_{6}\sqrt{k_{3}}}{S}, where

S = ∑_{n=0}^{∞}(−1)^{n} \frac{(6n)!(k_{2} + nk_{1})}{n!^{3}(3n)!(8k_{4}k_{5})^{n}}

The great advantages of this formula are that

1) It converges linearly, but very fast (more than 14 decimal digits per term).

2) The way it is written, all operations to compute S can be programmed very simply. This is why the constant 8k_{4}k_{5} appearing in the denominator has been written this way instead of 262537412640768000. This is how the Chudnovsky’s have computed several billion decimals.

An interesting new method was recently proposed by David Bailey, Peter Borwein and Simon Plouffe. It can compute the Nth hexadecimal digit of Pi efficiently without the previous N − 1 digits. The method is based on the formula:

π = ∑_{i=0}^{∞} \frac{1}{16^{i}} (\frac{4}{8i + 1} − \frac{2}{8i + 4} − \frac{1}{8i + 5} − \frac{1}{8i + 6})

in O(N) time and O(logN) space. (See references.) The following 160 character C program, written by Dik T. Winter at CWI, computes π

to 800 decimal digits.

int a=10000,b,c=2800,d,e,f[2801],g;main(){for(;b-c;)f[b++]=a/5;
for(;d=0,g=c*2;c-=14,printf("%.4d",e+d/a),e=d%a)for(b=c;d+=f[b]*a,
f[b]=d%--g,d/=g--,--b;d*=b);}

References

P. B. Borwein, and D. H. Bailey. Ramanujan, Modular Equations, and Approximations to π American Mathematical Monthly, vol. 96, no. 3 (March 1989), p. 201-220.

D. H. Bailey, P. B. Borwein, and S. Plouffe. A New Formula for Picking off Pieces of Pi, Science News, v 148, p 279 (Oct 28, 1995). also at http://www.cecm.sfu.ca/ pborwein

J.M. Borwein and P.B. Borwein. The arithmetic-geometric mean and fast computation of elementary functions. SIAM Review, Vol. 26, 1984, pp. 351-366.

J.M. Borwein and P.B. Borwein. More quadratically converging algorithms for π. Mathematics of Computation, Vol. 46, 1986, pp. 247-253.

Shlomo Breuer and Gideon Zwas Mathematical-educational aspects of the computation of π Int. J. Math. Educ. Sci. Technol., Vol. 15, No. 2, 1984, pp. 231-244.

David Chudnovsky and Gregory Chudnovsky. The computation of classical constants. Columbia University, Proc. Natl. Acad. Sci. USA, Vol. 86, 1989.

Classical Constants and Functions: Computations and Continued Fraction Expansions D.V.Chudnovsky, G.V.Chudnovsky, H.Cohn, M.B.Nathanson, eds. Number Theory, New York Seminar 1989-1990.

Y. Kanada and Y. Tamura. Calculation of π to 10,013,395 decimal places based on the Gauss-Legendre algorithm and Gauss arctangent relation. Computer Centre, University of Tokyo, 1983.

Morris Newman and Daniel Shanks. On a sequence arising in series for π. Mathematics of computation, Vol. 42, No. 165, Jan 1984, pp. 199-217.

E. Salamin. Computation of π using arithmetic-geometric mean. Mathematics of Computation, Vol. 30, 1976, pp. 565-570

David Singmaster. The legal values of π. The Mathematical Intelligencer, Vol. 7, No. 2, 1985.

Stan Wagon. Is π normal? The Mathematical Intelligencer, Vol. 7, No. 3, 1985.

A history of π. P. Beckman. Golem Press, CO, 1971 (fourth edition 1977)

π and the AGM - a study in analytic number theory and computational complexity. J.M. Borwein and P.B. Borwein. Wiley, New York, 1987.

4.2 Euler’s formula: e^{iπ} = −1

The definition and domain of exponentiation has been changed several times. The original operation x^{y} was only defined when y was a positive integer. The domain of the operation of exponentation has been extended, not so much because the original definition made sense in the extended domain, but because there were (almost) unique ways to extend exponentation which preserved many of what seemed to be the “important” properties of the original operation. So in part, these definitions are only convention, motivated by reasons of aesthetics and utility.

The original definition of exponentiation is, of course, that x^{y} = 1 ∗ x ∗ x ∗ ... ∗ x, where 1 is multiplied by x, y times. This is only a reasonable definition for y = 1, 2, 3, ... (It could be argued that it is reasonable when y = 0, but that issue is taken up in a different part of the FAQ). This operation has a number of properties, including

1. x^{1} = x

2. For any x, n, m, x^{n}x^{m} = x^{n+m}.

3. If x is positive, then x^{n} is positive.

Now, we can try to see how far we can extend the domain of exponentiation so that the above properties (and others) still hold. This naturally leads to defining the operation x^{y} on the domain x positive real; y rational, by setting x^{p/q} = the q^{th} root of x^{p}. This operation agrees with the original definition of exponentiation on their common domain, and also satisfies (1), (2) and (3). In fact, it is the unique operation on this domain that does so. This operation also has some other properties:

4. If x > 1, then x^{y} is an increasing function of y.

5. If 0 < x < 1, then x^{y} is a decreasing function of y.

Again, we can again see how far we can extend the domain of exponentiation while still preserving properties (1)-(5). This leads naturally to the following definition of x^{y} on the domain x positive real; y real:

If x > 1, x^{y} is defined to be sup_{q}{x^{q}}, where q runs over a ll rationals less than or equal to y.

If x < 1, x^{y} is defined to be inf_{q}{x^{q}}, where q runs over a ll rationals bigger than or equal to y.

If x = 1, x^{y} is defined to be 1.

Again, this operation satisfies (1)-(5), and is in fact the only operation on this domain to do so.

The next extension is somewhat more complicated. As can be proved using the methods of calculus or combinatorics, if we define e to be the number

e = 1 + 1/1! + 1/2! + 1/3! + ... = 2.71828...

it turns out that for every real number x, 6. e^{x} = 1 + x/1! + x^{2}/2! + x^{3}/3! + ...

e^{x} is also denoted exp(x). (This series always converges regardless of the value of x). One can also define an operation ln(x) on the positive reals, which is the inverse of the operation of exponentiation by e. In other words, exp(ln(x)) = x for all positive x. Moreover,

7. If x is positive, then x^{y} = exp(y ln(x)). Because of this, the natural extension of exponentiation to complex exponents, seems to be to define

exp(z) = 1 + z/1! + z^{2}/2! + z^{3}/3! + ...

for all complex z (not just the reals, as before), and to define x^{z} = exp(z ln(x)) when x is a positive real and z is complex. This is the only operation x^{y} on the domain x positive real, y complex which satisfies all of (1)-(7). Because of this and other reasons, it is accepted as the modern definition of exponentiation. From the identities

sin x = x − x^{3}/3! + x^{5}/5! − x^{7}/7! + ...
cos x = 1 − x^{2}/2! + x^{4}/4! − x^{6}/6! + ...

which are the Taylor series expansion of the trigonometric sine and cosine functions respectively. From this, one sees that, for any real x,

8. exp(ix) = cos x + i sin x.

Thus, we get Euler’s famous formula

e^{πi} = −1

and e^{2πi} = e^{0} = 1. One can also obtain the classical addition formulae for sine and cosine from (8) and (1).

All of the above extensions have been restricted to a positive real for the base. When the base x is not a positive real, it is not as clear-cut how to extend the definition of exponentiation. For example, (−1)^{1/2} could well be i or -i, (−1)^{1/3} could be −1, 1/2+\sqrt{(}3)i/2,

or 1/2 − \sqrt{(}3)i/2, and so on. Some values of x and y give infinitely many candidates for x^{y}, all equally plausible. And of course x = 0 has its own special problems. These problems can all be traced to the fact that the exp function is not injective on the complex plane, so that ln is not well defined outside the real line. There are ways around these difficulties (defining branches of the logarithm, for example), but we shall not go into this here.

The operation of exponentiation has also been extended to other systems like matrices and operators. The key is to define an exponential function by (6) and work from there. [Some reference on operator calculus and/or advanced linear algebra?]

References

Complex Analysis. Ahlfors, Lars V. McGraw-Hill, 1953.

4.3 What is 0^{0}

According to some Calculus textbooks, 0^{0} is an “indeterminate form”. When evaluating a limit of the form 0^{0}, then you need to know that limits of that form are called “indeterminate forms”, and that you need to use a special technique such as L’Hopital’s rule to evaluate them. Otherwise, 0^{0} = 1 seems to be the most useful choice for 0^{0}. This convention allows us to extend definitions in different areas of mathematics that otherwise would require treating 0 as a special case. Notice that 0^{0} is a discontinuity of the function x^{y}. More importantly, keep in mind that the value of a function and its limit need not be the same thing, and functions need not be continous, if that serves a purpose (see Dirac’s delta).

This means that depending on the context where 0^{0} occurs, you might wish to substitute it with 1, indeterminate or undefined/nonexistent.

Some people feel that giving a value to a function with an essential discontinuity at a point, such as x^{y} at (0, 0), is an inelegant patch and should not be done. Others point out correctly that in mathematics, usefulness and consistency are very important, and that under these parameters 0^{0} = 1 is the natural choice.

The following is a list of reasons why 0^{0} should be 1.

Rotando & Korn show that if f and g are real functions that vanish at the origin and are analytic at 0 (infinitely differentiable is not sufficient), then f(x)^{g(x)} approaches 1 as x approaches 0 from the right.

From Concrete Mathematics p.162 (R. Graham, D. Knuth, O. Patashnik):

Some textbooks leave the quantity 0^{0} undefined, because the functions x^{0} and 0^{x} have different limiting values when x decreases to 0. But this is a mistake. We must define x^{0} = 1 for all x, if the binomial theorem is to be valid when x = 0, y = 0, and/or x = −y. The theorem is too important to be arbitrarily restricted! By contrast, the function 0^{x} is quite unimportant.

Published by Addison-Wesley, 2nd printing Dec, 1988.

As a rule of thumb, one can say that 0^{0} = 1, but 0.0^{0.0} is undefined, meaning that when approaching from a different direction there is no clearly predetermined value to assign to 0.0^{0.0}; but Kahan has argued that 0.0^{0.0} should be 1, because if f(x), g(x) → 0 as x approaches some limit, and f(x) and g(x) are analytic functions, then f(x)^{g}(x) → 1.

The discussion on 0^{0} is very old, Euler argues for 0^{0} = 1 since a^{0} = 1 for a ≠ 0. The controversy raged throughout the nineteenth century, but was mainly conducted in the pages of the lesser journals: Grunert’s Archiv and Schlomilch’s Zeitschrift für Mathematik und Physik. Consensus has recently been built around setting the value of 0^{0} = 1.

On a discussion of the use of the function 0^{0^{x}} by an Italian mathematician named Guglielmo Libri.

[T]he paper [33] did produce several ripples in mathematical waters when it originally appeared, because it stirred up a controversy about whether 0^{0} is defined. Most mathematicians agreed that 0^{0} = 1, but Cauchy [5, page 70] had listed 0^{0} together with other expressions like 0/0 and ∞ − ∞ in a table of undefined forms. Libri’s justification for the equation 0^{0} = 1 was far from convincing, and a commentator who signed his name simply “S” rose to the attack [45]. August Möbius [36] defended Libri, by presenting his former professor’s reason for believing that 0^{0} = 1 (basically a proof that lim_{x→0+} x^{x} = 1). Möbius also went further and presented a supposed proof that lim_{x→0+} f(x)^{g(x)} whenever lim_{x→0+} f(x) = lim_{x→0+}g(x) = 0. Of course “S” then asked [3] whether Möbius knew about functions such as f(x) = e^{−1/x} and g(x) = x. (And paper

[36] was quietly omitted from the historical record when the collected words of Möbius were ultimately published.) The debate stopped there, apparently with the conclusion that 0^{0} should be undefined.

But no, no, ten thousand times no! Anybody who wants the binomial theorem (x + y)^{n} = ∑_{k=0}^{n} \binom{n}{k}x^{k}y^{n−k} to hold for at least one nonnegative integer n must believe that 0^{0} = 1, for we can plug in x = 0 and y = 1 to get 1 on the left and 0^{0} on the right.

The number of mappings from the empty set to the empty set is 0^{0}. It has to be 1.

On the other hand, Cauchy had good reason to consider 0^{0} as an undefined limiting form, in the sense that the limiting value of f(x)^{g(x)} is not known a priori when f(x) and g(x) approach 0 independently. In this much stronger sense, the value of 0^{0} is less defined than, say, the value of 0 + 0. Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side.

[3] Anonymous and S… Bemerkungen zu den Aufsatze überschrieben, ‘Beweis der Gleichung 0^{0} = 1, nach J. F. Pfaff’, im zweiten Hefte dieses Bandes, S. 134, Journal für die reine und angewandte Mathematik, 12 (1834), 292– 294.

[5] Œuvres Complètes. Augustin-Louis Cauchy. Cours d’Analyse de l’Ecole Royale Polytechnique (1821). Series 2, volume 3.

[33] Guillaume Libri. Mémoire sur les fonctions discontinues, Journal für die reine und angewandte Mathematik, 10 (1833), 303–316.

[36] A. F. Möbius. Beweis der Gleichung 0^{0} = 1, nach J. F. Pfaff. Journal für die reine und angewandte Mathematik,

12 (1834), 134–136.

[45] S… Sur la valeur de 0^{0}. Journal für die reine und angewandte Mathematik 11, (1834), 272–273.

References

Knuth. Two notes on notation. (AMM 99 no. 5 (May 1992), 403–422).

H. E. Vaughan. The expression ’0^{0}’. Mathematics Teacher 63 (1970), pp.111-112.

Kahan, W. Branch Cuts for Complex Elementary Functions or Much Ado about Nothing’s Sign Bit, The State of the Art in Numerical Analysis, editors A. Iserles and M. J. D. Powell, Clarendon Press, Oxford, pp. 165–212.

Louis M. Rotando and Henry Korn.

The Indeterminate Form 0^{0}. Mathematics Magazine,Vol. 50, No. 1 (January 1977), pp. 41-42. L. J. Paige,. A note on indeterminate forms. American Mathematical Monthly, 61 (1954), 189-190; reprinted in the Mathematical Association of America’s 1969 volume, Selected Papers on Calculus, pp. 210-211.

Baxley & Hayashi. A note on indeterminate forms. American Mathematical Monthly, 85 (1978), pp. 484-486.

Crimes and Misdemeanors in the Computer Algebra Trade. Notices of the American Mathematical Society, September 1991, volume 38, number 7, pp.778-785

4.3.1 Why is 0.9999 … = 1?

In modern mathematics, the string of symbols 0.9999 … is understood to be a shorthand for “the infinite sum 9/10 + 9/100 + 9/1000 + …”. This in turn is shorthand for “the limit of the sequence of real numbers 9/10, 9/10 + 9/100, 9/10 + 9/100 + 9/1000, …”. Using the well-known epsilon-delta definition of the limit (you can find it in any of the given references on analysis), one can easily show that this limit is 1. The statement that 0.9999 … = 1 is simply an abbreviation of this fact.

0.9999 … = ∑_{n=1}^{∞} \frac{9}{10^{n}} = lim_{m→∞} ∑_{n=1}^{m} \frac{9}{10^{n}}

Choose ε > 0. Suppose δ = 1/ − log_{10} ε, thus ε = 10^{−1/δ}. For every m > 1/δ we have that

|∑_{n=1}^{m} \frac{9}{10^{n}} − 1| = \frac{1}{10^{m}} < \frac{1}{10^{1/δ}} = ε

So by the ε − δ definition of the limit we have

lim_{m→∞} ∑_{n=1}^{m} \frac{9}{10^{n}} = 1

Not formal enough? In that case you need to go back to the construction of the number system. After you have constructed the reals (Cauchy sequences are well suited for this case, see [Shapiro75]), you can indeed verify that the preceding proof correctly shows 0.9999 … = 1.

An informal argument could be given by noticing that the following sequence of “natural” operations has as a consequence 0.9999 … = 1. Therefore it’s “natural” to assume 0.9999 … = 1.

x = 0.9999 …
10x = 10 · 0.9999 …
10x = 9.9999 …
10x − x = 9.9999 … − 0.9999 …
9x = 9
x = 1

Thus 0.9999 … = 1.

An even easier argument multiplies both sides of 0.3333 … = 1/3 by 3. The result is 0.9999 … = 3/3 = 1.

Another informal argument is to notice that all periodic numbers such as 0.46464646 … are equal to the period divided over the same number of 9s. Thus 0.46464646 … = 46/99. Applying the same argument to 0.9999 … = 9/9 = 1.

Although the three informal arguments might convince you that 0.9999 … = 1, they are not complete proofs. Basically, you need to prove that each step on the way is allowed and is correct. They are also “clumsy” ways to prove the equality since they go around the bush: proving 0.9999 … = 1 directly is much easier.

You can even have that while you are proving it the “clumsy” way, you get proof of the result in another way. For instance, in the first argument the first step is showing that 0.9999 … is real indeed. You can do this by giving the formal proof stated in the beginning of this FAQ question. But then you have 0.9999 … = 1 as corollary. So the rest of the argument is irrelevant: you already proved what you wanted to prove.

References

R.V. Churchill and J.W. Brown. Complex Variables and Applications. 5^{th} ed., McGraw-Hill, 1990.

E. Hewitt and K. Stromberg. Real and Abstract Analysis. Springer-Verlag, Berlin, 1965.

W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, 1976.

L. Shapiro. Introduction to Abstract Algebra. McGraw-Hill, 1975.

4.4 Name for f(x)^{f(x)} = x

Solving for f one finds a “continued fraction”-like answer

f(x) = \frac{log x}{log \frac{log x}{log \frac{log x}{log...}}}  (4.1)

This question has been repeated here from time to time over the years, and no one seems to have heard of any published work on it, nor a published name for it.

This function is the inverse of f(x) = x^{x}. It might be argued that such description is good enough as far as mathematical names go: ”the inverse of the function f(x) = x^{x}” seems to be clear and succint.

Another possible name is lx(x). This comes from the fact that the inverse of e^{x} is ln(x) thus the inverse of x^{x} could be named lx(x).

It’s not an analytic function.

The “continued fraction” form for its numeric solution is highly unstable in the region of its minimum at 1/e (because the graph is quite flat there yet logarithmic approximation oscillates wildly), although it converges fairly quickly elsewhere. To compute its value near 1/e, use the bisection method which gives good results. Bisection in other regions converges much more slowly than the logarithmic continued fraction form, so a hybrid of the two seems suitable. Note that it’s dual valued for the reals (and many valued complex for negative reals).

A similar function is a built-in function in MAPLE called W (x) or Lambert’s W function. MAPLE considers a solution in terms of W (x) as a closed form (like the erf function). W is defined as W (x)e^{W (x)} = x.

Notice that f(x) = exp(W (log(x))) is the solution to f(x)^{f}(x) = x

An extensive treatise on the known facts of Lambert’s W function is available for anonymous ftp at dragon.uwaterloo.ca at /cs-archive/CS-93-03/W.ps.Z.

4.5 Some Famous Mathematical Constants

A table of 120 known constants in math, such as Pi, e, sqrt(2), parking constant, Feigenbaum constant, etc. each of them is with references, and up to 1024 digits when possible can be found at the Centre for Experimental and Constructive Mathematics, Simon Fraser University at http://www.cecm.sfu.ca/projects/ISC.html

Another source of mathematical constants is:

http://www.mathsoft.com/asolve/constant/constant.html

maintained by Steve Finch.

Human Interest

5.1 Indiana bill sets the value of π to 3

The bill House Bill No. 246, Indiana State Legislature, 1897, reportedly set the value of π to an incorrect rational approximation.

The following is the text of the bill:

HOUSE BILL NO. 246

”A bill for an act introducing a new mathematical truth and offered as a contribution to education to be used only by the State of Indiana free of cost by paying any royalties whatever on the same, provided it is accepted and adopted by the official action of the legislature of 1897.

”Section 1. Be it enacted by the General Assembly of the State of Indiana: It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square on one side. The diameter employed as the linear unit according to the present rule in computing the circle’s area is entirely wrong, as it represents the circles area one and one-fifths times the area of a square whose perimeter is equal to the circumference of the circle. This is because one-fifth of the diameter fils to be represented four times in the circle’s circumference. For example: if we multiply the perimeter of a square by one-fourth of any line one-fifth greater than one side, we can, in like manner make the square’s area to appear one fifth greater than the fact, as is done by taking the diameter for the linear unit instead of the quadrant of the circle’s circumference.

”Section 2. It is impossible to compute the area of a circle on the diameter as the linear unit without trespassing upon the area outside the circle to the extent of including one-fifth more area than is contained within the circle’s circumference, because the square on the diameter produces the side of a square which equals nine when the arc of ninety degrees equals eight. By taking the quadrant of the circle’s circumference for the linear unit, we fulfill the requirements of both quadrature and rectification of the circle’s circumference. Furthermore, it has revealed the ratio of the chord and arc of ninety degrees, which is as seven to eight, and also the ratio of the diagonal and one side of a square which is as ten to seven, disclosing the fourth important fact, that the ratio of the diameter and circumference is as five-fourths to four; and because of these facts and the further fact that the rule in present use fails to work both ways mathematically, it should be discarded as wholly wanting and misleading in its practical applications.

”Section 3. In further proof of the value of the author’s proposed contribution to education, and offered as a gift to the State of Indiana, is the fact of his solutions of the trisection of the angle, duplication of the cube and quadrature having been already accepted as contributions to science by the American Mathematical Monthly, the leading exponent of mathematical thought in this country. And be it remembered that these noted problems had been long since given up by scientific bodies as unsolvable mysteries and above man’s ability to comprehend.”

Will E. Edington in an article published in the Proceedings of the Indiana Academy of Science describes the fate of the bill in the committees of the Indiana legislature. First it was referred to the House Committee on Canals, which was also referred to as the Committee on Swamp Lands. Notices of the bill appeared in the Indianapolis Journal and the Indianapolis Sentinel on Jan. 19, 1897, both of which described it a a bill telling how to square circles. On the same day, ”Representative M.B.Butler, of Steuben County, chairman of the Committee on Canals, submitted the following report:

”Your Committee on Canals, to which was referred House Bill No.246, entitled an act for the introduction of a mathematical truth, etc., has had the same under consideration and begs leave to report the same back to the House with the recommendation that said bill be referred to the Committee on Education.”

The next day, the following article appeared in the Indianapolis Sentinel:

”To SQUARE THE CIRCLE

”Claims Made That This Old Problem Has Been Solved. ”The bill telling how to square a circle, introduced in the House by Mr.Record, is not intended to be a hoax. Mr. Record knows nothing of the bill with the exception that he introduced it by request of Dr.Edwin Goodwin of Posey County, who is the author of the demonstration. The latter and State Superintendent of Public Instruction Geeting believe that it is the long-sought solution of the problem, and they are seeking to have it adopted by the legislature. Dr. Goodwin, the author, is a mathematician of note. He has it copyrighted and his proposition is that if the legislature will indorse the solution, he will allow the state to use the demonstration in its textbooks free of charge. The author is lobbying for the bill.”

On ”February 2, 1897, ...Representative S.E. Nicholson, of Howard County, chairman of the Committee on Education, reported to the House.

”Your Committee on Education, to which was referred House Bill No.246, entitled a a bill for an act entitled an act introducing a new mathematical truth, has had same under consideration, and begs leave to report the same back to the House with the recommendation that said bill do pass.

”The report was concurred in, and on February 8, 1897, it was brought up for the second reading, following which it was considered engrossed. Then ’Mr. Nicholson moved that the constitutional rule requiring bills to be read on three days be suspended, that the bill may be read a third time now.’ The constitutional rule was suspended by a vote of 72 to 0 and the bill was then read a third time. It was passed by a vote of 67 to 0, and the Clerk of the House was directed to inform the Senate of the passage of the bill.”

The newspapers reported the suspension of the consitutional rules and the unanimous passage of the bill matter-of-factly, except for one line in the Indianapolis Journal to the effect that ”this is the strangest bill that has ever passed an Indiana Assembly.”

The bill was referred to the Senate on Feb.10, 1897, and was read for the first time on Feb.11 and referred to the Committee on Temperance. ”On Feb.12 Senator Harry S. New, of Marion County, Chairman of the Committee on Temperance, made the following report to the Senate:

”Your committee on Temperance, to which was referred House Bill No.246, introduced by Mr.Record, has had the same under consideration and begs leave to report the same back to the Senate with the recommendation that said bill do pass.”

The Senate Journal mentions only that the bill was read a second time on Feb.12, 1897, that there was an unsuccessful attempt to amend the bill by strike out the enacting clause, and finally it was postponed indefinitely. That the bill was killed appears to be a matter of dumb luck rather than the superior education or wisdom of the Senate. It is true that the bill was widely ridiculed in Indiana and other states, but what actually brought about the defeat of the bill is recorded by Prof. C.A. Waldo in an article he wrote for the Proceedings of the Indiana Academy of Science in 1916. The reason he knows is that he happened to be at the State Capitol lobbying for the appropriation of the Indiana Academy of Science, on the day the Housed passed House Bill 246. When he walked in the found the debate on House Bill 246 already in progress. In his article, he writes (according to Edington):

”An ex-teacher from the eastern part of the state was saying: ’The case is perfectly simple. If we pass this bill which establishes a new and correct value for π, the author offers to our state without cost the use of his discovery and its free publication in our school text books, while everyone else must pay him a royalty.’”

The roll was then called and the bill passed its third and final reading in the lower house. A member then showed the writer [i.e. Waldo] a copy of the bill just passed and asked him if he would like an introduction to the learned doctor, its author. He declined the courtesy with thanks remarking that he was acquainted with as many crazy people as he cared to know.

”That evening the senators were properly coached and shortly thereafter as it came to its final reading in the upper house they threw out with much merriment the epoch making discovery of the Wise Man from the Pocket.”

The bill implies four different values for π and one for \sqrt{2}, as follows: π′ = 16/\sqrt{3}, 2\sqrt{5π/6}, 16\sqrt{2}/7, 16/5( 9.24, 3.236, 3.232, 3.2 respectively.) \sqrt{2}′ = 10/7.

It has been found that a circular area is to the square on a line equal to the quadrant of the circumference, as the area of an equilateral rectangle is to the square on one side.

π′ : (π′/2)^{2} = \sqrt{3}/4 : 1 i.e. π′ = 16/\sqrt{3} = 9.24.

The diameter employed as the linear unit according to the present rule in computing the circle’s area is entirely wrong, as it represents the circles area one and one-fifths times the area of a square whose perimeter is equal to the circumference of the circle. This is because one-fifth of the diameter fails to be represented four times in the circle’s circumference.

Bit tricky to decipher, but it seems to say (2π′/4)^{2}6/5 = π i.e. π′ = 2\sqrt{5π/6} = 3.236

Furthermore, it has revealed the ratio of the chord and arc of ninety degrees, which is as seven to eight,

\sqrt{2} : π/2 = 7 : 8 i.e. π = 16\sqrt{2}/7 = 3.232

and also the ratio of the diagonal and one side of a square which is as ten to seven

i.e. \sqrt{2} = 10/7 = 1.429

that the ratio of the diameter and circumference is as five-fourths to four

i.e. π = 16/5 = 3.2

5.2 Fields Medal

5.2.1 Historical Introduction

This is the original letter by Fields creating the endowment for the medals that bear his name. It is thought to have been written during the few months before his death. Notice that no mention is made about the age of the recipients (currently there is a 40 year-old limit), and that the medal should not be attached to any person, private or public, meaning that it shouldn’t bear anybody’s name.

It is proposed to found two gold medals to be awarded at successive International Mathematical Congress for outstanding achievements in mathematics. Because of the multiplicity of the branches of mathematics and taking into account the fact that the interval between such congresses is four years it is felt that at least two medals should be available. The awards would be open to the whole world and would be made by an International Committee.

The fund for the founding of the medals is constituted by balance left over after financing the Toronto congress held in 1924. This must be held in trust by the Government or by some body authorized by government to hold and invest such funds. It would seem that a dignified method for handling the matter and one which in this changing world should most nearly secure permanency would be for the Canadian Government to take over the fund and appoint as his custodian say the Prime Minister of the Dominion or the Prime Minister in association with the Minister of Finance. The medals would be struck at the Mint in Ottawa and the duty of the custodian would be simply to hand over the medals at the proper time to the accredited International Committee.

As things are at present a practical course of procedure would seem to be for the Executive Committee of a Congress to appoint a small international committee authorized to add to its number and call into consultation other mathematicians as it might deem expedient. The Committee would be expected to decide on the ones to whom the awards should be made thirty months in advance of the following Congress. Its decisions would be communicated to the President and Secretary of the Organizing Committee of the Congress, this Committee having the duty of communicating to the Prime Minister of Canada the names of the recipients in order that the medal might be prepared in time and forwarded to the president of the Organizing Committee. Immediately on the appointment of the Executive Committee of the Congress the medals would be handed over to its President. The presentation of the medals would constitute a special feature at some general meeting of the Congress.

In the above arrangements the role of the Organizing Committee might be taken over by the Executive of the International Mathematical Union at some time in the future when that organization has been generally accepted.

In coming to its decision the hands of the IC should be left as free as possible. It would be understood, however, that in making the awards while it was in recognition of work already done it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others.

In commenting on the work of the medalists it might be well to be conservative in one’s statements to avoid envidious comparisons explicit or implied. The Committee might ease matters by saying they have decided to make the awards along certain lines not alone because of the outstanding character of the achievement but also with a view to encouraging further development along these lines. In this connection the Committee might say that they had elected to select subjects in Analysis, in Geometry, in the Theory of Groups, in the Theory of Numbers etc. as the case might be. When the Committee had come to an agreement in this sense the claims for recognition of work done along the special lines in question could be considered in detail by two smaller groups or subcommittees with specialized qualifications who would have authority to take into consultation or add to the subcommittees other mathematicians of specialized knowledge.

With regard to the medals themselves, I might say that they should each contain at least 200 dollars worth of gold and be of a fair size, probably 7.5 centimeters in diameter. Because of the international character the language to be employed it would seem should be Latin or Greek? The design has still to be definitely determined. It will have to be decided on by artists in consultation with mathematicians. The suggestions made in the preceding are tentative and open to consideration on the part of mathematicians.

It is not contemplated to make an award until 1936 at the Congress following that at Zurich during which an international Medal Committee should be named.

The above programme means a new departure in the matter of international scientific cooperation and is likely to be the precursor of moves along like lines in other sciences than mathematics.

One would hear again emphasized the fact that the medals should be of a character as purely international and impersonal as possible. There should not be attached to them in any way the name of any country, institution or person.

Perhaps provision could be made as soon as possible after the appointment of the Executive of the Zurich Congress for the consideration by it of the subject of the medals, and the appointment without undue delay of a Committee and the awards of the medals to be made in connection with the Congress of 1936.

Suggestions with regard to the design of the medals will be welcome.

(signed) J.C. Fields Research Professor of Mathematics University of Toronto More information may also be found at

URL: http://www.math.toronto.edu/fields.html

5.2.2 Table of Awardees

Year Name  Birthplace  Country Age
1936 Ahlfors, Lars  Helsinki  Finland  29
1936 Douglas, Jesse  New York, NY USA  39
1950 Schwartz, Laurent Paris  France  35
1950 Selberg, Atle  Langesund  Norway  33
1954 Kodaira, Kunihiko Tokyo  Japan  39
1954 Serre, Jean-Pierre Bages  France  27
1958 Roth, Klaus  Breslau  Germany 32
1958 Thom, Rene  Montbeliard  France  35
1962 Hormander, Lars  Mjallby  Sweden  31
1962 Milnor, John  Orange, NJ  USA  31
1966 Atiyah, Michael  London  UK  37
1966 Cohen, Paul  Long Branch NJ USA  32
1966 Grothendieck, Alex. Berlin  Germany 38
1966 Smale, Stephen  Flint, MI  USA  36
1970 Baker, Alan  London  UK  31
1970 Hironaka, Heisuke Yamaguchi-ken Japan  39
1970 Novikov, Serge  Gorki  USSR  32
1970 Thompson, John  Ottawa, KA  USA  37
1974 Bombieri, Enrico  Milan  Italy  33
1974 Mumford, David  Worth, Sussex UK  37
1978 Deligne, Pierre  Brussels  Belgium  33
1978 Fefferman, Charles Washington DC USA  29
1978 Margulis, Gregori  Moscow  USSR  32
1978 Quillen, Daniel  Orange, NJ  USA  38
1982 Connes, Alain  Draguignan  France  35
1982 Thurston, William Washington DC USA  35
1982 Yau, Shing-Tung  Kwuntung  China  33
1986 Donaldson, Simon Cambridge  UK  27
1986 Faltings, Gerd  1954  Germany 32
1986 Freedman, Michael Los Angeles  USA  35

Year Name  Birthplace Country Age
1990 Drinfeld, Vladimir  Kharkov  USSR  36
1990 Jones, Vaughan  Gisborne  N Zealand 38
1990 Mori, Shigefumi  Nagoya  Japan  39
1990 Witten, Edward  Baltimore USA  38
1994 Pierre-Louis Lions  ????  France  38
1994 Jean-Chrisophe Yoccoz ????  France  37
1994 Jean Bourgain  Oostende Belge  40
1994 Efim Zelmanov  Novosibirsk Russia  39

Year Name  Institution  Country
1936 Ahlfors, Lars  Harvard University  USA
1936 Douglas, Jesse  MIT  USA
1950 Schwartz, Laurent Universite de Nancy  France
1950 Selberg, Atle  Institute for Advanced Study, Princeton USA
1954 Kodaira, Kunihiko Princeton University  USA
1954 Serre, Jean-Pierre College de France  France
1958 Roth, Klaus  University of London  UK
1958 Thom, Rene  University of Strasbourg  France
1962 Hormander, Lars University of Stockholm  Sweden
1962 Milnor, John Princeton University USA
1954 Serre, Jean-Pierre College de France  France
1958 Roth, Klaus  University of London  UK
1958 Thom, Rene  University of Strasbourg  France
1962 Hormander, Lars University of Stockholm  Sweden
1962 Milnor, John  Princeton University  USA
1966 Atiyah, Michael  Oxford University  UK
1966 Cohen, Paul  Stanford University  USA
1966 Grothendieck, Alex University of Paris  France
1966 Smale, Stephen  University of California at Berkeley  USA
1970 Baker, Alan  Cambridge University  UK
1970 Hironaka, Heisuke Harvard University  USA
1970 Novikov, Serge  Moscow University  USSR
1970 Thompson, John  University of Chicago  USA
1974 Bombieri, Enrico  Univeristy of Pisa  Italy
1974 Mumford, David  Harvard University  USA
1978 Deligne, Pierre  IHES  France
1978 Fefferman, Charles Princeton University  USA
1978 Margulis, Gregori InstPrblmInfTrans  USSR
1978 Quillen, Daniel  MIT  USA
1982 Connes, Alain  IHES  France
1982 Thurston, William Princeton University  USA
1982 Yau, Shing-Tung  Institute for Advanced Study, Princeton USA
1986 Donaldson, Simon Oxford University  UK
1986 Faltings, Gerd Princeton University USA
1986 Freedman, Michael University of California at San Diego  USA

Year Name  Institution  Country
1990 Drinfeld, Vladimir  Phys.Inst.Kharkov  USSR
1990 Jones, Vaughan  University of California at Berkeley  USA
1990 Mori, Shigefumi  University of Kyoto?  Japan
1990 Witten, Edward  Princeton/Institute for Advanced Study USA
1994 Pierre-Louis Lions Universite de Paris-Dauphine France
1994 Jean-Chrisophe Yoccoz Universite de Paris-Sud  France
1994 Jean Bourgain  Institute for Advanced Study  USA
1994 Efim Zelmanov  University of Wisconsin  USA

References

International Mathematical Congresses, An Illustrated History 1893-1986. Donald J.Alberts, G. L. Alexanderson and Constance Reid. Revised Edition, Including 1986, Springer Verlag, 1987.

Tropp, Henry S. The origins and history of the Fields Medal. Historia Mathematica, 3(1976), 167-181.

5.3 Erdos Number

Form an undirected graph where the vertices are academics, and an edge connects academic X to academic Y if X has written a paper with Y. The Erdos number of X is the length of the shortest path in this graph connecting X with Erdos.

Erdos has Erdos number 0. Co-authors of Erdos have Erdos number 1. Einstein has Erdos number 2, since he wrote a paper with Ernst Straus, and Straus wrote many papers with Erdos.

The Extended Erdos Number applies to co-authors of Erdos. For People who have authored more than one paper with Erdos, their Erdos number is defined to be 1/# papersco-authored.

Why people care about it?

Nobody seems to have a reasonable answer...

Who is Paul Erdos?

Paul Erdos was an Hungarian mathematician. He obtained his PhD from the University of Manchester and spent most of his efforts tackling ”small” problems and conjectures related to graph theory, combinatorics, geometry and number theory.

He was one of the most prolific publishers of papers; and was also and indefatigable traveller.

Paul Erdös died on September 20, 1996.

At this time the number of people with Erdos number 2 or less is estimated to be over 4750, according to Professor Jerrold W. Grossman archives. These archives can be consulted via anonymous ftp at vela.acs.oakland.edu under the directory pub/math/erdos or on the Web at http://www.acs.oakland.edu/ grossman/erdoshp.html. At this time it contains a list of all co-authors of Erdos and their co-authors.

On this topic, he writes

Let E_{1} be the subgraph of the collaboration graph induced by people with Erdős number 1. We found that E_{1} has 451 vertices and 1145 edges. Furthermore, these collaborators tended to collaborate a lot, especially among themselves. They have an average of 19 other collaborators (standard deviation 21), and only seven of them collaborated with no one except Erdős. Four of them have over 100 coauthors. If we restrict our attention just to E_{1}, we still find a lot of joint work. Only 41 of these 451 people have collaborated with no other persons with Erdős number 1 (i.e., there are 41 isolated vertices in E_{1}), and E_{1} has four components with two vertices each. The remaining 402 vertices in E_{1} induce a connected subgraph. The average vertex degree in E_{1} is 5, with a standard deviation of 6; and there are four vertices with degrees of 30 or higher. The largest clique in E_{1} has seven vertices, but it should be noted that six of these people and Erdős have a joint seven-author paper. In addition, there are seven maximal 6-cliques, and 61 maximal 5-cliques. In all, 29 vertices in E_{1} are involved in cliques of order 5 or larger. Finally, we computed that the diameter of E_{1} is 11 and its radius is 6.

Three quarters of the people with Erdős number 2 have only one co-author with Erdős number 1 (i.e., each such person has a unique path of length 2 to p). However, their mean number of Erdős number 1 co-authors is 1.5, with a standard deviation of 1.1, and the count ranges as high as 13.

Folklore has it that most active researchers have a finite, and fairly small, Erdős number. For supporting evidence, we verified that all the Fields and Nevanlinna prize winners during the past three cycles (1986–1994) are indeed in the Erdős component, with Erdős number at most 9. Since this group includes people working in theoretical physics, one can conjecture that most physicists are also in the Erdős component, as are, therefore, most scientists in general. The large number of applications of graph theory to the social sciences might also lead one to suspect that many researchers in other academic areas are included as well. We close with two open questions about C, restricted to mathematicians, that such musings suggest, with no hope that either will ever be answered satisfactorily: What is the diameter of the Erdős component, and what is the order of the second largest component?

References

Caspar Goffman. And what is your Erdos number? American Mathematical Monthly, v. 76 (1969), p. 791.

Tom Odda (alias for Ronald Graham) On Properties of a Well- Known Graph, or, What is Your Ramsey Number? Topics in Graph Theory (New York, 1977), pp. [166-172].

5.4 Why is there no Nobel in mathematics?

Nobel prizes were created by the will of Alfred Nobel, a notable Swedish chemist.

One of the most common –and unfounded– reasons as to why Nobel decided against a Nobel prize in math is that [a woman he proposed to/his wife/his mistress] [rejected him because of/cheated him with] a famous mathematician. Gosta Mittag-Leffler is often claimed to be the guilty party.

There is no historical evidence to support the story.

For one, Mr. Nobel was never married.

There are more credible reasons as to why there is no Nobel prize in math. Chiefly among them is simply the fact he didn’t care much for mathematics, and that it was not considered a practical science from which humanity could benefit (a chief purpose for creating the Nobel Foundation).

Further, at the time there existed already a well known Scandinavian prize for mathematicians. If Nobel knew about this prize he may have felt less compelled to add a competing prize for mathematicians in his will.

[...] As professor ordinarius in Stockholm, Mittag-Leffler began a 30-year career of vigorous mathematical activity. In 1882 he founded the Acta Mathematica, which a century later is still one of the world’s leading mathematical journals. Through his influence in Stockholm he persuaded King Oscar II to endow prize competitions and honor various distinguished mathematicians all over Europe. Hermite, Bertrand, Weierstrass, and Poincare were among those honored by the King. [...]

Source: ”The Mathematics of Sonya Kovalevskaya” by Roger Cooke (Springer-Verlag, New York etc., 1984, II.5.2, p. 90-91:

Here are some relevant facts:

• Nobel never married, hence no “wife”. (He did have a mistress, a Viennese woman named Sophie Hess.)

• Gosta Mittag-Leffler was an important mathematician in Sweden in the late 19thearly 20th century. He was the founder of the journal Acta Mathematica, played an important role in helping the career of Sonya Kovalevskaya, and was eventually head of the Stockholm Hogskola, the precursor to Stockholms Universitet. However, it seems highly unlikely that he would have been a leading candidate for an early Nobel Prize in mathematics, had there been one – there were guys like Poincare and Hilbert around, after all.

• There is no evidence that Mittag-Leffler had much contact with Alfred Nobel (who resided in Paris during the latter part of his life), still less that there was animosity between them for whatever reason. To the contrary, towards the end of Nobel’s life Mittag-Leffler was engaged in “diplomatic” negotiations to try to persuade Nobel to designate a substantial part of his fortune to the Hogskola. It seems hardly likely that he would have undertaken this if there was prior bad blood between them. Although initially Nobel seems to have intended to do this, eventually he came up with the Nobel Prize idea – much to the disappointment of the Hogskola, not to mention Nobel’s relatives and Fraulein Hess.

• According to the very interesting study by Elisabeth Crawford, “The Beginnings of the Nobel Institution”, Cambridge Univ. Press, 1984, pages 52-53:

Although it is not known how those in responsible positions at the Hogskola came to believe that a large bequest was forthcoming, this indeed was the expectation, and the disappointment was keen when it was announced early in 1897 that the Hogskola had been left out of Nobel’s final will in 1895. Recriminations followed, with both Pettersson and Arrhenius [academic rivals of Mittag-Leffler in the administration of the Hogskola] letting it be known that Nobel’s dislike for Mittag-Leffler had brought about what Pettersson termed the ‘Nobel Flop’. This is only of interest because it may have contributed to the myth that Nobel had planned to institute a prize in mathematics but had refrained because of his antipathy to Mittag-Leffler or –in another version of the same story– because of their rivalry for the affections of a woman....

However, Sister Mary Thomas a Kempis discovered a letter by R. C. Archibald in the archives of Brown University and discussed its contents in ”The Mathematics Teacher” (1966, pp.667-668). Archibald had visited Mittag-Leffler and, on his report, it would seem that M-L *believed* that the absence of a Nobel Prize in mathematics was due to an estrangement between the two men. (This at least is the natural reading, but not the only possible one.)

• A final speculation concerning the psychological element. Would Nobel, sitting down to draw up his testament, presumably in a mood of great benevolence to mankind, have allowed a mere personal grudge to distort his idealistic plans for the monument he would leave behind?

Nobel, an inventor and industrialist, did not create a prize in mathematics simply because he was not particularly interested in mathematics or theoretical science. His will speaks of prizes for those “inventions or discoveries” of greatest practical benefit to mankind. (Probably as a result of this language, the physics prize has been awarded for experimental work much more often than for advances in theory.)

However, the story of some rivalry over a woman is obviously much more amusing, and that’s why it will probably continue to be repeated.

References

Mathematical Intelligencer, vol. 7 (3), 1985, p. 74.

The Beginnings of the Nobel Institution. Elisabeth Crawford. Cambridge Univ. Press, 1984.

5.5 International Mathematics Olympiad and Other Competitions

From the IMO home page:

The International Mathematics Olympiad (IMO) is an annual mathematics competition for highschool students. It is one of the International Science Olympiads. The first IMO was held in Romania in 1959. The usual size of an official delegation to an IMO is (a maximum of) six student competitors and (a maximum of) two leaders. There is no official “team”. The student competitors write two papers, on consecutive days, each paper consisting of three questions. Each question is worth seven marks.

You can check results and other info at

http://www.win.tue.nl/win/ioi/imo/

5.6 Who is N. Bourbaki?

A group of mostly French mathematicians which began meeting in the 1930s, aiming to write a thorough unified account of all mathematics. They had tremendous influence on the way math is done since. For a very accessible sampler see Dieudonne Mathematics: The Music Of Reason (Orig. Pour L’honneur De L’esprit Humain).

The founding is described in Andre Weil’s autobiography, titled something like “memoir of an apprenticeship” (orig. Souvenirs D’apprentissage). There is a usable book Bourbaki by J. Fang. Liliane Beaulieu has a book forthcoming, which you can sample in “A Parisian Cafe and Ten Proto-Bourbaki Meetings 1934-1935” in the Mathematical Intelligencer 15 no.1 (1993) 27-35.

The history behind Bourbaki is also described in Scientific American, May 1957.

Mathematical Trivia

6.1 Names of Large Numbers

Naming for 10^{k}.

k     American      European    SI--Prefix

-33                                revo
-30                                tredo
-27                                syto
-24                                fito
-21                                ento
-18   quintillionth                atto
-15   Quadrillionth                femto
-12   trillionth                   pico
-9    Billionth                    nano
-6    Millionth                    micro
-3    Thousandth                   milli
-2    Hundredth                    centi
-1    Tenth                        deci
1     Ten                          deca
2     Hundred                      hecto
3     Thousand                     kilo
4     Myriad
6     Million       Million        mega
9     Billion       Milliard       giga
12    Trillion      Billion        tera

15    Quadrillion   Billiard       peta
18    Quintillion   Trillion       exa
21    Sextillion    Trilliard      hepa
24    Septillion    Quadrillion    otta
27    Octillion     Quadrilliard   nea
30    Nonillion     Quintillion    dea
      (Noventillion)
33    Decillion     Quintilliard   una
36    Undecillion          Sextillion
39    Duodecillion         Sextilliard
42    tredecillion         Septillion
45     quattuordecillion   Septilliard
48   quindecillion         Octillion
51   sexdecillion          Octilliard
54   septendecillion       Nonillion
                          (Noventillion)
57   octodecillion         Nonilliard
                          (Noventilliard)
60   novemdecillion        Decillion
63  VIGINTILLION           Decilliard
6*n   (2n-1)-illion n-illion
6*n+3 (2n)-illion   n-illiard
100   Googol        Googol
303   CENTILLION
600                 CENTILLION
10^100 Googolplex   Googolplex
%From: balden@wimsey.com (Bruce Balden)
%Date: Fri, 11 Oct 1996 12:46:39 GMT
Chinese System

1       yi4
10      shi2
100     bai3
1000     qian2
10000   wan4

10^6     yi bai3 wan (i.e. 100 times wan)
10^8     yi1
10^12   ???

The American system is used in:
      US,
      ...
The European system is used in:
      Austria,
      Belgium,
      Chile,
      Germany,
      the Netherlands,
      Italy (see exception)
      Scandinavia

%Date: Mon, 25 Aug 1997 22:47:48 -0700
%From: Torbjorn Larsson <ekatla@eka.ericsson.se>
%Subject: Sci.math FAQ
Note that all prefixes are to be spelled with a leading small letter. (As are
all SI units, even those that honors persons by using their names.)
- All prefixes with n < 0 should have a small letter abbreviation.
Eg. 1 picoampere = 1 pA. (SI unit rule explanation: person name unit
is abreviated using a capital letter)
- All prefixes with n > 0 should have a large letter abbreviation.
Eg. 1 gigameter = 1 Gm. (SI unit rule explanation: non-person name unit
is abreviated in lower case).  _Except_the mass unit: 1 kilogram is
abreviated as kg (compare to Km. for kilometer).

hv@cix.compulink.co.uk (Hugo van der Sanden):
  To the best of my knowledge, the House of Commons decided to adopt the
  US definition of billion quite a while ago - around 1970? - since which
  it has been official government policy.

dik@cwi.nl (Dik T. Winter):
The interesting thing about all this is that originally the French used
billion to indicate 10^9, while much of the remainder of Europe used
billion to indicate 10^12.  I think the Americans have their usage from
the French.  And the French switched to common European usage in 1948.
gonzo@ing.puc.cl (Gonzalo Diethelm):
Other countries (such as Chile, my own, and I think
most of Latin America) use billion to mean 10^12, trillion to mean
10^18, etc. What is the usage distribution over the world population,
anyway?

Famous Problems in Mathematics

7.1 The Four Colour Theorem

Theorem 2 (Four Colour Theorem) Every planar map with regions of simple borders can be coloured with 4 colours in such a way that no two regions sharing a non-zero length border have the same colour.

An equivalent combinatorial interpretation is

Theorem 3 (Four Colour Theorem) Every loopless planar graph admits a vertex-colouring with at most four different colours.

This theorem was proved with the aid of a computer in 1976. The proof shows that if aprox. 1,936 basic forms of maps can be coloured with four colours, then any given map can be coloured with four colours. A computer program coloured these basic forms. So far nobody has been able to prove it without using a computer. In principle it is possible to emulate the computer proof by hand computations.

The known proofs work by way of contradiction. The basic thrust of the proof is to assume that there are counterexamples, thus there must be minimal counterexamples in the sense that any subset of the graphic is four colourable. Then it is shown that all such minimal counterexamples must contain a subgraph from a set basic configurations.

But it turns out that any one of those basic counterexamples can be replaced by something smaller, while preserving the need for five colours, thus contradicting minimality.

The number of basic forms, or configurations has been reduced to 633.

A recent simplification of the Four Colour Theorem proof, by Robertson, Sanders, Seymour and Thomas, has removed the cloud of doubt hanging over the complex original proof of Appel and Haken.

The programs can now be obtained by ftp and easily checked over for correctness. Also the paper part of the proof is easier to verify. This new proof, if correct, should dispel all reasonable criticisms of the validity of the proof of this theorem.

References

K. Appel and W. Haken. Every planar map is four colorable. Bulletin of the American Mathematical Society, vol. 82, 1976 pp.711-712.

K. Appel and W. Haken. Every planar map is four colorable. Illinois Journal of Mathematics, vol. 21, 1977, pp. 429-567.

N. Robertson, D. Sanders, P. Seymour, R. Thomas The Four Colour Theorem Preprint, February 1994. Available by anonymous ftp from ftp.math.gatech.edu, in directory /pub/users/thomas/fcdir/npfc.ps.

The Four Color Theorem: Assault and Conquest T. Saaty and Paul Kainen. McGraw-Hill, 1977. Reprinted by Dover Publications 1986.

7.2 The Trisection of an Angle

Theorem 4 The trisection of the angle by an unmarked ruler and compass alone is in general not possible.

This problem, together with Doubling the Cube, Constructing the regular Heptagon and Squaring the Circle were posed by the Greeks in antiquity, and remained open until modern times.

The solution to all of them is rather inelegant from a geometric perspective. No geometric proof has been offered [check?], however, a very clever solution was found using fairly basic results from extension fields and modern algebra.

It turns out that trisecting the angle is equivalent to solving a cubic equation. Constructions with ruler and compass may only compute the solution of a limited set of such equations, even when restricted to integer coefficients. In particular, the equation for θ = 60 degrees cannot be solved by ruler and compass and thus the trisection of the angle is not possible.

It is possible to trisect an angle using a compass and a ruler marked in 2 places.

Suppose X is a point on the unit circle such that ^{}X̸OE is the angle we would like to “trisect”. Draw a line AX through a point A on the x-axis such that |AB| = 1 (which is the same as the radius of the circle), where B is the intersection-point of the line AX with the circle.

Let θ be ^{}B̸AO. Then ^{}B̸OA = θ, and ^{}X̸BO = ^{}B̸XO = 2θ

Since the sum of the internal angles of a triangle equals π radians (180 degrees) we have ^{}X̸BO + ^{}B̸XO + ^{}B̸OX = π, implying 4θ + ^{}B̸OX = π. Also, we have that A̸OB +^{}B̸OX +^{}X̸OE = π, implying θ +^{}B̸OX +^{}X̸OE = π. Since both quantities are equal to π we obtain

4θ +^{}B̸OX = θ +^{}B̸OX +^{}X̸OE

Figure 7.1: Trisection of the Angle with a marked ruler

From which

3θ = ^{}X̸OE

follows. QED.

7.3 Which are the 23 Hilbert Problems?

The original was published in German in a couple of places. A translation was published by the AMS in 1902. This article has been reprinted in 1976 by the American Mathematical Society (see references).

The AMS Symposium mentioned at the end contains a series of papers on the thencurrent state of most of the Problems, as well as the problems.

The URL contains the list of problems, and their current status:

http://www.astro.virginia.edu/ eww6n/math/Hilbert’sProblems.html Mathematical Developments Arising from Hilbert Problems, volume 28 of Proceedings of Symposia in Pure Mathematics, pages 1–34, Providence, Rhode Island. American Mathematical Society, 1976.

D. Hilbert. Mathematical problems. Lecture delivered before the International Congress of Mathematicians at Paris in 1900. Bulletin of the American Mathematical Society, 8:437–479, 1902.

7.4 Unsolved Problems

7.4.1 Does there exist a number that is perfect and odd?

A given number is perfect if it is equal to the sum of all its proper divisors. This question was first posed by Euclid in ancient Greece. This question is still open. Euler proved that if N is an odd perfect number, then in the prime power decomposition of N, exactly one exponent is congruent to 1 mod 4 and all the other exponents are even. Furthermore, the prime occurring to an odd power must itself be congruent to 1 mod 4. A sketch of the proof appears in Exercise 87, page 203 of Underwood Dudley’s Elementary Number Theory. It has been shown that there are no odd perfect numbers < 10^{300}.

7.4.2 Collatz Problem

Take any natural number m > 0. n := m; repeat if (n is odd) then n := 3 ∗ n + 1; else n := n/2; until (n == 1)

Conjecture 1 For all positive integers m, the program above terminates.

The conjecture has been verified for all numbers up to 5.6 ∗ 10^{13}.

References

Unsolved Problems in Number Theory. Richard K Guy. Springer, Problem E16.

Elementary Number Theory. Underwood Dudley. 2nd ed.

G.T. Leavens and M. Vermeulen 3x+1 search programs ] Comput. Math. Appl.

vol. 24 n. 11 (1992), 79-99.

7.4.3 Goldbach’s conjecture

This conjecture claims that every even integer bigger equal to 4 is expressible as the sum of two prime numbers. It has been tested for all values up to 4.10^{10} by Sinisalo.

7.4.4 Twin primes conjecture

There exist an infinite number of positive integers p with p and p + 2 both prime. See the largest known twin prime section. There are some results on the estimated density of twin primes.

Mathematical Games

8.1 The Monty Hall problem

This problem has rapidly become part of the mathematical folklore.

The American Mathematical Monthly, in its issue of January 1992, explains this problem carefully. The following are excerpted from that article.

Problem:

A TV host shows you three numbered doors (all three equally likely), one hiding a car and the other two hiding goats. You get to pick a door, winning whatever is behind it. Regardless of the door you choose, the host, who knows where the car is, then opens one of the other two doors to reveal a goat, and invites you to switch your choice if you so wish. Does switching increases your chances of winning the car?

If the host always opens one of the two other doors, you should switch. Notice that 1/3 of the time you choose the right door (i.e. the one with the car) and switching is wrong, while 2/3 of the time you choose the wrong door and switching gets you the car.

Thus the expected return of switching is 2/3 which improves over your original expected gain of 1/3.

Even if the hosts offers you to switch only part of the time, it pays to switch. Only in the case where we assume a malicious host (i.e. a host who entices you to switch based in the knowledge that you have the right door) would it pay not to switch.

There are several ways to convince yourself about why it pays to switch. Here’s one. You select a door. At this time assume the host asks you if you want to switch before he opens any doors. Even though the odds that the door you selected is empty are high (2/3), there is no advantage on switching as there are two doors, and you don’t know thich one to switch to. This means the 2/3 are evenly distributed, which as good as you are doing already. However, once Monty opens one of the two doors you selected, the chances that you selected the right door are still 1/3 and now you only have one door to choose from if you switch. So it pays to switch.

References

L. Gillman The Car and the Goats American Mathematical Monthly, January 1992, pp. 3-7.

8.2 Master Mind

For the game of Master Mind it has been proven that no more than five moves are required in the worst case.

One such algorithm was published in the Journal of Recreational Mathematics; in ’70 or ’71 (I think), which always solved the 4 peg problem in 5 moves. Knuth later published an algorithm which solves the problem in a shorter number of moves - on average - but can take six guesses on certain combinations.

In 1994, Kenji Koyama and Tony W. Lai found, by exhaustive search that 5625/1296 = 4.340 is the optimal strategy in the expected case. This strategy may take six guesses in the worst case. A strategy that uses at most five guesses in the worst case is also shown. This strategy requires 5626/1296 = 4.341 guesses.

References

Donald E. Knuth. The Computer as Master Mind. J. Recreational Mathematics, 9 (1976-77), 1-6.

Kenji Koyama, Tony W. Lai. An optimal Mastermind Strategy. J. Recreational Mathematics, 1994.

Axiom of Choice and Continuum Hypothesis

9.1 The Axiom of Choice

There are several equivalent formulations:

• The Cartesian product of nonempty sets is nonempty, even if the product is of an infinite family of sets.

• Given any set S of mutually disjoint nonempty sets, there is a set C containing a single member from each element of S. C can thus be thought of as the result of “choosing” a representative from each set in S. Hence the name.

9.1.1 Relevance of the Axiom of Choice

THE AXIOM OF CHOICE

There are many equivalent statements of the Axiom of Choice. The following version gave rise to its name:

For any set X there is a function f, with domain X\0, so that f(x) is a member of x for every nonempty x in X.

Such an f is called a “choice function” on X. [Note that X\0 means X with the empty set removed. Also note that in Zermelo-Fraenkel set theory all mathematical objects are sets so each member of X is itself a set.]

The Axiom of Choice (AC) is one of the most discussed axioms of mathematics, perhaps second only to Euclid’s parallel postulate. The axioms of set theory provide a foundation for modern mathematics in the same way that Euclid’s five postulates provided a foundation for Euclidean geometry, and the questions surrounding AC are the same as the questions that surrounded Euclid’s Parallel Postulate:

1. Can it be derived from the other axioms?

2. Is it consistent with the other axioms?

3. Should we accept it as an axiom?

For many sets, including any finite set, the first six axioms of set theory (abbreviated ZF) are enough to guarantee the existence of a choice function but there do exist sets for which AC is required to show the existence of a choice function. The existence of such sets was proved in 1963 by Paul Cohen. This means that AC cannot be derived from the other six axioms; in other words “AC is independent of ZF.” This answers question [1] posed above.

The question of whether AC is consistent with the other axioms (question [2] above) was answered by Goedel in 1938. Goedel showed that if the other axioms are consistent then AC is consistent with them. This is a “relative consistency” proof which is the best we can hope for because of Goedel’s Second Incompleteness Theorem.

The third question, “Should we accept it as an axiom?”, moves us into the realm of philosophy. Today there are three major schools of thought concerning the use of AC:

1. Accept it as an axiom and use it without hesitation.

2. Accept it as an axiom but use it only when you cannot find a proof without it.

3. AC is unacceptable.

Most mathematicians today belong to school A. Mathematicians who are in school B are usually there because of a belief in Occam’s Razor (use as few assumptions as possible when explaining something) or an interest in metamathematics. There are a growing number of people moving to school C, especially computer scientists who work on automated reasoning using constructive type theories.

Underlying the schools of thought about the use of AC are views about truth and the nature of mathematical objects. Three major views are platonism, constructivism, and formalism.

Platonism

A platonist believes that mathematical objects exist independent of the human mind, and a mathematical statement, such as AC, is objectively either true or false. A platonist accepts AC only if it is objectively true, and probably falls into school A or C depending on her belief. If she isn’t sure about AC’s truth then she may be in school B so that once she finds out the truth about AC she will know which theorems are true.

Constructivism

A constructivist believes that the only acceptable mathematical objects are ones that can be constructed by the human mind, and the only acceptable proofs are constructive proofs. Since AC gives no method for constructing a choice set constructivists belong to school C.

Formalism

A formalist believes that mathematics is strictly symbol manipulation and any consistent theory is reasonable to study. For a formalist the notion of truth is confined to the context of mathematical models, e.g., a formalist would say ”The parallel postulate is false in Riemannian geometry.” but she wouldn’t say ”The parallel postulate is false.” A formalist will probably not allign herself with any school. She will comfortably switch between A, B, and C depending on her current interests.

So: Should you accept the Axiom of Choice? Here are some arguments for and against it.

Against

• It’s not as simple, aesthetically pleasing, and intuitive as the other axioms.

• It is equivalent to many statements which are not intuitive such as ”Every set can be well ordered.” How, for example, would you well order the reals?

• With it you can derive non-intuitive results, such as the existence of a discontinuous additive function, the existence of a non-measurable set of reals, and the Banach-Tarski Paradox (see the next section of the sci.math FAQ).

• It is nonconstructive - it conjures up a set without providing any sort of procedure for its construction.

For

The acceptance of AC is based on the belief that our intuition about finite sets can be extended to infinite sets. The main argument for accepting it is that it is useful. Many important, intuitively plausible theorems are equivalent to it or depend on it. For example these statements are equivalent to AC:

• Every vector space has a basis.

• Trichotomy of Cardinals: For any cardinals k and l, either k < l or k = l or k > l.

• Tychonoff’s Theorem: The product of compact spaces is compact in the product topology.

• Zorn’s Lemma: Every nonempty partially ordered set P in which each chain has an upper bound in P has a maximal element.

And these statements depend on AC (i.e., they cannot be proved in ZF without AC):

• The union of countably many countable sets is countable.

• Every infinite set has a denumerable subset.

• The Loewenheim-Skolem Theorem: Any first-order theory which has a model has a denumerable model.

• The Baire Category Theorem: The reals are not the union of countably many nowhere dense sets (i.e., the reals are not meager).

• The Ultrafilter Theorem: Every Boolean algebra has an ultrafilter on it.

Alternatives to AC

• Accept only a weak form of AC such as the Denumerable Axiom of Choice (every denumerable set has a choice function) or the Axiom of Dependent Choice.

• Accept an axiom that implies AC such as the Axiom of Constructibility (V = L) or the Generalized Continuum Hypothesis (GCH).

• Adopt AC as a logical axiom (Hilbert suggested this with his epsilon axiom). If set theory is done in such a logical formal system the Axiom of Choice will be a theorem.

• Accept a contradictory axiom such as the Axiom of Determinacy.

• Use a completely different framework for mathematics such as Category Theory. Note that within the framework of Category Theory Tychonoff’s Theorem can be proved without AC (Johnstone, 1981).

Test Yourself: When is AC necessary?

If you are working in Zermelo-Fraenkel set theory without the Axiom of Choice, can you choose an element from...

1. a finite set?

2. an infinite set?

3. each member of an infinite set of singletons (i.e., one-element sets)?

4. each member of an infinite set of pairs of shoes?

5. each member of inifinite set of pairs of socks?

6. each member of a finite set of sets if each of the members is infinite?

7. each member of an infinite set of sets if each of the members is infinite?

8. each member of a denumerable set of sets if each of the members is infinite?

9. each member of an infinite set of sets of rationals?

10. each member of a denumerable set of sets if each of the members is denumberable?

11. each member of an infinite set of sets if each of the members is finite?

12. each member of an infinite set of finite sets of reals?

13. each member of an infinite set of sets of reals?

14. each member of an infinite set of two-element sets whose members are sets of reals?

The answers to these questions with explanations are accessible through http://www.jazzie.com/ii/math/index.html

References

Benacerraf, Paul and Putnam, Hilary. ”Philosophy of Mathematics: Selected Readings, 2nd edition.” Cambridge University Press, 1983.

Dauben, Joseph Warren. ”Georg Cantor: His Mathematics and Philosophy of the Infinite.” Princeton University Press, 1979.

A. Fraenkel, Y. Bar-Hillel, and A. Levy with van Dalen, Dirk. ”Foundations of Set Theory, Second Revised Edition.” North-Holland, 1973.

Johnstone, Peter T. ”Tychonoff’s Theorem without the Axiom of Choice.” Fundamenta Mathematica 113: 21-35, 1981.

Leisenring, Albert C. ”Mathematical Logic and Hilbert’s Epsilon-Symbol.” Gordon and Breach, 1969.

Maddy, ”Believing the Axioms, I”, J. Symb. Logic, v. 53, no. 2, June 1988, pp. 490-500, and ”Believing the Axioms II” in v.53, no. 3.

Moore, Gregory H. ”Zermelo’s Axiom of Choice: Its Origins, Development, and Influence.” Springer-Verlag, 1982.

Rubin, Herman and Rubin, Jean E. ”Equivalents of the Axiom of Choice II.” North-Holland, 1985.

This section of the FAQ is Copyright (c) 1994 Nancy McGough. Send comments and or corrections relating to this part to nancym@ii.com. The most up to date version of this section of the sci.math FAQ is accesible through http://www.jazzie.com/ii/math/index.html

9.2 Cutting a sphere into pieces of larger volume

Is it possible to cut a sphere into a finite number of pieces and reassemble into a solid of twice the volume?

This question has many variants and it is best answered explicitly.

Given two polygons of the same area, is it always possible to dissect one into a finite number of pieces which can be reassembled into a replica of the other?

Dissection theory is extensive. In such questions one needs to specify • What is a “piece”? (polygon? Topological disk? Borel-set? Lebesgue-measurable set?

Arbitrary?)

• How many pieces are permitted (finitely many? countably? uncountably?)

• What motions are allowed in “reassembling” (translations? rotations? orientationreversing maps? isometries? affine maps? homotheties? arbitrary continuous images? etc.)

• How the pieces are permitted to be glued together. The simplest notion is that they must be disjoint. If the pieces are polygons [or any piece with a nice boundary] you can permit them to be glued along their boundaries, ie the interiors of the pieces disjoint, and their union is the desired figure.

Some dissection results

• We are permitted to cut into finitely many polygons, to translate and rotate the pieces, and to glue along boundaries; then yes, any two equal-area polygons are equidecomposable.

This theorem was proven by Bolyai and Gerwien independently, and has undoubtedly been independently rediscovered many times. I would not be surprised if the Greeks knew this.

The Hadwiger-Glur theorem implies that any two equal-area polygons are equi-decomposable using only translations and rotations by 180 degrees.

• Theorem 5 (Hadwiger-Glur, 1951) Two equal-area polygons P,Q are equi-decomposable by translations only, iff we have equality of these two functions: ϕ_{P} () = ϕ_{Q}()

Here, for each direction v (ie, each vector on the unit circle in the plane), let ϕ_{P} (v) be the sum of the lengths of the edges of P which are perpendicular to v, where for such an edge, its length is positive if v is an outward normal to the edge and is negative if v is an inward normal to the edge.

• In dimension 3, the famous “Hilbert’s third problem” is:

If P and Q are two polyhedra of equal volume, are they equi-decomposable by means of translations and rotations, by cutting into finitely many subpolyhedra, and gluing along boundaries?

The answer is no and was proven by Dehn in 1900, just a few months after the problem was posed. (Ueber raumgleiche polyeder, Goettinger Nachrichten 1900, 345-354). It was the first of Hilbert’s problems to be solved. The proof is nontrivial but does not use the axiom of choice.

References

Hilbert’s Third Problem. V.G. Boltianskii. Wiley 1978.

• Using the axiom of choice on non-countable sets, you can prove that a solid sphere can be dissected into a finite number of pieces that can be reassembled to two solid spheres, each of same volume of the original. No more than nine pieces are needed.

The minimum possible number of pieces is five. (It’s quite easy to show that four will not suffice). There is a particular dissection in which one of the five pieces is the single center point of the original sphere, and the other four pieces A, A′, B, B′ are such that A is congruent to A′ and B is congruent to B′. [See Wagon’s book].

This construction is known as the Banach-Tarski paradox or the Banach-Tarski-Hausdorff paradox (Hausdorff did an early version of it). The “pieces” here are non-measurable sets, and they are assembled disjointly (they are not glued together along a boundary, unlike the situation in Bolyai’s thm.) An excellent book on Banach-Tarski is:

The Banach-Tarski Paradox. Stan Wagon. Cambridge University Press, 985

Robert M. French. The Banach-Tarski theorem. The Mathematical Intelligencer, 10 (1988) 21-28.

The pieces are not (Lebesgue) measurable, since measure is preserved by rigid motion. Since the pieces are non-measurable, they do not have reasonable boundaries. For example, it is likely that each piece’s topological-boundary is the entire ball.

The full Banach-Tarski paradox is stronger than just doubling the ball. It states:

• Any two bounded subsets (of 3-space) with non-empty interior, are equi-decomposable by translations and rotations.

This is usually illustrated by observing that a pea can be cut up into finitely pieces and reassembled into the Earth.

The easiest decomposition “paradox” was observed first by Hausdorff:

• The unit interval can be cut up into countably many pieces which, by translation only, can be reassembled into the interval of length 2.

This result is, nowadays, trivial, and is the standard example of a non-measurable set, taught in a beginning graduate class on measure theory.

• Theorem 6 There is a finite collection of disjoint open sets in the unit cube in R^{3} which can be moved by isometries to a finite collection of disjoint open sets whose union is dense in the cube of size 2 in R^{3}.

This result is by Foreman and Dougherty.

• A square cannot be rearranged into a disk, if one is allowed finitely many pieces with analytic boundaries, glued at edges.

• A square can be rearranged into a disk, with translations only, if one is allowed to use finitely many pieces with unconstrained shape (not necessarily connected), and disjoint assembly.

References

Boltyanskii. Equivalent and equidecomposable figures. in Topics in Mathematics published by D.C. HEATH AND CO., Boston.

Dubins, Hirsch and ? Scissor Congruence American Mathematical Monthly.

“Banach and Tarski had hoped that the physical absurdity of this theorem would encourage mathematicians to discard AC. They were dismayed when the response of the math community was ‘Isn’t AC great? How else could we get such counterintuitive results?’ ”

9.3 The Continuum Hypothesis

A basic reference is Godel’s “What is Cantor’s Continuum Problem?”, from 1947 with a 1963 supplement, reprinted in Benacerraf and Putnam’s collection Philosophy of Mathematics. This outlines Godel’s generally anti-CH views, giving some “implausible” consequences of CH.

”I believe that adding up all that has been said one has good reason to suspect that the role of the continuum problem in set theory will be to lead to the discovery of new axioms which will make it possible to disprove Cantor’s conjecture.”

At one stage he believed he had a proof that C = ℵ_{2} from some new axioms, but this turned out to be fallacious. (See Ellentuck, “Godel’s Square Axioms for the Continuum”, Mathematische Annalen 1975.)

Maddy’s “Believing the Axioms”, Journal of Symbolic Logic 1988 (in 2 parts) is an extremely interesting paper and a lot of fun to read. A bonus is that it gives a non-set-theorist who knows the basics a good feeling for a lot of issues in contemporary set theory.

Most of the first part is devoted to “plausible arguments” for or against CH: how it stands relative to both other possible axioms and to various set-theoretic “rules of thumb”. One gets the feeling that the weight of the arguments is against CH, although Maddy says that many “younger members” of the set-theoretic community are becoming more sympathetic to CH than their elders. There’s far too much here for me to be able to go into it in much detail.

Some highlights from Maddy’s discussion, also incorporating a few things that other people sent me:

1. Cantor’s reasons for believing CH aren’t all that persuasive today.

2. Godel’s proof of the consistency of CH shows that CH follows from ZFC plus the Axiom of Constructibility (V = L, roughly that the set-theoretic universe = the constructible universe). However, most set-theorists seem to find Constructiblity implausible and much too restrictive. It’s an example of a “minimizing” principle, which tends to cut down on the number of sets admitted to one’s universe. Apparently “maximizing” principles meet with much more sympathy from set theorists. Such principles are more compatible with ¬CH than with CH.

3. If GCH is true, this implies that ℵ_{0} has certain unique properties: e.g. that it’s that cardinal before which GCH is false and after which it is true. Some would like to believe that the set-theoretic universe is more “uniform” (homogeneous) than that, without this kind of singular occurrence. Such a “uniformity” principle tends to imply ¬GCH.

4. Most of those who disbelieve CH think that the continuum is likely to have very large cardinality, rather than ℵ_{2} (as Godel seems to have suggested). Even Cohen, a professed formalist, argues that the power set operation is a strong operation that should yield sets much larger than those reached quickly by stepping forward through the ordinals:

”This point of view regards C as an incredibly rich set given to us by a bold new axiom, which can never be approached by any piecemeal process of construction.”

5. There are also a few arguments in favour of CH, e.g. there’s an argument that ¬CH is restrictive (in the sense of (2) above). Also, CH is much easier to force (Cohen’s method) than ¬CH. And CH is much more likely to settle various outstanding results than is ¬CH, which tends to be neutral on these results.

6. Most large cardinal axioms (asserting the existence of cardinals with various properties of hugeness: these are usually derived either from considering the hugeness of ℵ_{0} compared to the finite cardinals and applying uniformity, or from considering the hugeness of V (the set-theoretic universe) relative to all sets and applying “reflection”) don’t seem to settle CH one way or the other.

7. Various other axioms have some bearing. Axioms of determinacy restrict the class of sets of reals that might be counterexamples to CH. Various forcing axioms (e.g. Martin’s axiom), which are “maximality” principles (in the sense of (2) above), imply ¬CH. The strongest (Martin’s maximum) implies that C = ℵ_{2}. Of course the “truth” or otherwise of all these axioms is controversial.

8. Freiling’s principle about “throwing darts at the real line” is a seemingly very plausible principle, not involving large cardinals at all, from which ¬CH immediately follows. Freiling’s paper (JSL 1986) is a good read. More on this at the end of this message.

Of course we have conspicuously avoided saying anything about whether it’s even reasonable to suppose that CH has a determinate truth-value. Formalists will argue that we may choose to make it come out whichever way we want, depending on the system we work in. On the other hand, the mere fact of its independence from ZFC shouldn’t immediately lead us to this conclusion – this would be assigning ZFC a privileged status which it hasn’t necessarily earned. Indeed, Maddy points out that various axioms within ZFC (notably the Axiom of Choice, and also Replacement) were adopted for extrinsic reasons (e.g. “usefulness”) as well as for “intrinsic” reasons (e.g. “intuitiveness”). Further axioms, from which CH might be settled, might well be adopted for such reasons.

One set-theorist correspondent said that set-theorists themselves are very loathe to talk about “truth” or “falsity” of such claims. (They’re prepared to concede that 2 + 2 = 4 is true, but as soon as you move beyond the integers trouble starts. e.g. most were wary even of suggesting that the Riemann Hypothesis necessarily has a determinate truth-value.) On the other hand, Maddy’s contemporaries discussed in her paper seemed quite happy to speculate about the “truth” or “falsity” of CH.

The integers are not only a bedrock, but also any finite number of power sets seem to be quite natural Intuitively are also natural which would point towards the fact that CH may be determinate one way or the other. As one correspondent suggested, the question of the determinateness of CH is perhaps the single best way to separate the Platonists from the formalists.

And is it true or false? Well, CH is somewhat intuitively plausible. But after reading all this, it does seem that the weight of evidence tend to point the other way.

The following is from Bill Allen on Freiling’s Axiom of Symmetry. This is a good one to run your intuitions by.

Let A be the set of functions mapping Real Numbers into countable sets of Real Numbers. Given a function f in A, and some arbitrary real numbers x and y, we see that x is in f(y) with probability 0, i.e. x is not in f(y) with probability 1. Similarly, y is not in f(x) with probability 1. Let AX be the axiom which states

“for every f in A, there exist x and y such that x is not in f(y) and y is not in f(x)”

The intuitive justification for AX is that we can find the x and y by choosing them at random.

In ZFC, AX = not CH. proof: If CH holds, then well-order R as r_{0}, r_{1}, ...., r_{x}, ... with x < ℵ_{1}. Define f(r_{x}) as {r_{y} : y ≥ x}. Then f is a function which witnesses the falsity of AX.

If CH fails, then let f be some member of A. Let Y be a subset of R of cardinality ℵ_{1}. Then Y is a proper subset. Let X be the union of all the sets f(y) with y in Y, together with Y. Then, as X is an ℵ_{1} union of countable sets, together with a single ℵ_{1} size set Y, the cardinality of X is also ℵ_{1}, so X is not all of R. Let a be in R X, so that a is not in f(y) for any y in Y. Since f(a) is countable, there has to be some b in Y such that b is not in f(a). Thus we have shown that there must exist a and b such that a is not in f(b) and b is not in f(a). So AX holds.

Freiling’s proof, does not invoke large cardinals or intense infinitary combinatorics to make the point that CH implies counter-intuitive propositions. Freiling has also pointed out that the natural extension of AX is AXL (notation mine), where AXL is AX with the notion of countable replaced by Lebesgue Measure zero. Freiling has established some interesting Fubini-type theorems using AXL.

See “Axioms of Symmetry: Throwing Darts at the Real Line”, by Freiling, Journal of Symbolic Logic, 51, pages 190-200. An extension of this work appears in ”Some properties of large filters”, by Freiling and Payne, in the JSL, LIII, pages 1027-1035.

The section above was excerpted from a posting from David Chalmers, of Indiana University.

See also

Nancy McGough’s *Continuum Hypothesis article* or its *mirror*.

http://www.jazzie.com/ii/math/ch/

http://www.best.com/ ii/math/ch/

Formulas of General Interest

10.1 How to determine the day of the week, given the month, day and year

First a brief explanation: In the Gregorian Calendar, over a period of four hundred years, there are 97 leap years and 303 normal years. Each normal year, the day of January 1 advances by one; for each leap year it advances by two.

303 + 97 + 97 = 497 = 7 ∗ 71

As a result, January 1 year N occurs on the same day of the week as January 1 year N + 400. Because the leap year pattern also recurs with a four hundred year cycle, a simple table of four hundred elements, and single modulus, suffices to determine the day of the week (in the Gregorian Calendar), and does it much faster than all the other algorithms proposed. Also, each element takes (in principle) only three bits; the entire table thus takes only 1200 bits; on many computers this will be less than the instructions to do all the complicated calculations proposed for the other algorithms.

Incidental note: Because 7 does not divide 400, January 1 occurs more frequently on some days than others! Trick your friends! In a cycle of 400 years, January 1 and March 1 occur on the following days with the following frequencies:

           Sun      Mon     Tue     Wed     Thu     Fri     Sat
    Jan 1: 58       56      58      57      57      58      56
    Mar 1: 58       56      58      56      58      57      57

Of interest is that (contrary to most initial guesses) the occurrence is not maximally flat. In the Mathematical Gazette, vol. 53,, pp.127-129, it is shown that the 13th of the month

is more likely to be a Friday than any other day.The author is a 13 year old S.R.Baxter.

The Gregorian calendar was introduced in 1582 in parts of Europe; it was adopted in 1752 in Great Britain and its colonies, and on various dates in other countries. It replaced the Julian Calendar which has a four-year cycle of leap years; after four years January 1 has advanced by five days. Since 5 is relatively prime to 7, a table of 4 ∗ 7 = 28 elements is necessary for the Julian Calendar.

There is still a 3 day over 10,000 years error which the Gregorian calendar does not take into account. At some time such a correction will have to be done but your software will probably not last that long!

Here is a standard method suitable for mental computation:

1. Take the last two digits of the year.

2. Divide by 4, discarding any fraction.

3. Add the day of the month.

4. Add the month’s key value: JFM AMJ JAS OND 144 025 036 146

5. Subtract 1 for January or February of a leap year.

6. For a Gregorian date, add 0 for 1900’s, 6 for 2000’s, 4 for 1700’s, 2 for 1800’s; for other years, add or subtract multiples of 400.

7. For a Julian date, add 1 for 1700’s, and 1 for every additional century you go back.

8. Add the last two digits of the year.

9. Divide by 7 and take the remainder.

Now 1 is Sunday, the first day of the week, 2 is Monday, and so on.

The following formula, which is for the Gregorian calendar only, may be more convenient for computer programming. Note that in some programming languages the remainder operation can yield a negative result if given a negative operand, so mod 7 may not translate to a simple remainder.

W = (k + ⌊2.6m − 0.2⌋ − 2C + Y + ⌊Y/4⌋ + ⌊C/4⌋)mod7

where ⌊ ⌋ denotes the integer floor function, k is day (1 to 31) m is month (1 = March, ..., 10 = December, 11 = Jan, 12 = Feb) Treat Jan & Feb as months of the preceding year C is century (1987 has C = 19) Y is year (1987 has Y = 87 except Y = 86 for Jan & Feb) W is week day (0 = Sunday, ..., 6 = Saturday)

Here the century and 400 year corrections are built into the formula. The ⌊2.6m − 0.2⌋ term relates to the repetitive pattern that the 30-day months show when March is taken as the first month.

The following short C program works for a restricted range, it returns 0 for Monday, 1 for Tuesday, etc.

dow(m,d,y){y-=m<3;return(y+y/4-y/100+y/400+"-bed=pen+mad."[m]+d)%7;}

The program appeared was posted by sakamoto@sm.sony.co.jp (Tomohiko Sakamoto) on comp.lang.c on March 10th, 1993.

A good mnemonic rule to help on the computation of the day of the week is as follows. In any given year the following days come on the same day of the week:

4/4
6/6
8/8
10/10
12/12

to remember the next four, remember that I work from 9-5 at a 7-11 so

9/5
5/9
7/11
11/7

and the last day of Feb.

”In 1995 they come on Tuesday. Every year this advances one other than leap-years which advance 2. Therefore for 1996 the day will be Thursday, and for 1997 it will be Friday. Therefore ordinarily every 4 years it advances 5 days. There is a minor correction for the century since the century is a leap year iff the century is divisible by 4. Therefore 2000 is a leap year, but 1900, 1800, and 1700 were not.”

Even ignoring the pattern over for a period of years this is still useful since you can generally figure out what day of the week a given date is on faster than someone else can look it up with a calender if the calender is not right there. (A useful skill that.)

References

Winning Ways for your mathematical plays. Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy London ; Toronto : Academic Press, 1982. Mathematical Carnival. Martin Gardner. New York : Knopf, c1975. Elementary Number Theory and its applications. Kenneth Rosen. Reading, Mass. ; Don Mills, Ont. : Addison-Wesley Pub. Co., c1993. p. 156. Michael Keith and Tom Craver. The Ultimate Perpetual Calendar? Journal of Recreational Mathematics, 22:4, pp. 280-282, 19

10.2 Symbolic Computation Packages

This is not a comprehensive list. There are other Computer Algebra packages available that may better suit your needs. There is an Available Packages listing maintained at UC Berkeley. (The list can be obtained from math.berkeley.edu via anonymous ftp).

The ¡A HREF=”http://symbolicnet.mcs.kent.edu/”¿ Symbolic Computation Network¡/a¿ contains lots of useful information.

A: Maple
Purpose: Symbolic and numeric computation, mathematical
programming, and mathematical visualization.
Contact: Waterloo Maple Software,
450 Phillip Street
Waterloo, Ontario
N2L 5J2
Phone (519)747-2373
FAX   (519)747-5284
email:  info@maplesoft.on.ca

A: DOE-Macsyma
Purpose: Symbolic and mathematical manipulations.
Contact: National Energy Software Center
Argonne National Laboratory 9700 South Cass Avenue
Argonne, Illinois 60439
Phone: (708) 972-7250

A: Pari
Purpose: Number-theoretic computations and simple numerical
analysis.
Available for most 32-bit machines, including 386+387 and 486.
This is a copyrighted but free package, available by ftp from
math.ucla.edu (128.97.4.254) and ftp.inria.fr (128.93.1.26).
Contact: questions about pari can be sent to pari@math.u-bordeaux.fr
and for the Macintosh versions to bernardi@mathp7.jussieu.fr

A: Mathematica

Purpose: Mathematical computation and visualization,
symbolic programming.
Contact: Wolfram Research, Inc.
100 Trade Center Drive Champaign,
IL 61820-7237
Phone: 1-800-441-MATH
A: Macsyma
Purpose: Symbolic numerical and graphical mathematics.
Contact: Macsyma Inc.
20 Academy Street
Arlington, MA 02174
tel: 617-646-4550
fax: 617-646-3161
email: info-macsyma@macsyma.com

A: Matlab
Purpose: `matrix laboratory' for tasks involving
matrices, graphics and general numerical computation.
Contact: The MathWorks, Inc.
21 Prime Park Way
Natick, MA 01760
508-653-1415
info@mathworks.com

A: Cayley/Magma
A: Cayley/Magma
Cayley is no longer being licenced or supported.
It has been superseded by a new and more powerful system
called Magma.
Purpose: Computation in algebraic, geometric and
combinatorial structures such as groups, rings, fields,
algebras, modules, graphs and codes.
Available for: SUN 3, SUN 4, SUN 10 (SUNOS 4.x and Solaris 2)
DECstation (Ultrix), DEC Alpha (OSF/1), IBM RS6000 (AIX),
HP9000/700 (HP-UX), Apollo M680x0, SGI, 486/Pentium (MS-DOS).
Contact: Computational Algebra Group
School of Mathematics and Statistics
University of Sydney

NSW 2006
Australia
Phone:  +61 2 351 3338
Fax: +61 2 351 4534
URL: http://www.maths.usyd.edu.au:8000/comp/magma/Overview.html
magma@maths.su.oz.au
A: Axiom
Purpose: Symbolic programming, symbolic and numeric computation,
         mathematical visualisation.
Contact: The Numerical Algorithms Group Ltd
Wilkinson House
Jordan Hill Road
Oxford
OX2 8DR
UK
email:  infodesk@nag.co.uk
Phone: +44 1865 311744
Fax:   +44 1865 311755

10.3 Formula for the Surface Area of a sphere in Euclidean N-Space

This is equivalent to the volume of the N-1 solid which comprises the boundary of an N-Sphere.

The volume of a ball is the easiest formula to remember: It’s r^{N} \frac{π^{N/2}}{(N/2)!}. The only hard part is taking the factorial of a half-integer. The real definition is that x! = Γ(x + 1), but if you want a formula, it’s:

(1/2 + n)! = \sqrt{π} \frac{(2n + 2)!}{(n + 1)!4^{n+1}}

To get the surface area, you just differentiate to get N \frac{π^{N/2}}{(N/2)!}r^{N−1}.

There is a clever way to obtain this formula using Gaussian integrals. First, we note that the integral over the line of e^{−x^{2}} is \sqrt{π}. Therefore the integral over N-space of e^{−x_{1}^{2}−x_{2}^{2}−...−x_{N}^{2}} is \sqrt{π}^{n}. Now we change to spherical coordinates. We get the integral from 0 to infinity of V r^{N−1}e^{−r^{2}}, where V is the surface volume of a sphere. Integrate by parts repeatedly to get the desired formula.

It is possible to derive the volume of the sphere from “first principles”.

10.4 Formula to compute compound interest.

Here’s a formula which can be used in 123, Excel, Wings and Dynaplan:

------- Input this data -------------------------------
principal amount = E9                  ( in dollars )
Amortization Period = d10              ( in years ie 6 mon = .5 )
Payments / year = D11                  ( 12 = monthly, 52 = weekly )
Published Interest rate = D12          ( ie 9 % = 0.09 )
Times per year Int calculated = d13    ( CDN mortgage use 2
                                   US mortgage use 12
                                   all other loans use 12 )
----- Calculate the proper rate of interest -----------
e14 = Effective annual rate = EXP(D13*LN(1+(D12/D13)))-1
e15 = Interest rate per payment = (EXP(LN(E14+1)/(D10*D11))-1)*D10*D11
e17 = Payments = APMT(E9,E15/D11,D10*D11) ( both these functions are
               = PMT (E9,E15/D11,D10*D11) ( identical,diff spreadsheet)
      APMT( principal amount,interest rate per period,# periods )
      ( this is a standard function on any true commercial spreadsheet)
      OR use the following if done using a calculator
    = Payments = P*I/[1-(I+1)^-T]
               = E9*(E15/D11)/(1-((E15/D11) +1)**(-1*D10*D11))
Total interest cost = E17*D10*D11-E9
-- Use these formulas if you wish to generate an amortization table --
always add up to 'Payments (e17)'
Interest per payment  = current balance * ( E15 / D11 )
Principal per payment = current balance - Interest per payment
new current balance   = current balance - Principal per payment -
                        (extra payment)
   keep repeating until 'new current balance' = 0

Derivation of Compound Interest Rate Formula

Suppose you deposited a fixed payment into an interest bearing account at regular intervals, say monthly, at the end of each month. How much money would there be in the account at the end of the nth month (at which point you’ve made n payments)? Let i be the monthly interest rate as a fraction of principle. Let x be the amount deposited each month. Let n be the total number of months. Let p[k] be the principle after k months.

So the recursive formula is:

p[n] = x + ((1 + i)p[n − 1])eq1

This yields the summation:

p[n] = ∑_{k=0}^{n−1} x(1 + i)^{k}

The way to solve this is to multiply through by (1 + i) and subtract the original equation from the resulting equation. Observe that all terms in the summation cancel except the last term of the multiplied equation and the first term of the original equation:

πp[n] = x((1 + i)^{n} − 1)

or

p[n] = x((1 + i)^{n} − 1)/i

Now suppose you borrow p at constant interest rate i. You make monthly payments of x. It turns out that this problem is identical to taking out a balloon loan of p (that is it’s all due at the end of some term) and putting payments of x into a savings account. At the end of the term you use the principle in the savings account to pay off the balance of the loan. The loan and the savings account, of course, must be at the same interest rate. So what we want to know is: what monthly payment is needed so that the balance of the savings account will be identical to the balance of the balloon loan after n payments?

The formula for the principal of the balloon loan at the end of the nth month is:

p[n] = p[0](1 + i)^{n}

So we set this expression equal to the expression for the the savings account, and we get:

p[0](1 + i)^{n} = x((1 + i)^{n} − 1)/i

or solving for x:

x = p[0](1 + i)^{n}i/((1 + i)^{n} − 1)

If (1 + i)^{n} is large enough (say greater than 5), here is an approximation for determining n from x, p, and i:

n ≈ −ln(ln(x/(ip)))/ln(1 + i)

The above approximation is based upon the following approximation:

ln(y − 1) ≈ lny − 1/y

Which is within 2

For example, a $100000 loan at 1% monthly, paying $1028.61 per month should be paid in 360 months. The approximation yields 358.9 payments.

If this were your 30 year mortgage and you were paying $1028.61 per month and you wanted to see the effect of paying $1050 per month, the approximation tells you that it would be paid off in 303.5 months (25 years and 3.5 months). If you stick 304 months into the equation for x, you get $1051.04, so it is fairly close. This approximation does not work, though, for very small interest rates or for a small number of payments. The rule is to get a rough idea first of what (1 + i)^{n} is. If that is greater than 5, the approximation works pretty well. In the examples given, (1 + i)^{n} is about 36.

Finding i given n, x, and p is not as easy. If i is less than 5% per payment period, the following equation approximately holds for i:

i = −(1/n)ln(1 − ip/x)

There is no direct solution to this, but you can do it by Newton-Raphson approximation. Begin with a guess, i[0]. Then apply:

i[k + 1] = i[k] − \frac{x(1 − i[k]p/x)(ni[k] + ln(1 − i[k]p/x))}{xn(1 − i[k]p/x) − p}

You must start with i too big, because the equation for i has a solution at i = 0, and that’s not the one you want to end up with.

Example: Let the loan be for p = $10000, x = $50 per week for 5 years (n = 260). Let i[0] = 20% per annum or 0.3846% per week. Since i must be a fraction rather than a percent, i[0] = 0.003846. Then, applying eq 11:

i[1] = 0.003077
i[2] = 0.002479
i[3] = 0.002185
i[4] = 0.002118
i[5] = 0.002115

The series is clearly beginning to converge here.

To get i[5] as an annual percentage rate, multiply by 52 weeks in a year and then by 100%, so i[5] = 10.997% per annum. Substituting i[5] back into eq 7, we get x = $50.04, so it works pretty well.

References

The theory of interest. Stephen G. Kellison. Homewood, Ill., R. D. Irwin, 1970.

References, General Bibliography and Textbooks

The following books have been recommended by several readers. The number of recommendations is in brackets.

• Algebra Lang, Serge. Algebra. 2nd ed. Addison-Wesley Pub. Co., 1984. [2] Halmos, Linear Algebra. [1] Birkhoff, McLane, Algebra van der Waerden. Algebra Atiyah & MacDonald. Introduction to Commutative Algebra

• Complex Analysis Ahlfors, Lars Valerian. Complex analysis : an introduction to the theory of analytic functions of one complex variable. 3rd ed. New York; Toronto : McGraw-Hill, c1979. [2] Conway, John B. Functions of one complex variable [by] John B. Conway. [New York] Springer-Verlag New York, 1973. [1] Priestley, Introduction to complex analysis

• Real & Complex Analysis Titchmarsh. Theory of Functions Boas. Primer of Real Functions Polya & Szego. Problems & Theorems in Analysis Rudin, Walter. Principles of mathematical analysis. 3d ed. New York : McGraw-Hill, 1976. [2] Rudin, Walter. ”Functional Analysis” Royden, H. L. Real analysis. 3rd ed. New York, Macmillan ; London : Collier Macmillan, 1988. [1] Hewitt, Edwin, 1920. Real and abstract analysis : a modern treatment of the theory of functions of a real variable. New York : Springer-Verlag, 1969. [2] Dieudonne’. Foundations of Analysis Courant & Hilbert. Mathematical Methods of Physics.

• Geometry David Hilbert. Foundations of Geometry 2nd English edition, tr. by Leo Unger, publ. by Open Court, 1971. Neumann, Stoy & Thompson. Groups and Geometry [1]

• Number Theory Hardy, Littlewood. Samuel, ”Algebraic Theory of Numbers” Hardy & Wright

• History of Mathematics Morris Kline Mathematical Thought from Ancient to Modern Times

• Topology Guillemin, Victor and Alan Pollack: Differential Topology. Spivak, Michael: A Comprehensive Introduction to Differential Geometry, Vol. I Morgan, Frank: Riemannian Geometry: A Beginner’s Guide Milnor, ”Topology from the Differentiable Viewpoint” R. Engelking. General Topology. Kuratowski. Topology. Copson. Metric Spaces. Greenberg, Martin and (?) Harper: Algebraic Topology: An Introduction. Kelly, General topology

• Calculus Hardy, Course of Pure Mathematics.[2] Landau. Differential & Integral Calculus.

Courant & John. Introduction to Calculus & Analysis, vol.1. Spivak. Calculus on Manifolds.

• Probability Feller, Introduction to probability theory

• Statistics Silvey, Statistical inference

• Measure Theory Weir, Integration and measure

• General Courant & Robins [2] What is Mathematics. Oxford University Press. 1969

The Sci.Math FAQ Team

The sci.math FAQ, which was initally edited and compiled by Alex Lopez-Ortiz is now a distributed effort of scientists in over five countries. At this time, the FAQ contains sections maintained by Alex Lopez-Ortiz (alopez-o@barrow.uwaterloo.ca), Nancy McGough (nancym@ii.com), and Hans de Vreught (J.P.M.deVreught@cs.tudelft.nl). Several others sections are in the works, on the hands of other volunteers.

If you wish to collaborate, send mail to alopez-o@barrow.uwaterloo.ca.

12.1 Copyright Notice

Copyright (c) 1993, 1994, 1995 A. Lopez-Ortiz

This FAQ is Copyright (C) 1994, 1995 by Alex Lopez-Ortiz. This text, in whole or in part, may not be sold in any medium, including, but not limited to electronic, CD-ROM, or published in print, without the explicit, written permission of Alex Lopez-Ortiz.

Individual sections are Copyright (C) 1994, 1995 of their individual maintainers.

——————————— Questions and Answers Edited and Compiled by:

Alex Lopez-Ortiz alopez-o@barrow.UWaterloo.ca Department of Computer Science University of Waterloo Waterloo, Ontario Canada