KMB, 14/10/99

M1F Foundations of Analysis—Problem Sheet 2

If you do this sheet then you will probably be getting quite good at writing down proofs. Questions marked with a † might be trickier than the others, so don’t get disheartened if you can’t do one of them!

1) For the following statements, decide whether they are true or false, and give a proof or a counterexample as required.

(a) The product of two irrational numbers is irrational.

(b) The product of two irrational numbers is rational.

(c) If n is an integer, then n^{2} is a multiple of 3 if and only if n is a multiple of 3.

(d) \sqrt{3} is irrational.

2)

(a) Find positive integers a and b such that \frac{a}{b} = 0.\overline{12} = 0.121212 . . .. (b)^{†} The decimal expansion of \frac{100}{9899} begins 0.010102030508132134 . . .. If you

read the digits of this decimal expansion in pairs, you get 01 01 02 03 05 08 and so on, which are the first few terms in the Fibonacci sequence. Can you explain why this is happening? Hint: let x = \frac{100}{9899} = 0.0101020305 . . .. Can you write down an equation relating x, 100x and 10000x?

3) Are the following statements true or false? Give proofs, disproofs, or counterexamples—do not just write down “TFT” or whatever.

a) If x and y are real, and x < y, then x^{2} < y^{2}. b)^{†} If x is real then x^{4} + 2x^{3} + 3x^{2} + 2x + 3 is positive [Hint: the simplest

approach does not involve calculus.]

c) A polynomial of degree n with complex coefficients always has exactly n distinct complex roots.

4)

a) Prove that \sqrt{2} + \sqrt{3} is irrational.

b) Prove that 1 + \sqrt{2} + \sqrt{3} is irrational.

5)^{†} Let x be a real number such that for all positive real numbers y, we have x ≤ y. Prove that x ≤ 0.

6) In this question, a “square” means the square of an integer. So the squares are {0, 1, 4, 9, … }. Proofs or counterexamples required to:

a) Every non-negative integer is a square.

b) Every non-negative integer is the sum of two squares.

c) Every non-negative integer is the sum of three squares.