KMB, 28/10/99

M1F Foundations of Analysis—Problem Sheet 4

1) Find all real numbers x such that 0 ≤ x ≤ 1 and

\sqrt{2x} + \sqrt{1 − x} = 1.

2) The conjugate of a complex number z = a + bi is \overline{z} = a − bi.

(a) Prove that |z|^{2} = z\overline{z}.

(b) Prove that if z and w are complex numbers then \overline{zw} = \overline{z}\overline{w}.

(c) Prove that if z = (r, θ) then \overline{z} = (r, −θ). 3)

(a) Find the real and imaginary parts of (1 + i)^{100}.

(b) If you know the binomial theorem, then use part (a) to evaluate the sum \binom{100}{0} − \binom{100}{2} + \binom{100}{4} − \binom{100}{6} + … + \binom{100}{100}.

(c) If you want a challenge, find a purely combinatorial proof of (b) (that is, don’t use (a) or indeed anything about the complex numbers; just use facts about binomial coefficients.)

4) By considering \frac{1+i}{\sqrt{3}+i}, or otherwise, show that cos(π/12) = \frac{\sqrt{6}+\sqrt{2}}{4}. Is this irrational?

5)

(a) Find all ten 10th roots of i. Which one is nearest to i in the Argand diagram?

(b) Let z be a non-zero complex number. Prove that the three cube roots of z are the vertices of an equilateral triangle on the Argand diagram.

6) Here’s another cosine evaluated explicitly. Let ζ = e^{2πi/5} be one of the 5th roots of 1.

a) Show that 1 + ζ + ζ^{2} + ζ^{3} + ζ^{4} = 0.

b) Let α = ζ + ζ^{4}, and let β = ζ^{2} + ζ^{3}. Prove that α and β are the roots of the polynomial X^{2} + X − 1.

c) Deduce that cos(2π/5) = \frac{\sqrt{5}−1}{4}.