38 CHAPTER 2. THE REAL AND COMPLEX NUMBERS

6. If z,w are complex numbers prove zw = z w. Show that z1 · · ·zm = z1 · · ·zm. Alsoverify that ∑

mk=1 zk = ∑

mk=1 zk. In words this says the conjugate of a product equals

the product of the conjugates and the conjugate of a sum equals the sum of the con-jugates.

7. Suppose p(x) = anxn +an−1xn−1 + · · ·+a1x+a0 where all the ak are real numbers.Suppose also that p(z) = 0 for some z ∈ C. Show it follows that p(z) = 0 also.

8. I claim that 1 = −1. Here is why.−1 = i2 =√−1√−1 =

√(−1)2 =

√1 = 1. This

is clearly a remarkable result but is there something wrong with it? If so, what iswrong? Hint: When we push symbols without consideration of their meaning, wecan accomplish many strange and wonderful but false things.

9. De Moivre’s theorem of Problem 4 is really a grand thing. I plan to use it now for ra-tional exponents, not just integers. 1 = 1(1/4) = (cos2π + isin2π)1/4 = cos(π/2)+isin(π/2) = i. Therefore, squaring both sides it follows 1 = −1 as in the previousproblem. What does this tell you about De Moivre’s theorem? Is there a profounddifference between raising numbers to integer powers and raising numbers to noninteger powers?

10. Review Problem 4 at this point. Now here is another question: If n is an integer, is italways true that (cosθ − isinθ)n = cos(nθ)− isin(nθ)? Explain.

11. Suppose you have any polynomial in cosθ and sinθ . By this I mean an expressionof the form ∑

mα=0 ∑

nβ=0 aαβ cosα θ sinβ

θ where aαβ ∈C. Can this always be writtenin the form ∑

m+nγ=−(n+m)

bγ cosγθ +∑n+mτ=−(n+m)

cτ sinτθ? Explain.

12. Does there exist a subset of C, C+ which satisfies 2.4.1 - 2.4.3? Hint: You mightreview the theorem about order. Show−1 cannot be inC+. Now ask questions about−i and i. In mathematics, you can sometimes show certain things do not exist. It isvery seldom you can do this outside of mathematics. For example, does the LochNess monster exist? Can you prove it does not?

13. Show that if a/b is irrational, then {ma+nb}m,n∈Z is dense in R, each an irra-tional number. If a/b is rational, show that {ma+nb}m,n∈Z is not dense. Hint:From Theorem 2.16.1 there exist integers, ml ,nl such that |mla+nlb| < 2−l . LetPl ≡ ∪k∈Z {k (mla+nlb)} . Thus this is a collection of numbers which has succes-sive numbers 2−l apart. Then consider ∪l∈NPl . In case the ratio is rational and{ma+nb}m,n∈Z is dense, explain why there are relatively prime integers p,q suchthat p/q = a/b is rational and {mp+nq}m,n∈Z would be dense. Isn’t this last acollection of integers?

14. This problem will show, as a special case, that the rational numbers are dense in R.Referring to the proof of Theorem 2.16.1.

(a) Suppose α ∈ (0,1) and is irrational. Show that if N is a positive integer, thenthere are integers m,n such that 0 < n ≤ N and |nα−m| < 1

N12 (1+α) < 1

N .

Thus∣∣α− m

n

∣∣< 1nN ≤

1n2 .