- Let z = 5 + i9. Find z
^{−1}. - Let z = 2 + i7 and let w = 3 − i8. Find zw,z + w,z
^{2}, and w∕z. - If z is a complex number, show there exists ω a complex number with = 1 and ωz =.
- For those who know about the trigonometric functions from calculus or
trigonometry
^{3}, De Moivre’s theorem saysfor n a positive integer. Prove this formula by induction. Does this formula continue to hold for all integers, n, even negative integers? Explain.

- Using De Moivre’s theorem from Problem 4, derive a formula for sinand one for cos. Hint: Use Problem 18 on Page 59 and if you like, you might use Pascal’s triangle to construct the binomial coefficients.
- If z,w are complex numbers prove zw = zw and then show by induction that
z
_{1}z_{m}= z_{1}z_{m}. Also verify that ∑_{k=1}^{m}z_{k}= ∑_{k=1}^{m}z_{k}. 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 conjugates. - Suppose p= a
_{n}x^{n}+ a_{n−1}x^{n−1}++ a_{1}x + a_{0}where all the a_{k}are real numbers. Suppose also that p= 0 for some z ∈ ℂ. Show it follows that p= 0 also. - I claim that 1 = −1. Here is why.
This is clearly a remarkable result but is there something wrong with it? If so, what is wrong?

- De Moivre’s theorem of Problem 4 is really a grand thing. I plan to use it now for
rational exponents, not just integers.
Therefore, squaring both sides it follows 1 = −1 as in the previous problem. What does this tell you about De Moivre’s theorem? Is there a profound difference between raising numbers to integer powers and raising numbers to non integer powers?

- Review Problem 4 at this point. Now here is another question: If n is an integer, is it
always true that
^{n}= cos− isin? Explain. - Suppose you have any polynomial in cosθ and sinθ. By this I mean an expression of the
form ∑
_{α=0}^{m}∑_{β=0}^{n}a_{αβ}cos^{α}θ sin^{β}θ where a_{αβ}∈ ℂ. Can this always be written in the form ∑_{γ=−(n+m ) }^{m+n}b_{γ}cosγθ + ∑_{τ=−(n+m ) }^{n+m}c_{τ}sinτθ? Explain. - Does there exist a subset of ℂ, ℂ
^{+}which satisfies 2.4.1 - 2.4.3? Hint: You might review the theorem about order. Show −1 cannot be in ℂ^{+}. Now ask questions about −i and i. In mathematics, you can sometimes show certain things do not exist. It is very seldom you can do this outside of mathematics. For example, does the Loch Ness monster exist? Can you prove it does not? - Show that if a∕b is irrational, then
_{m,n∈ℤ}is dense in ℝ. If a∕b is rational, show that_{m,n∈ℤ}is not dense. Hint: From Theorem 2.15.1 there exist integers, m_{l},n_{l}such that< 2^{−l}. Let P_{l}≡∪_{k∈ℤ}. Thus this is a collection of numbers which has successive numbers 2^{−l}apart. Then consider ∪_{l∈ℕ}P_{l}.

Download PDFView PDF