- 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. - Give the complete solution to x
^{4}+ 16 = 0. - Graph the complex cube roots of −8 in the complex plane. Do the same for the four fourth roots of −16.
- If z is a complex number, show there exists ω a complex number with = 1 and ωz =.
- De Moivre’s theorem says
^{n}= r^{n}for n a positive integer. Does this formula continue to hold for all integers, n, even negative integers? Explain. - You already know formulas for cosand sinand these were used to prove De Moivre’s theorem. Now using De Moivre’s theorem, derive a formula for sinand one for cos. Hint: Use the binomial theorem.
- If z and w are two complex numbers and the polar form of z involves the angle θ while
the polar form of w involves the angle ϕ, show that in the polar form for zw the angle
involved is θ + ϕ. Also, show that in the polar form of a complex number, z, r = .
- Factor x
^{3}+ 8 as a product of linear factors. - Write x
^{3}+ 27 in the formwhere x^{2}+ ax + b cannot be factored any more using only real numbers. - Completely factor x
^{4}+ 16 as a product of linear factors. - Factor x
^{4}+ 16 as the product of two quadratic polynomials each of which cannot be factored further without using complex numbers. - 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: −1 = i
^{2}==== 1 . This is clearly a remarkable result but is there something wrong with it? If so, what is wrong? - De Moivre’s theorem 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?

- Show that ℂ cannot be considered an ordered field. Hint: Consider i
^{2}= −1. Recall that 1 > 0 by Proposition 1.4.2. - Say a + ib < x + iy if a < x or if a = x, then b < y. This is called the lexicographic order. Show that any two different complex numbers can be compared with this order. What goes wrong in terms of the other requirements for an ordered field.
- With the order of Problem 18, consider for n ∈ ℕ the complex number 1 −. Show that with the lexicographic order just described, each of 1 − in is an upper bound to all these numbers. Therefore, this is a set which is “bounded above” but has no least upper bound with respect to the lexicographic order on ℂ.

Download PDFView PDF