Intro to Complex Numbers

Question 1

What is the complex conjugate of a complex number z = a + bi?

Answer 1

z bar = a - bi

Question 2

What is the complex conjugate of zw?

Answer 2

zw bar = z bar w bar

Question 3

z zbar =

Answer 3

|z|²

Question 4

What is the triangle inequality?

Answer 4

|z+w| ≤ |z| + |w|

Question 5

arg(zw) =

Answer 5

argz + arg w

Question 6

What is De Moivre’s Theorem?

Answer 6

If z ∈ S¹ and n ∈ Z, then
arg(zⁿ) = n arg z.
Equivalently, for θ ∈ R and n ∈ Z, we have
(cos θ + isin θ)ⁿ = cos(nθ) + isin(nθ)

Question 7

Prove De Moivre’s Theorem

Answer 7

Fix z ∈ S¹ and let θ = arg z.
For n ≥ 0, we use induction on n.
n = 0: We have z⁰ = 1 so arg(z⁰) = 0, and n arg z = 0.
inductive step: Suppose the result holds for some n ≥ 0, so arg(zⁿ) = n arg z.
Then, 
arg(zⁿ⁺¹) = arg(zⁿ) + arg z
= n arg z + arg z
= (n + 1) arg z
so the result holds for n + 1.
For n < 0, we use the result for positive values. Fix n < 0, and let
m = −n, so m > 0. Then, as above, arg(w
m) = m arg w for all w ∈ S¹.
But zⁿ = (z⁻¹)ᵐ = zbarᵐ and arg(zbar) = − arg z, so
arg(zⁿ) = arg(zbarᵐ)
= m arg zbar
= m(− arg z)
= (−m) arg z
= n arg z.

Question 8

Define root of unity

What is the arg of an nth root of unity?

Answer 8

If z ∈ C, n ∈ Z>0 and zⁿ= 1, then we say that z is a root of unity (an nth root of unity).
argz = 2kπ/n for some k ∈ Z

Question 9

What is a primitive root of unity?

Answer 9

If z ∈ C is an n
th root of unity and zᵐ ≠ 1 for 1 ≤ m ≤ n − 1,
then we say that z is a primitive nth root of unity

Question 10

Let z ∈ C, let n be an integer with n ≥ 2. If z is an nth

root of unity and z ≠ 1, then zⁿ⁻¹ + zⁿ⁻² + · · · + z + 1 = [ ]

Answer 10

zⁿ⁻¹ + zⁿ⁻² + · · · + z + 1 = 0

Question 11

Euler’s formula

Answer 11

e^(iθ) = cosθ + isinθ

Question 12

A complex polynomial of degree n has at most [ ] roots in C.

Answer 12

A complex polynomial of degree n has at most n roots in C.

Question 13

What is the fundamental theorem of algebra?

Answer 13

Every complex polynomial
with degree n has exactly n roots (counted with multiplicity). That is, if p is a monic polynomial with complex coefficients and degree n, then p(x) =
(x − α1)· · ·(x − αn) for some α1, . . . , αn ∈ C (not necessarily distinct).