Intro to Complex Numbers Flashcards

1
Q

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

A

z bar = a - bi

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2
Q

What is the complex conjugate of zw?

A

zw bar = z bar w bar

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3
Q

z zbar =

A

|z|²

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4
Q

What is the triangle inequality?

A

|z+w| ≤ |z| + |w|

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5
Q

arg(zw) =

A

argz + arg w

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6
Q

What is De Moivre’s Theorem?

A

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)

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7
Q

Prove De Moivre’s Theorem

A
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.
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8
Q

Define root of unity

What is the arg of an nth root of unity?

A

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

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9
Q

What is a primitive root of unity?

A

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

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10
Q

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 = [ ]

A

zⁿ⁻¹ + zⁿ⁻² + · · · + z + 1 = 0

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11
Q

Euler’s formula

A

e^(iθ) = cosθ + isinθ

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12
Q

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

A

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

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13
Q

What is the fundamental theorem of algebra?

A

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).

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