Complex Numbers Flashcards

1
Q

What is the value of i?

A

i = √(-1)

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2
Q

What is z (usually)?

A

A complex number

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3
Q

What form does complex number take?

A

z = a + ib
a,b ∈ R

a = Re(z)
b = Im(z)
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4
Q

What is the complex conjugate (z ̅) of z=a+ib?

A

z ̅ = a-ib

Re(z ̅) = Re(z)
Im(z ̅) = -Im(z)

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5
Q

What is the modulus |z| of z = a + ib?

A

|z| = √((a^2)+(b^2))

Think of it like the Pythagorean theorem with |z| being the hypotenuse and a,b being the other sides.

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6
Q

Scalar Multiplication:

For z = a+ib, what does 4z equal?

A

4z = 4a+4ib

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7
Q

For z = a + bi, what is the result of z * |z|?

A

z * |z| = a^2 + b^2

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8
Q

Complex Multiplication:

What is the result of z = u * v where u and v are complex numbers?

A
Re(z) = (Re(u) * Re(v)) - (Im(u) * Im(v))
Im(z) = (Re(u) * Im(v)) + (Im(u) * Re(v))

Try grid method multiplication to see how this works subbing in the value of (i^2) = -1

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9
Q

Complex Division:

What is the value of z = u/v where u and v are both complex numbers and v != 0?

A

z = u * (v ̅ / |v|^2)

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10
Q

How is a complex number represented as a matrix?

A

Matrix z = | a -b |
| b a |

I dunno how well that’ll show up when using the flashcards so maybe look at an image of the notes too

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11
Q

With a complex number in matrix form, how do you get the complex conjugate?

A

Perform a transposition of the matrix

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12
Q

With a complex number in matrix form, how do you get the modulus of the complex number?

A

Find the determinant of the matrix

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13
Q

How is a complex number represented as a 2-vector on an argand diagram?

A

z = a+bi

z = (a,b) (but vertical since its a vector)

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14
Q

With a complex number in vector form, how do you find the complex conjugate?

A

The reflection in the x-axis (the Re axis)

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15
Q

With a complex number in vector form, how do you find the modulus of the complex number?

A

The magnitude (size) of the vector

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16
Q

What flaws does vector form for complex numbers have?

A

Cannot do multiplication and division.

17
Q

How is a complex number represented in Polar Form?

A

z = (r,θ)

where:
r = |z|
θ = arg z = cos^-1(Re(z)/|z|) = sin^-1(Im(z)/|z|)
The angle is in radians and can be found from trigonometric calculations when viewing the complex number as a vector on an argand diagram

18
Q

How is a complex number represented in Euler form?

A

z = r * (e^iθ) = r(cosθ + isinθ)

where:
r = |z|
θ = arg z = cos^-1(Re(z)/|z|) = sin^-1(Im(z)/|z|)
The angle is in radians and can be found from trigonometric calculations when viewing the complex number as a vector on an argand diagram

19
Q

What relation does the Euler form give us?

A

(cosθ + isinθ)^α = cos(αθ) + isin(αθ)

20
Q

What downside does Euler’s form have which requires us to use principal values for arg z?

A

Because cos and sin are cyclical in nature, Each complex number has an infinite number of representations as arg z can take infinitely many values to represent the same thing.
To combat this we use the principal value where
0 <= arg z <= 2π

21
Q

Complex Powers:

What is the result of u^v where u∈C and v∈Q+ (v is positive rational)?

A

Representing u as u = |u|(e^(iarg u))) and v as v = 1/k:
u^(1/k) = (|u|^(1/k)) * e^((i*arg u)/k)

See slides/notes for better view of this as text like this doesn’t show superscripts.

22
Q

Can complex numbers be ordered?

A

No

23
Q

For complex number v = a + ib, what is the value of w = 1/v?

A

w = (v ̅ / |v|^2)