Complex numbers

Recall the properties of imaginary numbers

Rectangular form

z = a + bi

a is called "real part" -> Re(z)
b is called "imaginary part" -> Im(z)

if a == 0 you have a pure imaginary number

z = bi

if b == 0 you have a real number

z = a

Operations

Add and subtract complex numbers by separating the imaginary parts

(-2 + 17i) + (6-8i)
(-2 + 6) + (17i - 8i)
(-2 + 6) + (17 - 8)i
4 + 9i

(21 + 16i) - (-9 + 4i)
(21 + 9) + (16i - 4i)
30 + 12i

Multiply and divide complex numbers like polynomials FOIL, and multiply by the conjugate if you need to.

Division: multiply the numerator and denominator by the conjugate.

When doing these operations, always remember i^2 is -1.

Find the absolute value with sqt(Re(z)^2 + Im(z)^2). This is just using the Pythagorean theory to find the distance from the origin.

A complex number can be written in exponential form using Euler's number e ≅ 2.718

z = re^i𝛳 // how to find r and 𝛳 down below

Complex Conjugation

Written as a bar over the complex number, you just flip the sign in front of the imaginary part.

______
3 + 2i -> 3 - 2i

So it's just reflection on the complex plane

Graphing

Complex numbers are graphed on the complex plane (also called the argand plane), which looks like a rectangular plane but the y axis is the imaginary value and the x axis is the real value.

You can find the distance between two complex numbers by finding the distance between the real parts and imaginary parts independently and then using the Pythagorean theory.

Same with midpoints: find the midpoint of the real part and then the midpoint of the imaginary part.

To convert a complex number into polar form, r is the absolute value and use trig functions to find 𝛳.

r = |z| = sqt(a^2 + b^2)

sin = opp/hyp
cos = adj/hyp
tan = opp/adj

tan𝛳 = a/b
arctan(tan𝛳) = arctan(b/a)
𝛳 = arctan(b/a)

also:
cos𝛳 = a/r, so a = rcos𝛳
sin𝛳 = b/r, so b = rsin𝛳

So the polar form is:
z = a + bi
z = rcos𝛳 + (rsin𝛳)i
z = r(cos𝛳 + isin𝛳)

Polar operations

Multiplication

To multiply two complex numbers in polar form, you pull out both r values, foil the parentheses, and then group together the real and imaginary parts - using the sum identities for sin and cosine to simplify.

When you find the product, it's distance from the origin is the product of the original distances: r = r * r, and its angle is the sum of the original angles 𝛳 = 𝛳 + 𝛳

A formula for this operation would look like this: (using 1 and 2 as labels here)

z1 * z2 = r1r2[cos(𝛳1 + 𝛳2) + isin(𝛳1 + 𝛳2)]

Division

Similarly, with division you have two options:

z1/z2 = r1/r2[cos(𝛳1 - 𝛳2) + isin(𝛳1 - 𝛳2)]

or

z1/z2 = (r1/r2)(e^(i(𝛳1 - 𝛳2)))

Exponents

Using De Moivre's theorem we find z^n = r^n[cos(n𝛳) + isin(n𝛳)]

If you want to find the power of a rectangular complex number, just convert it to polar form first.

You can reverse the direction and find the roots of a complex number with:

nsqt(z) = nsqt(r)[cos((𝛳+2pik)/n) + isin((𝛳 + 2pik)/n)]

or replace (2pi) with 360 for degrees

Gauss Theorem

If you have a polynomial equation to term n, you have n roots and some can be complex.

x^3 = 1

has 3 roots, one is x = 1 the other two are complex

x^n = 1

has n roots

Complex roots of unity

x is a n'th root of unity if x^n = 1

It's a set of n vectors of length one separated by angles of 2pi/n