Complex Numbers — Formula Sheet

i = sqrt(-1)

i^2 = -1

i is a solution of x^2 + 1 = 0

i^0 = 1

i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1

i^n = i^r, where r is the remainder on dividing n by 4

i^n = 1 / i^(-n)

z = a + ib

Im z = b

C = {a + ib : a, b in R}

R subset of C

z bar = a - ib

|z| = sqrt(a^2 + b^2)

z times z bar = |z|^2

z^(-1) = a/(a^2 + b^2) - [b/(a^2 + b^2)] i

1/z = a/(a^2 + b^2) - [b/(a^2 + b^2)] i

i = sqrt(-1) and i^2 = -1.

i^0 = 1.

i = √-1 and i^2 = -1

It is a solution of x^2 + 1 = 0.

Two complex numbers a1 + ib1 and a2 + ib2 are equal exactly when a1 = a2 and b1 = b2.

If Im z = 0, the number is purely real.