z = a + bi, where a = Re(z), b = Im(z), i^2 = -1

a + bi = c + di iff a = c and b = d

(a+bi) + (c+di) = (a+c) + (b+d)i

Geometric meaning: vector addition (head-to-tail) on the complex plane

(a+bi) - (c+di) = (a-c) + (b-d)i

Geometric meaning: vector difference = add the negative vector

(a+bi)(c+di) = (ac-bd) + (ad+bc)i

Polar insight: multiply moduli, add arguments

Geometric meaning: scaling by r2 and rotation by θ2

(a+bi)/(c+di) = ((a+bi)(c-di))/((c+di)(c-di)) = ((ac+bd) + (bc-ad)i)/(c^2+d^2), with c+di ≠ 0

Polar insight: divide moduli, subtract arguments

If z = a + bi, then z̄ = a - bi

z z̄ = a^2 + b^2 = |z|^2

overline(z1 ± z2) = z̄1 ± z̄2

overline(z1 z2) = z̄1 z̄2

overline(z1/z2) = z̄1/z̄2, z2 ≠ 0

Reflection across the real axis on the complex plane

Point (a, b) maps to (a, -b)

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

Distance from the origin to the point (a, b)

Right triangle: legs a and b, hypotenuse |z|

|z| ≥ 0, and |z| = 0 iff z = 0

|z1 z2| = |z1||z2|

|z1/z2| = |z1|/|z2|, z2 ≠ 0

Triangle inequality: |z1 + z2| ≤ |z1| + |z2|

z = a + bi corresponds to point (a, b) on the Argand plane

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

θ = arg(z) (angle from the positive real axis)

z = r(cosθ + i sinθ)

Conversions

z = r e^{iθ} (Euler’s formula link)

Axes: real (horizontal) and imaginary (vertical)

Vector from origin to (a, b)

Angle θ at the origin and radius r

Multiply: r multiplies, θ adds

Divide: r divides, θ subtracts