Complex Numbers in Cartesian Form

Definition

i² = −1. A complex number is z = a + bi, with real part a and imaginary part b.

Arithmetic

• Add and subtract real and imaginary parts separately.
• Multiply by expanding and using i² = −1: (2 + 3i)(1 − i) = 2 − 2i + 3i − 3i² = 5 + i.
• Divide by multiplying top and bottom by the conjugate z* = a − bi:

(3 + i)/(1 − i) = (3 + i)(1 + i)/2 = (2 + 4i)/2 = 1 + 2i.

Conjugate facts

• z z* = a² + b², a real number.
• z + z* = 2a.

Equating parts

If a + bi = c + di, then a = c and b = d. Useful for solving equations such as finding √(3 + 4i): (x + yi)² = 3 + 4i gives ±(2 + i).

Quadratics with real coefficients

If Δ < 0 the roots are complex conjugates: x² − 4x + 13 = 0 gives x = 2 ± 3i.

For polynomials with real coefficients, non-real roots come in conjugate pairs.

The Argand diagram

Plot a + bi as the point (a, b).

• Modulus abs(z) = √(a² + b²): distance from the origin.
• Argument arg z: angle from the positive real axis.

abs(3 + 4i) = 5.

Quick summary

• i² = −1; multiply out and simplify.
• Divide using the conjugate.
• Real polynomials have conjugate pairs of complex roots.