Perform complex arithmetic in Cartesian form and use conjugate properties to divide and simplify.

What this dot point is asking

This dot point is the foundation that every later complex number skill stands on. Before you can use polar form, De Moivre or factorisation, you must be able to do arithmetic in Cartesian form without slips. The single idea that drives all of it is that $i^2 = -1$, so any power of $i$ collapses to one of $1, i, -1, -i$.

Powers of the imaginary unit

The powers of $i$ cycle with period four: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, then repeat. To simplify a high power, divide the exponent by four and use the remainder. For example $i^{23} = i^{20}\cdot i^3 = (i^4)^5 \cdot i^3 = 1 \cdot (-i) = -i$.

Addition, subtraction and multiplication

For $z = a + ib$ and $w = c + id$:

Multiplication uses ordinary expansion, replacing $i^2$ with $-1$:

You do not need to memorise that final formula. It is faster and safer to expand the brackets each time and tidy up.

The conjugate

Conjugates obey clean rules that examiners love to test:

Two further identities are worth knowing: $z + \bar z = 2\operatorname{Re}(z)$ and $z - \bar z = 2i\operatorname{Im}(z)$. These let you extract the real or imaginary part without separating $z$ into pieces by hand.

Division by realising the denominator

To divide, multiply numerator and denominator by the conjugate of the denominator. This turns the denominator into a real number, after which you split into real and imaginary parts.

Equating real and imaginary parts

Two complex numbers are equal exactly when their real parts match and their imaginary parts match. This turns one complex equation into two real equations, a technique used constantly when solving for unknowns.

Why this matters

Fluent Cartesian arithmetic is what lets you switch to polar form with confidence and check polar answers by converting back. Practise division until realising the denominator is automatic, because it appears inside almost every harder complex number question.