Perform addition, subtraction, multiplication and division of complex numbers in Cartesian form using the conjugate

What this dot point is asking

SCSA wants fluent calculator-free arithmetic in Cartesian form. You must identify real and imaginary parts, carry out all four operations, use the conjugate, and simplify a quotient to the form $a + ib$.

Real and imaginary parts

Every complex number is written $z = x + iy$ where $x$ and $y$ are real. We call $x = \operatorname{Re}(z)$ the real part and $y = \operatorname{Im}(z)$ the imaginary part. Note that $\operatorname{Im}(z)$ is the real number $y$, not $iy$. Two complex numbers are equal exactly when their real parts match and their imaginary parts match, which is the basis of equating coefficients.

Addition and subtraction

These act on the parts separately:

Geometrically this is vector addition of the points $(a, b)$ and $(c, d)$ in the Argand plane.

Multiplication

Expand using the distributive law and replace $i^2$ by $-1$:

A useful special case is squaring: $(a + ib)^2 = (a^2 - b^2) + i(2ab)$. Powers of $i$ cycle with period four: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, so to simplify $i^n$ you reduce $n$ modulo $4$.

The conjugate

The conjugate has the key property that

a non-negative real number equal to $|z|^2$. Conjugation distributes over the operations: $\overline{z + w} = \bar z + \bar w$ and $\overline{zw} = \bar z\, \bar w$. Also $z + \bar z = 2\operatorname{Re}(z)$ and $z - \bar z = 2i\operatorname{Im}(z)$.

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:

Then split into real and imaginary parts to land in the form $a + ib$.