Represent complex numbers in Cartesian and polar form, perform arithmetic, and interpret modulus, argument and conjugate geometrically on the Argand plane

What this dot point is asking

NESA wants you to work fluently with complex numbers in both Cartesian form $z = x + iy$ and polar (modulus-argument) form $z = r(\cos\theta + i\sin\theta)$. You must add, subtract, multiply and divide them, find the modulus, argument and conjugate, and read all of these geometrically on the Argand plane.

Cartesian arithmetic

For $z = a + bi$ and $w = c + di$:

Multiplication uses $i^2 = -1$:

Division is performed by multiplying numerator and denominator by the conjugate of the denominator, which "realises" the denominator:

The denominator $c^2 + d^2 = |w|^2$ is real and positive whenever $w \neq 0$.

The Argand plane

The Argand plane represents $z = x + iy$ as the point $(x, y)$ or, equivalently, the position vector from the origin to that point. The horizontal axis is the real axis and the vertical axis the imaginary axis. Addition of complex numbers obeys the parallelogram law, exactly like vector addition.

Modulus and argument

The modulus is the distance from the origin:

The argument $\arg z = \theta$ is the angle measured anticlockwise from the positive real axis to the vector representing $z$. It satisfies $\tan\theta = \dfrac{y}{x}$, but you must choose the quadrant using the signs of $x$ and $y$. The principal argument is taken in $(-\pi, \pi]$.

Key identities:

The conjugate

The conjugate $\bar z = x - iy$ is the reflection of $z$ in the real axis. Useful properties include $\overline{z + w} = \bar z + \bar w$, $\overline{zw} = \bar z\,\bar w$, and $z + \bar z = 2\operatorname{Re}(z)$, while $z - \bar z = 2i\operatorname{Im}(z)$.

Polar (modulus-argument) form

Writing $x = r\cos\theta$ and $y = r\sin\theta$ where $r = |z|$ gives

In polar form, multiplication multiplies the moduli and adds the arguments, while division divides the moduli and subtracts the arguments. This makes polar form the natural setting for powers and roots.