Complex number arithmetic and the Argand plane in HSC Mathematics Extension 2: Cartesian and polar form, modulus, argument, conjugate and geometric interpretation

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. This is the foundation dot point of the whole complex-numbers course: modulus-argument form, de Moivre's theorem, loci and complex polynomials are all written in this language, so the fluency built here is repaid in every later topic.

Why complex numbers exist

Real numbers run out when you try to solve $x^2 = -1$, because no real number squares to a negative. Defining a single new number $i$ with $i^2 = -1$ fixes this, and remarkably it fixes everything at once: once $i$ is allowed, every polynomial equation has a solution (the fundamental theorem of algebra). A complex number combines a real part and an imaginary part into $z = x + iy$, where $x = \operatorname{Re}(z)$ and $y = \operatorname{Im}(z)$ are both real. Note that $\operatorname{Im}(z) = y$ is the real coefficient of $i$, not $iy$ itself: the imaginary part of $2 + 3i$ is $3$, not $3i$.

The real numbers sit inside the complex numbers as the special case $y = 0$, and the purely imaginary numbers are the case $x = 0$. This is why the two axes of the Argand plane are the real axis and the imaginary axis: a complex number is genuinely two-dimensional, carrying twice the information of a real number.

Cartesian arithmetic

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

Addition and subtraction work componentwise, exactly like adding vectors: combine the real parts, then combine the imaginary parts. Multiplication uses ordinary expansion together with $i^2 = -1$:

You never have to memorise this formula; just expand the brackets and replace $i^2$ with $-1$. 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 $w \neq 0$, so the quotient lands cleanly in the form $x + iy$.

The Argand plane

The Argand plane represents $z = x + iy$ as the point $(x, y)$ or, equivalently, the position vector (arrow) from the origin to that point. The horizontal axis is the real axis and the vertical axis the imaginary axis. Every fact about a complex number has a picture here: the modulus is a length, the argument is an angle, the conjugate is a reflection, addition is a parallelogram, and multiplication is a rotation combined with a scaling. Sketching the point before doing algebra is the single habit that prevents the most common Extension 2 mistakes, especially picking the wrong quadrant for an argument.

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, each of which has a clean geometric reading:

The first turns "modulus squared" into a product you can compute by multiplying out; the second and third are the algebra behind the rotate-and-scale picture of multiplication below.

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 $z - \bar z = 2i\operatorname{Im}(z)$. The single most useful fact is

a real number. This is exactly why multiplying by the conjugate realises a denominator: it converts a complex divisor into its squared modulus.

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, and it is developed fully in the modulus-argument-form and de Moivre dot points.

Addition is the parallelogram law

Because real and imaginary parts add separately, complex addition is vector addition: $z + w$ is found by the head-to-tail (triangle) rule or, equivalently, by completing the parallelogram on $z$ and $w$. The sum is the diagonal of that parallelogram from the origin. The other diagonal, drawn from the head of $w$ to the head of $z$, represents the difference $z - w$. Recognising this turns many "find the fourth vertex" or "show this is a rhombus" geometry questions into one-line vector statements.

The figure below builds $z + w$ for $z = 3 + i$ and $w = 1 + 2i$ one move at a time, finishing with the full parallelogram.

Stage 1, plot the two numbers as vectors. Draw $z = 3 + i$ and $w = 1 + 2i$ as arrows from the origin $O$. Nothing has been combined yet; these are the two numbers we want to add.

Stage 2, slide w to the head of z. A complex number is a free vector, so move a copy of $w$ until its tail sits at the head of $z$. Walking along $z$ and then along $w$ lands you at the sum.

Stage 3, the sum is the diagonal. Completing the parallelogram, the sum $z + w$ is the diagonal from $O$ to the far corner. Reading off coordinates, $z + w = (3 + 1) + (1 + 2)i = 4 + 3i$, which matches adding the components directly.

Multiplication is rotate and scale

Multiplication has a strikingly different picture from addition. Since $|zw| = |z||w|$ and $\arg(zw) = \arg z + \arg w$, multiplying $z$ by $w$ scales the length of $z$ by the factor $|w|$ and rotates it anticlockwise by the angle $\arg w$. Multiplication does not slide points around as addition does; it spins and stretches them about the origin.

Below, $z = 2 + i$ is multiplied by $w = 1 + i$. Since $|w| = \sqrt2$ and $\arg w = \tfrac{\pi}{4}$ ($45^\circ$), the result $zw$ is $z$ rotated $45^\circ$ anticlockwise and stretched by $\sqrt2$. Expanding confirms it: $(2 + i)(1 + i) = 2 + 2i + i + i^2 = 1 + 3i$.

Geometry of conjugation, negation and multiplying by i

Three transformations of $z = x + iy$ appear constantly, and each is a rigid rotation or reflection about the origin:

• Conjugation $z \mapsto \bar z$ reflects the point in the real axis (it negates the imaginary part, and so negates the argument).
• Negation $z \mapsto -z$ rotates it by $180^\circ$ about the origin.
• Multiplying by $i$ rotates the point $90^\circ$ anticlockwise without changing its distance from the origin, because $|i| = 1$ and $\arg i = \tfrac{\pi}{2}$.

Reading these geometrically is often faster than algebra: $iz$ is always perpendicular to $z$, which is why $z$ and $iz$ form two adjacent sides of a square in many locus problems.

Powers of $i$ and the imaginary unit

The imaginary unit $i$ satisfies $i^2 = -1$, and its powers cycle with period four: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, then the pattern repeats. To simplify a high power such as $i^{23}$, divide the exponent by $4$ and keep the remainder: $23 = 4(5) + 3$, so $i^{23} = i^3 = -i$. This cyclic behaviour is just the rotate-by-$90^\circ$ picture above repeated: $i = \cos\tfrac{\pi}{2} + i\sin\tfrac{\pi}{2}$, so each power turns the point a further quarter turn anticlockwise about the origin, returning to the start after four steps.

Equality and solving for real unknowns

Two complex numbers are equal exactly when their real parts are equal and their imaginary parts are equal. This lets you turn one complex equation into two real simultaneous equations. For example, if $(a + bi)(2 - i) = 5 + 5i$ with $a, b$ real, expand the left side to $(2a + b) + (2b - a)i$, then equate parts: $2a + b = 5$ and $2b - a = 5$. Solving gives $a = 1$, $b = 3$. Splitting into real and imaginary parts is the standard route for finding square roots of a complex number and for any problem that supplies a Cartesian target.

How exam questions ask about complex arithmetic

The wording varies, but each phrasing maps to one routine:

• "Express in the form $x + iy$" with a fraction: realise the denominator (multiply by the conjugate of the denominator), then simplify.
• "Simplify $z^2$" or a product: expand the brackets and replace every $i^2$ with $-1$.
• "Find $|z|$" or "the modulus": compute $\sqrt{x^2 + y^2}$; for a product or quotient, use $|zw| = |z||w|$ or $|z/w| = |z|/|w|$ as a shortcut.
• "Find $\arg z$": sketch the point, find the reference angle from $\tan^{-1}|y/x|$, then fix the quadrant from the signs of $x$ and $y$; give the principal value in $(-\pi, \pi]$.
• "Find the conjugate" or a symbol $\bar z$: change the sign of the imaginary part.
• "On an Argand diagram, plot / show that ...": read addition as the parallelogram and multiplication by a unit number as a rotation; "multiply by $i$" means rotate $90^\circ$.
• "Find real $a$ and $b$ such that ...": equate real and imaginary parts to get two simultaneous equations.