Modulus-argument form in HSC Mathematics Extension 2: polar form, the principal argument, and the geometry of multiplication and division of complex numbers

What this dot point is asking

NESA wants you to express a complex number by its distance from the origin and the angle it makes with the positive real axis, the modulus-argument (polar) form. You must convert between Cartesian and polar form, identify the principal argument, and use polar form to multiply and divide complex numbers, understanding these as scaling and rotation in the Argand plane. This is the engine room of the whole module: once multiplication is "multiply the lengths, add the angles", de Moivre's theorem, roots of unity and rotation loci all become almost mechanical.

Modulus and argument

For $z = x + iy$, the modulus is the distance from the origin,

and the argument $\theta = \arg z$ is the angle measured anticlockwise from the positive real axis to the ray from the origin to $z$. The principal argument is the value in the interval $(-\pi, \pi]$. The Cartesian and polar coordinates are linked by

so that

This is often abbreviated $z = r\operatorname{cis}\theta$, where $\operatorname{cis}\theta = \cos\theta + i\sin\theta$. The figure shows the geometry behind these formulas: the modulus $r$ is the hypotenuse of a right triangle, and $x = r\cos\theta$ and $y = r\sin\theta$ are its legs.

Converting from Cartesian to polar form

To put $z = x + iy$ into polar form, compute $r = \sqrt{x^2 + y^2}$, then find $\theta$ in the correct quadrant. A dependable routine is: find the reference angle $\alpha = \tan^{-1}\left|\dfrac{y}{x}\right|$ (always a first-quadrant angle), then adjust for the quadrant of $z$:

• first quadrant ($x > 0, y > 0$): $\theta = \alpha$;
• second quadrant ($x < 0, y > 0$): $\theta = \pi - \alpha$;
• third quadrant ($x < 0, y < 0$): $\theta = -(\pi - \alpha) = \alpha - \pi$;
• fourth quadrant ($x > 0, y < 0$): $\theta = -\alpha$.

Purely real or imaginary numbers are special cases you should know by sight: $\arg(\text{positive real}) = 0$, $\arg(\text{negative real}) = \pi$, $\arg(\text{positive imaginary}) = \tfrac{\pi}{2}$, and $\arg(\text{negative imaginary}) = -\tfrac{\pi}{2}$.

Finding the argument correctly

The relation $\tan\theta = \dfrac{y}{x}$ does not by itself fix $\theta$, because the calculator inverse tangent only returns angles in $(-\tfrac{\pi}{2}, \tfrac{\pi}{2})$. You must use the signs of $x$ and $y$ to place $z$ in the correct quadrant. For example, $z = -1 + i$ lies in the second quadrant, so its argument is $\dfrac{3\pi}{4}$, not $-\dfrac{\pi}{4}$. Always sketch the point in the Argand plane before committing to an argument.

Multiplication: scaling and rotation

If $z_1 = r_1\operatorname{cis}\theta_1$ and $z_2 = r_2\operatorname{cis}\theta_2$, then expanding the product and using the angle-sum identities $\cos(\theta_1 + \theta_2) = \cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2$ and $\sin(\theta_1 + \theta_2) = \sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2$,

So multiplying by $z_2$ scales the modulus by $r_2$ and rotates the point anticlockwise by $\theta_2$. In particular, multiplying by $i = \operatorname{cis}\tfrac{\pi}{2}$ is a rotation by $90^\circ$ with no change in length, and multiplying by a real $k > 0$ is a pure scaling.

The figure below builds the product $z_1 z_2$ for $z_1 = 2\operatorname{cis}30^\circ$ and $z_2 = 1.5\operatorname{cis}45^\circ$, showing the lengths multiplying and the angles adding.

Stage 1, two numbers as vectors. Plot $z_1 = 2\operatorname{cis}30^\circ$ (length $2$, angle $30^\circ$) and $z_2 = 1.5\operatorname{cis}45^\circ$ (length $1.5$, angle $45^\circ$). Each is a vector from the origin with a known modulus and argument.

Stage 2, multiply the lengths and add the angles. The product has modulus $2 \times 1.5 = 3$ and argument $30^\circ + 45^\circ = 75^\circ$, so $z_1 z_2 = 3\operatorname{cis}75^\circ$. The result is a longer vector swung round to a steeper angle, never found by adding the two arrows.

Division: subtraction of arguments

Dividing reverses the process:

The modulus of the quotient is the ratio of moduli, and the argument is the difference of arguments. These rules make repeated multiplication and powers far easier than expanding in Cartesian form, and they lead directly into de Moivre's theorem. A useful special case is the reciprocal: $\dfrac{1}{z} = \dfrac{1}{r}\operatorname{cis}(-\theta)$, so taking a reciprocal inverts the length and reflects the angle.

The exponential (Euler) form

Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ lets the polar form be written compactly as $z = r e^{i\theta}$. In this form the multiplication and division rules become the ordinary index laws: $r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = r_1 r_2 e^{i(\theta_1 + \theta_2)}$ and $\frac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$. Because the exponential form obeys the index laws automatically, it is the most efficient notation for powers and roots, and it is the natural bridge to de Moivre's theorem $(r e^{i\theta})^n = r^n e^{in\theta}$.

Why the geometry matters

Reading multiplication as scaling-and-rotation is not just elegant; it is the fastest way to answer many Extension 2 questions. Multiplying a point by $\operatorname{cis}\alpha$ rotates it about the origin by $\alpha$ without changing its distance, so a problem asking you to rotate a complex number by $90^\circ$ is solved instantly by multiplying by $i$. Recognising that conjugation reflects $z$ in the real axis (negating the argument) and that $z \bar z = |z|^2$ rounds out the geometric toolkit you carry into curves, regions and locus problems.

How exam questions ask about modulus-argument form

• "Write ... in modulus-argument form" or "express in polar/exponential form": find $r$ and the principal argument $\theta$, then write $r\operatorname{cis}\theta$ or $r e^{i\theta}$.
• "Find $zw$" or "$z/w$" with numbers given in polar form: multiply or divide the moduli and add or subtract the arguments; reduce to the principal range only if asked.
• "Find the modulus and argument of $zw$" from Cartesian inputs: use $|zw| = |z||w|$ and $\arg(zw) = \arg z + \arg w$ instead of expanding the product.
• "Show that multiplying by ... rotates ...": identify the multiplier as $\operatorname{cis}\alpha$ and state the rotation angle $\alpha$.
• "Hence find $z^n$": this is de Moivre in disguise; raise the modulus to the power and multiply the argument by $n$.
• "Convert back to $x + iy$": evaluate $r\cos\theta$ and $r\sin\theta$ once the argument is reduced into a single revolution.