Complex numbers - TCE Mathematics Specialised (Tasmania)

What this dot point is asking

Complex numbers extend the real numbers by introducing the imaginary unit $i$, defined by $i^2 = -1$. Once we allow $i$, every polynomial equation has a solution, which is the content of the Fundamental Theorem of Algebra. This dot point is about being completely comfortable with the two standard representations and knowing when to use each.

Cartesian form

In Cartesian (rectangular) form we write $z = x + iy$, where $x = \operatorname{Re}(z)$ is the real part and $y = \operatorname{Im}(z)$ is the imaginary part. Addition and subtraction are done component by component:

Multiplication uses the distributive law together with $i^2 = -1$:

The conjugate of $z = x+iy$ is $\bar z = x - iy$. It is the key tool for division, because $z\bar z = x^2 + y^2$ is always a non-negative real number. To divide, multiply top and bottom by the conjugate of the denominator:

Modulus and argument

The modulus is $|z| = r = \sqrt{x^2 + y^2}$, the distance from the origin to $z$ on the Argand diagram. The argument $\arg(z) = \theta$ is the angle measured anticlockwise from the positive real axis. The principal argument is taken in $(-\pi, \pi]$. Be careful: $\theta = \arctan(y/x)$ only directly when $z$ is in the first or fourth quadrant; otherwise adjust by $\pm\pi$ using the signs of $x$ and $y$.

Polar and exponential form

In polar form $z = r(\cos\theta + i\sin\theta)$, often abbreviated $r\operatorname{cis}\theta$, and in exponential form $z = re^{i\theta}$ via Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$.

De Moivre's theorem

For any integer $n$,

This makes powers trivial in polar form and is the route to finding roots. The $n$ solutions of $z^n = w$ where $w = R\operatorname{cis}\phi$ are

These $n$ roots lie equally spaced on a circle of radius $R^{1/n}$.

Choosing the right form

Use Cartesian form for addition and subtraction, and for stating exact real and imaginary parts. Use polar or exponential form for multiplication, division, powers and roots. The exponential form $z = re^{i\theta}$ is the same information as polar form, and it makes the multiplication and division rules obvious because exponents add and subtract: $r_1 e^{i\theta_1}\cdot r_2 e^{i\theta_2} = r_1 r_2 e^{i(\theta_1 + \theta_2)}$. Switching at the right moment is what separates fast, clean solutions from long algebraic struggles, so practise converting both ways until it is automatic.

The fundamental theorem in practice

Allowing $i$ guarantees solutions to equations that have none over the reals. For instance $z^2 + 2z + 5 = 0$ has discriminant $4 - 20 = -16 < 0$, so over the reals it has no solution. Using $\sqrt{-16} = 4i$, the quadratic formula gives $z = \dfrac{-2 \pm 4i}{2} = -1 \pm 2i$, a conjugate pair. This is the bridge to factorising polynomials over the complex field, where every quadratic factor with a negative discriminant splits into two complex linear factors.