Complex numbers

3.9 Complex numbers

What a complex number is

$\mathrm{i}$ is defined by $\mathrm{i}^2 = -1$. A complex number has the form $z = x + \mathrm{i}y$, where $x$ and $y$ are real.

Arithmetic in Cartesian form

Add, subtract and multiply as with brackets, replacing $\mathrm{i}^2$ with $-1$. To divide, multiply the top and bottom by the conjugate of the bottom.

$\frac{(1 + 2\mathrm{i})(3 + \mathrm{i})}{(3 - \mathrm{i})(3 + \mathrm{i})} = \frac{3 + \mathrm{i} + 6\mathrm{i} - 2}{9 + 1} = \frac{1 + 7\mathrm{i}}{10} = 0.1 + 0.7\mathrm{i}$

Conjugate roots

The pair gives a real quadratic factor: $z^2 - (\text{sum})z + (\text{product})$.

$1 - 2\mathrm{i}$ is also a root. Their sum is 2 and product is 5, so $z^2 - 2z + 5$ is a factor. Dividing gives

$(z^2 - 2z + 5)(z + 1) = 0$

The roots are $1 + 2\mathrm{i}$, $1 - 2\mathrm{i}$ and $-1$.

The Argand diagram

A complex number $x + \mathrm{i}y$ is drawn at the point $(x, y)$. The modulus is the distance from the origin, and the argument is the angle measured from the positive real axis, with $-\pi < \theta \leq \pi$.

Polar and exponential forms

$z = r(\cos\theta + \mathrm{i}\sin\theta) = r\mathrm{e}^{\mathrm{i}\theta}$

where $r = \lvert z\rvert$ and $\theta = \arg z$. These forms make multiplication and division simple:

$zw = 8\mathrm{e}^{\mathrm{i}\frac{\pi}{2}} = 8\mathrm{i} \qquad \frac{z}{w} = \frac{1}{2}\mathrm{e}^{\mathrm{i}\frac{\pi}{6}}$

Keep the argument of the answer between $-\pi$ and $\pi$, adding or subtracting $2\pi$ if needed.

Square roots

Write the root as $x + \mathrm{i}y$, square it and compare parts.

$(x + \mathrm{i}y)^2 = x^2 - y^2 + 2\mathrm{i}xy$, so $x^2 - y^2 = 3$ and $2xy = 4$.

Put $y = \dfrac{2}{x}$: $x^2 - \dfrac{4}{x^2} = 3$, so $x^4 - 3x^2 - 4 = 0$ and $x^2 = 4$ (the value $x^2 = -1$ is rejected, since $x$ is real).

So $x = \pm 2$, and the roots are $2 + \mathrm{i}$ and $-2 - \mathrm{i}$.

Geometric effects

Loci

An inequality gives a region:

• $\lvert z - a\rvert \leq r$ is the inside of the circle.
• $\lvert z - a\rvert < \lvert z - b\rvert$ is the side of the bisector nearer $a$.
• $\alpha \leq \arg(z - a) \leq \beta$ is the wedge between two half-lines.

The centre is 5 from the origin, and the radius is 2. So the least value is $5 - 2 = 3$ and the greatest is $5 + 2 = 7$.