Argand diagrams — complex numbers as geometry

A complex number $z = x + iy$ has two parts, both ordinary real numbers:

• the real part $x$, written $\operatorname{Re} z$;
• the imaginary part $y$, written $\operatorname{Im} z$. (It's the number $y$, not $iy$.)

Two numbers make a point, so plot $z$ at $(x, y)$. The grid you plot on is an Argand diagram:

• the horizontal axis is the real axis (real part);
• the vertical axis is the imaginary axis (imaginary part).

So $3 + 4i$ sits 3 right and 4 up. $-2i$ is the same as $0 - 2i$, so it sits on the imaginary axis, 2 down. A plain real number like 5 sits on the real axis.

It often helps to picture $z$ as an arrow (a vector) from the origin $O$ to its point.

Two measurements describe where the arrow points:

• The modulus $\lvert z \rvert$ is the length of the arrow, the distance from $O$ to the point. By Pythagoras, $\lvert z \rvert = \sqrt{x^2 + y^2}$.
• The argument $\arg z$ is the angle the arrow makes with the positive real axis, measured anticlockwise (clockwise counts as negative). Cambridge uses the range $-\pi < \arg z \leqslant \pi$, in radians.

To find the argument, sketch the point first so you know which quadrant it is in. Then find the angle in a right-angled triangle using $\tan^{-1}\left\lvert \tfrac{y}{x} \right\rvert$, and adjust:

• top right (first quadrant): $\arg z = \alpha$, where $\alpha = \tan^{-1}\left\lvert \tfrac{y}{x} \right\rvert$;
• top left (second quadrant): $\arg z = \pi - \alpha$;
• bottom left (third quadrant): $\arg z = -(\pi - \alpha)$;
• bottom right (fourth quadrant): $\arg z = -\alpha$.

To add complex numbers, add the real parts and add the imaginary parts:

$(a + ib) + (c + id) = (a + c) + i(b + d).$

On the diagram that's exactly vector addition: walk along the arrow for $z_1$, then from where you land, walk along the arrow for $z_2$. You finish at $z_1 + z_2$.

Drawn both ways round ($z_1$ then $z_2$, or $z_2$ then $z_1$), the two routes make a parallelogram, and $z_1 + z_2$ is its far corner.

$z_1 - z_2$ is the arrow that goes from the point $z_2$ to the point $z_1$.

Why? Because $z_2 + (z_1 - z_2) = z_1$. In words: start at $z_2$, move by $z_1 - z_2$, and you arrive at $z_1$.

So its length, $\lvert z_1 - z_2 \rvert$, is the distance between the two points. This idea is the key to the loci lesson that comes later.