The complex plane, modulus and argument

The Argand diagram

• Complex numbers can be plotted as points on a special graph called the Argand diagram (or complex plane).
• The horizontal axis is the real part; the vertical axis is the imaginary part.
• So $z = a + bi$ is plotted at the point $(a, b)$.

The modulus

• The modulus $|z|$ is the distance from the origin to the point — how "big" the complex number is.
• By Pythagoras, for $z = a + bi$:

$|z| = \sqrt{a^2 + b^2}$

• The modulus is always a real number that is zero or positive.

The argument

• The argument $\arg(z)$ is the angle the line from the origin to $z$ makes with the positive real axis, measured anticlockwise.
• Find it using $\tan\theta = \dfrac{b}{a}$, but check which quadrant the point is in to get the correct angle.
• The argument is usually given in the range $-\pi < \theta \le \pi$ (the principal argument).