Modulus and Argument

A complex number \(z = a + ib\) can be represented as a point \((a, b)\) in a plane called the Argand plane.

Here:

• The horizontal axis is the real axis.
• The vertical axis is the imaginary axis.

So the number \(3 + 4i\) is the point \((3, 4)\), and \(-2 - i\) is the point \((-2, -1)\).

This geometric view helps in understanding modulus and argument easily.

The modulus of a complex number \(z = a + ib\) is the distance of the point \((a, b)\) from the origin.

Definition:

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

It tells how far the number lies from the origin in the Argand plane.

Examples:

• For \(3 + 4i\), \(|z| = \sqrt{3^2 + 4^2} = 5\)
• For \(-5 + 12i\), \(|z| = \sqrt{25 + 144} = 13\)

The argument of a complex number \(z = a + ib\) is the angle \(\theta\) made by the line joining \((a, b)\) to the origin, measured from the positive real axis.

It is written as \(\arg(z)\).

The principal argument, denoted \(\text{Arg}(z)\), is the unique value of the argument in the interval:

\(-\pi < \text{Arg}(z) \le \pi\)

Example:

The point \((1, 1)\) has argument \(\theta = \frac{\pi}{4}\), so:

\(\arg(z) = \text{Arg}(z) = \frac{\pi}{4}\).

The value of \(\theta = \arg(z)\) depends on the signs of \(a\) and \(b\). Here are the rules:

Examples:

• For \(z = -3 + 3i\) (Quadrant II), \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\).
• For \(z = 2 - 2i\) (Quadrant IV), \(\theta = -\frac{\pi}{4}\).

A complex number \(z = a + ib\) has useful relationships with its modulus and conjugate.

• \(z \cdot \overline{z} = |z|^2\)
• \(\overline{z} = a - ib\)
• \(\left| \overline{z} \right| = |z|\)

Example: For \(z = 3 + 4i\):

• \(\overline{z} = 3 - 4i\)
• \(z \cdot \overline{z} = 9 + 16 = 25 = |z|^2\)

Some geometric places (loci) of complex numbers can be described using modulus or argument conditions.