How Big and Which Way?
Just like polar coordinates, every complex number can be described by a distance and an angle.
- Modulus (\(|z|\)): The distance from zero. \(|z| = \sqrt{a^2 + b^2}\).
- Argument (\(\arg z\)): The angle \(\theta\) from the positive real axis. \(\tan\theta = b/a\).
Worked Examples
Example 1: Finding the Modulus
Find the modulus of \(z = 3 + 4i\).
- \(|z| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\).
Example 2: Finding the Argument
Find the argument of \(z = 1 + i\).
- \(\tan\theta = 1/1 = 1 \to \theta = 45^\circ\) (\(\pi/4\)).
The Bridge to Quantum Mechanics
In Quantum Mechanics, the Modulus is the most important number because its square (\(|z|^2\)) is the Probability. If the modulus of a wavefunction at point X is 0.7, then the probability of finding the particle there is \(0.7^2 = 0.49\) (or 49%). The Argument is the "Quantum Phase." While the modulus tells us if the particle is there, the argument tells us how it will Interfere with other particles. These two numbers together define everything you can possibly know about a quantum state.