AcademyComplex Arithmetic
Academy
Modulus
Level 1 - Math I (Physics) topic page in Complex Arithmetic.
Modulus (Absolute Value)
The modulus of a complex number represents its distance from the origin in the complex plane.
Definition
For \(z = a + bi\):
Definition
\[|z| = \sqrt{a^2 + b^2}\]
The modulus is always a non-negative real number.
Geometric Interpretation
The modulus is the length of the vector representing \(z\) in the complex plane. By the Pythagorean theorem:
Pythagorean
\[|z|^2 = a^2 + b^2\]
Properties
Positive
\[|z| \geq 0 \quad \text{and} \quad |z| = 0 \iff z = 0\]
Product
\[|z_1 z_2| = |z_1| \cdot |z_2|\]
Quotient
\[\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} \quad (z_2 \neq 0)\]
Connection to Conjugate
Conjugate
\[|z|^2 = z \cdot z^*\]
This provides an alternative formula: \(|z| = \sqrt{z \cdot z^*}\)
In Polar Form
If \(z = re^{i\theta}\), then:
PolarMod
\[|z| = r\]
The modulus in polar form is simply \(r\), the distance from origin.
Example
Example
\[|3 + 4i| = \sqrt{9 + 16} = \sqrt{25} = 5\]