Modulus of a complex number

The modulus measures the “size” of a complex number: geometrically it is the distance of the point $(a;b)$ from the origin.

Property — Modulus

The modulus of $z=a+ib$ is $|z| = \sqrt{a^2 + b^2}.$ It holds that $|z|^2 = z\cdot\bar{z}$. For every $z$ we have $|z|\ge 0$, with equality if and only if $z=0$.

The relation $|z|^2 = z\cdot\bar z$ links the modulus to the conjugate: multiplying a number by its conjugate always gives a non-negative real number, equal to the square of the modulus. It is precisely this property that makes rationalisation possible in division.

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Topics: Complex numbers
Concepts: Conjugate · Modulus
Skills: Calculate