The Absolute Value of a Complex Number

The Absolute Value of a Complex Number

Given any complex number $z$, we can compute its absolute value as described in the following definition:

Definition: Let $z = a + bi$ be a complex number. Then the Absolute Value of $z$ denoted $\mid z \mid = \sqrt{a^2 + b^2} = \sqrt{\Re (z) ^2 + \Im (z)^2}$.

For example, consider the complex number $z = 2 + 3i$. Then $\mid z \mid = \sqrt{2^2 + 3^3} = \sqrt{13}$. Geometrically, the absolute value of a complex number represents the length of vector representing that complex number on a 2-dimensional grid with a real axis and imaginary axis:

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Let's now look at some properties regarding the absolute value of a complex number.

Theorem 1: Let $z, z' \in \mathbb{C}$ be complex numbers $z = a + bi$ and $z' = a' + b'i$. Then:
a) $z \bar{z} = \mid z \mid$ where $\bar{z} = a - bi$ is the Complex Conjugate of $z$.
b) $\mid zz' \mid = \mid z \mid \mid z' \mid$.
  • Proof a) $z\bar{z} = (a + bi)(a -bi) = a^2 - abi + abi + b^2 = a^2 + b^2 = \mid z \mid$.
  • Proof b) We will not prove part b) as it is rather lengthy, but the reader is advised to construct a proof. $\blacksquare$
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