Product of Complex Number with Conjugate

Theorem

Let $z = a + i b \in \C$ be a complex number.

Let $\overline z$ denote the complex conjugate of $z$.

Then:

$z \overline z = a^2 + b^2 = \cmod z^2$

and thus is wholly real.

By the definition of a complex number, let $z = a + i b$ where $a$ and $b$ are real numbers.

Then:

$\ds z \overline z$

$=$

$\ds \paren {a + i b} \paren {a - i b}$

Definition of Complex Conjugate

$\ds $

$=$

$\ds a^2 + a \cdot i b + a \cdot \paren {-i b} + i \cdot \paren {-i} \cdot b^2$

Definition of Complex Multiplication

$\ds $

$=$

$\ds a^2 + i a b - i a b + b^2$

$\ds $

$=$

$\ds a^2 + b^2$

$\ds $

$=$

$\ds \paren {\sqrt {a^2 + b^2} }^2$

$\ds $

$=$

$\ds \cmod z^2$

Definition of Complex Modulus

As $a^2 + b^2$ is wholly real, the result follows.

$\blacksquare$

1957: E.G. Phillips: Functions of a Complex Variable (8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 2$. Conjugate Complex Numbers
1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.3$ Complex conjugation: $(3)$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conjugate (of a complex number)
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): conjugate (of a complex number)