Product of Complex Number with Conjugate
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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.
Proof
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.
$\blacksquare$
Sources
- 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)