Modulus in Terms of Conjugate

Theorem

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

Let $\cmod z$ be the modulus of $z$.

Let $\overline z$ be the conjugate of $z$.

Then:

$\cmod z = \sqrt {z \overline z}$

Let $z = a + i b$.

Then:

$\ds z \overline z$

$=$

$\ds a^2 + b^2$

$\ds $

$=$

$\ds \cmod z^2$

Definition of Complex Modulus

$\ds \leadsto \ \ $

$\ds \cmod z$

$=$

$\ds \sqrt {z \overline z}$

taking square root of both sides

$\blacksquare$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: $(1.9)$
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms
1998: Yoav Peleg, Reuven Pnini and Elyahu Zaarur: Quantum Mechanics ... (previous) ... (next): Chapter $2$: Mathematical Background: $2.1$ The Complex Field $C$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 4$: Complex Numbers: Complex Plane: $4.6.$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conjugate (of a complex number)