Complex Modulus of Product of Complex Numbers
Theorem
Let $z_1, z_2 \in \C$ be complex numbers.
Let $\cmod z$ be the modulus of $z$.
Then:
- $\cmod {z_1 z_2} = \cmod {z_1} \cdot \cmod {z_2}$
General Result
Let $z_1, z_2, \ldots, z_n \in \C$ be complex numbers.
Let $\cmod z$ be the modulus of $z$.
Then:
- $\cmod {z_1 z_2 \cdots z_n} = \cmod {z_1} \cdot \cmod {z_2} \cdots \cmod {z_n}$
Proof 1
Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$, where $x_1, y_1, x_2, y_2 \in \R$.
| \(\ds \cmod {z_1 z_2}\) | \(=\) | \(\ds \sqrt {\paren {x_1 x_2 - y_1 y_2}^2 + \paren {x_1 y_2 + x_2 y_1}^2}\) | Definition of Complex Modulus, Definition of Complex Multiplication | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {\paren {x_1^2 x_2^2 + y_1^2 y_2^2 - 2 x_1 x_2 y_1 y_2} + \paren {x_1^2 y_2^2 + x_2^2 y_1^2 + 2 x_1 x_2 y_1 y_2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {x_1^2 x_2^2 + y_1^2 y_2^2 + x_1^2 y_2^2 + x_2^2 y_1^2}\) |
| \(\ds \cmod {z_1} \cdot \cmod {z_2}\) | \(=\) | \(\ds \sqrt {x_1^2 + y_1^2} \sqrt {x_2^2 + y_2^2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {\paren {x_1^2 + y_1^2} \paren {x_2^2 + y_2^2} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {x_1^2 x_2^2 + y_1^2 y_2^2 + x_1^2 y_2^2 + x_2^2 y_1^2}\) |
$\blacksquare$
Proof 2
Let $\overline z$ denote the complex conjugate of $z$.
Then:
| \(\ds \cmod {z_1 z_2}\) | \(=\) | \(\ds \sqrt {\paren {z_1 z_2} \overline {\paren {z_1 z_2} } }\) | Modulus in Terms of Conjugate | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {z_1 \overline {z_1} z_2 \overline {z_2} }\) | Product of Complex Conjugates, Complex Multiplication is Commutative | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sqrt {z_1 \overline {z_1} } \sqrt {z_2 \overline {z_2} }\) | Power of Product | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cmod {z_1} \cmod {z_2}\) |
$\blacksquare$
Proof 3
Let:
- $z_1 = r_1 \paren {\cos \theta_1 + i \sin \theta_1}$
- $z_2 = r_2 \paren {\cos \theta_2 + i \sin \theta_2}$
Then:
| \(\ds \cmod {z_1 z_2}\) | \(=\) | \(\ds \cmod {r_1 \paren {\cos \theta_1 + i \sin \theta_1} r_2 \paren {\cos \theta_2 + i \sin \theta_2} }\) | Definition of Polar Form of Complex Number | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cmod {r_1 r_2 \paren {\map \cos {\theta_1 + \theta_2} + i \map \sin {\theta_1 + \theta_2} } }\) | Product of Complex Numbers in Polar Form | |||||||||||
| \(\ds \) | \(=\) | \(\ds r_1 r_2\) | Definition of Polar Form of Complex Number | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cmod {z_1} \cmod {z_2}\) | Definition of Polar Form of Complex Number |
$\blacksquare$
Also see
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: $(1.11)$
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Multiplication and Division: $3.7.11$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Morphisms
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Absolute Value: $1$
- 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.3$ Complex conjugation