Inverse for Complex Multiplication

Theorem

Each element $z = x + i y$ of the set of non-zero complex numbers $\C_{\ne 0}$ has an inverse element $z^{-1}$ under the operation of complex multiplication:

$\forall z \in \C_{\ne 0}: \exists z^{-1} \in \C_{\ne 0}: z \times z^{-1} = 1 + 0 i = z^{-1} \times z$

This inverse can be expressed as:

$\dfrac 1 z := \dfrac {x - i y} {x^2 + y^2} = \dfrac {\overline z} {z \overline z}$

where $\overline z$ is the complex conjugate of $z$.

Proof

$\ds \paren {x + i y} \frac {x - i y} {x^2 + y^2}$

$=$

$\ds \frac {\paren {x \cdot x - y \cdot \paren {-y} } + i \paren {x \cdot \paren {-y} + x \cdot y} } {x^2 + y^2}$

$\ds $

$=$

$\ds \frac {\paren {x^2 + y^2} + 0 i} {x^2 + y^2}$

$\ds $

$=$

$\ds 1 + 0 i$

Similarly for $\dfrac {x - i y} {x^2 + y^2} \paren {x + i y}$.

So the inverse of $x + i y \in \struct {\C_{\ne 0}, \times}$ is $\dfrac {x - i y} {x^2 + y^2}$.

As $x^2 + y^2 > 0 \iff x, y \ne 0$ the inverse is defined for all $z \in \C: z \ne 0 + 0 i$.

$\Box$

From the definition, the complex conjugate $\overline z$ of $z = x + i y$ is $x - i y$.

From the definition of the modulus of a complex number, we have:

$\cmod z = \sqrt {a^2 + b^2}$

From Modulus in Terms of Conjugate, we have that:

$\cmod z^2 = z \overline z$

Hence the result:

$\dfrac 1 z = \dfrac {\overline z} {z \overline z}$

$\blacksquare$

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Sources

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: $(1.15)$
1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.7$ Complex Numbers and Functions: Powers: $3.7.25$
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Subgroups
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System: $9$
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Axiomatic Foundations of Complex Numbers: $78$
1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.2$ The algebraic structure of the complex numbers
1998: Yoav Peleg, Reuven Pnini and Elyahu Zaarur: Quantum Mechanics ... (previous) ... (next): Chapter $2$: Mathematical Background: $2.1$ The Complex Field $C$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): complex number
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): inverse of a complex number

Inverse for Complex Multiplication

Theorem

Proof

Examples

Example: $\dfrac 1 {1 + i}$

Example: $\dfrac 1 {3 + 2 i}$

Sources