Complex Multiplication Identity is One

Theorem

Let $\C_{\ne 0}$ be the set of complex numbers without zero.

The identity element of $\struct {\C_{\ne 0}, \times}$ is the complex number $1 + 0 i$.

Proof

\(\ds \paren {x + i y} \paren {1 + 0 i}\)

\(=\)

\(\ds \paren {x \cdot 1 - y \cdot 0} + i \paren {x \cdot 0 + y \cdot 1}\)

\(\ds \)

\(=\)

\(\ds \paren {x + i y}\)

and similarly:

\(\ds \paren {1 + 0 i} \paren {x + i y}\)

\(=\)

\(\ds \paren {1 \cdot x - 0 \cdot y} + i \paren {0 \cdot x + 1 \cdot y}\)

\(\ds \)

\(=\)

\(\ds \paren {x + i y}\)

$\blacksquare$

Sources

• 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
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System: $7$
• 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