Complex Multiplication is Commutative

Theorem

The operation of multiplication on the set of complex numbers $\C$ is commutative:

$\forall z_1, z_2 \in \C: z_1 z_2 = z_2 z_1$

Proof

From the definition of complex numbers, we define the following:

\(\ds z\)

\(:=\)

\(\ds \tuple {x_1, y_1}\)

\(\ds w\)

\(:=\)

\(\ds \tuple {x_2, y_2}\)

where $x_1, x_2, y_1, y_2 \in \R$.

Then:

\(\ds z_1 z_2\)

\(=\)

\(\ds \tuple {x_1, y_1} \tuple {x_2, y_2}\)

Definition 2 of Complex Number

\(\ds \)

\(=\)

\(\ds \tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + x_2 y_1}\)

Definition of Complex Multiplication

\(\ds \)

\(=\)

\(\ds \tuple {x_2 x_1 - y_2 y_1, x_1 y_2 + x_2 y_1}\)

Real Multiplication is Commutative

\(\ds \)

\(=\)

\(\ds \tuple {x_2 x_1 - y_2 y_1, x_2 y_1 + x_1 y_2}\)

Real Addition is Commutative

\(\ds \)

\(=\)

\(\ds \tuple {x_2, y_2} \tuple {x_1, y_1}\)

Definition of Complex Multiplication

\(\ds \)

\(=\)

\(\ds z_2 z_1\)

Definition 2 of Complex Number

$\blacksquare$

Examples

Example: $\paren {2 - 3 i} \paren {4 + 2 i} = \paren {4 + 2 i} \paren {2 - 3 i}$

Example: $\paren {2 - 3 i} \paren {4 + 2 i}$

$\paren {2 - 3 i} \paren {4 + 2 i} = 14 - 8 i$

Example: $\paren {4 + 2 i} \paren {2 - 3 i}$

$\paren {4 + 2 i} \paren {2 - 3 i} = 14 - 8 i$

As can be seen:

$\paren {2 - 3 i} \paren {4 + 2 i} = \paren {4 + 2 i} \paren {2 - 3 i}$

$\blacksquare$

Also see

• Complex Multiplication is Associative

Sources

• 1957: E.G. Phillips: Functions of a Complex Variable (8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 1$. Complex Numbers: $\text {(iii)}$ The fundamental operations $\text {(b)}$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System: $4$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Axiomatic Foundations of Complex Numbers: $76 \ \text{(b)}$
• 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