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
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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): complex number