Complex Multiplication Distributes over Addition
Jump to navigation
Jump to search
Theorem
The operation of multiplication on the set of complex numbers $\C$ is distributive over the operation of addition.
- $\forall z_1, z_2, z_3 \in \C:$
- $z_1 \paren {z_2 + z_3} = z_1 z_2 + z_1 z_3$
- $\paren {z_2 + z_3} z_1 = z_2 z_1 + z_3 z_1$
Proof
From the definition of complex numbers, we define the following:
\(\ds z_1\) | \(:=\) | \(\ds \tuple {x_1, y_1}\) | ||||||||||||
\(\ds z_2\) | \(:=\) | \(\ds \tuple {x_2, y_2}\) | ||||||||||||
\(\ds z_3\) | \(:=\) | \(\ds \tuple {x_3, y_3}\) |
where $x_1, x_2, x_3, y_1, y_2, y_3 \in \R$.
Thus:
\(\ds z_1 \paren {z_2 + z_3}\) | \(=\) | \(\ds \tuple {x_1, y_1} \paren {\tuple {x_2, y_2} + \tuple {x_3, y_3} }\) | Definition 2 of Complex Number | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x_1, y_1} \tuple {x_2 + x_3, y_2 + y_3}\) | Definition of Complex Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x_1 \paren {x_2 + x_3} - y_1 \paren {y_2 + y_3}, x_1 \paren {y_2 + y_3} + y_1 \paren {x_2 + x_3} }\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x_1 x_2 + x_1 x_3 - y_1 y_2 - y_1 y_3, x_1 y_2 + x_1 y_3 + y_1 x_2 + y_1 x_3}\) | Real Multiplication Distributes over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {\paren {x_1 x_2 - y_1 y_2}\ + \paren {x_1 x_3 - y_1 y_3}, \paren {x_1 y_2 + y_1 x_2} + \paren {x_1 y_3 + y_1 x_3} }\) | Real Addition is Commutative | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2} + \tuple {x_1 x_3 - y_1 y_3, x_1 y_3 + y_1 x_3}\) | Definition of Complex Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x_1, y_1} \tuple {x_2, y_2} + \tuple {x_1, y_1} \tuple {x_3, y_3}\) | Definition of Complex Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds z_1 z_2 + z_1 z_3\) | Definition 2 of Complex Number |
The result $\paren {z_2 + z_3} z_1 = z_2 z_1 + z_3 z_1$ follows directly from the above, and the fact that Complex Multiplication is Commutative.
$\blacksquare$
Examples
Example: $\paren {-1 + 2 i} \paren {\paren {7 - 5 i} + \paren {-3 + 4 i} } = \paren {-1 + 2 i} \paren {7 - 5 i} + \paren {-1 + 2 i} \paren {-3 + 4 i}$
Example: $\paren {-1 + 2 i} \paren {\paren {7 - 5 i} + \paren {-3 + 4 i} }$
- $\paren {-1 + 2 i} \paren {\paren {7 - 5 i} + \paren {-3 + 4 i} } = -2 + 9 i$
Example: $\paren {-1 + 2 i} \paren {7 - 5 i} + \paren {-1 + 2 i} \paren {-3 + 4 i}$
- $\paren {-1 + 2 i} \paren {7 - 5 i} + \paren {-1 + 2 i} \paren {-3 + 4 i} = -2 + 9 i$
As can be seen:
- $\paren {-1 + 2 i} \paren {\paren {7 - 5 i} + \paren {-3 + 4 i} } = \paren {-1 + 2 i} \paren {7 - 5 i} + \paren {-1 + 2 i} \paren {-3 + 4 i}$
$\blacksquare$
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 $\text {(c)}$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System: $6$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Solved Problems: Axiomatic Foundations of Complex Numbers: $15$
- 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