Definition:Complex Number/Definition 2
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Definition
A complex number is an ordered pair $\tuple {x, y}$ where $x, y \in \R$ are real numbers, on which the operations of addition and multiplication are defined as follows:
Complex Addition
Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers.
Then $\tuple {x_1, y_1} + \tuple {x_2, y_2}$ is defined as:
- $\tuple {x_1, y_1} + \tuple {x_2, y_2}:= \tuple {x_1 + x_2, y_1 + y_2}$
Complex Multiplication
Let $\tuple {x_1, y_1}$ and $\tuple {x_2, y_2}$ be complex numbers.
Then $\tuple {x_1, y_1} \tuple {x_2, y_2}$ is defined as:
- $\tuple {x_1, y_1} \tuple {x_2, y_2} := \tuple {x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}$
Scalar Product
Let $\tuple {x, y}$ be a complex numbers.
Let $m \in \R$ be a real number.
Then $m \tuple {x, y}$ is defined as:
- $m \tuple {x, y} := \tuple {m x, m y}$
The set of all complex numbers is denoted $\C$.
Also see
- Results about complex numbers can be found here.
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
- 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
- 1990: H.A. Priestley: Introduction to Complex Analysis (revised ed.) ... (previous) ... (next): $1$ The complex plane: Complex numbers $\S 1.1$ Complex numbers and their representation
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: complex number