Definition:Complex Number/Definition 2
Convert this into an "axiomatic definition" -- the addition and multiplication operations define the complex-number nature of these entities, rather than being a derivable result of them. The contents of "Addition" and "Multiplication" links on this page are the basis of the proof that this definition of Complex Numbers is equivalent to Definition 1.
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Contents
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
