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.
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
Definition
A complex number is an ordered pair $\left({x, y}\right)$ where $x, y \in \R$ are real numbers, on which the operations of addition and multiplication are defined as follows:
Complex Addition
Let $\left({x_1, y_1}\right)$ and $\left({x_2, y_2}\right)$ be complex numbers.
Then $\left({x_1, y_1}\right) + \left({x_2, y_2}\right)$ is defined as:
- $\left({x_1, y_1}\right) + \left({x_2, y_2}\right):= \left({x_1 + x_2, y_1 + y_2}\right)$
Complex Multiplication
Let $\left({x_1, y_1}\right)$ and $\left({x_2, y_2}\right)$ be complex numbers.
Then $\left({x_1, y_1}\right) \left({x_2, y_2}\right)$ is defined as:
- $\left({x_1, y_1}\right) \left({x_2, y_2}\right) := \left({x_1 x_2 - y_1 y_2, x_1 y_2 + y_1 x_2}\right)$
The set of all complex numbers is denoted $\C$.
Also see
- Results about complex numbers can be found here.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory
- 1964: Murray R. Spiegel: Theory and Problems of Complex Variables ... (previous) ... (next): $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): $\text{II}$: Subgroups
