Definition:Complex Number/Definition 2

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This page has been identified as a candidate for refactoring of advanced complexity.
In particular: 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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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

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