Definition:Complex Number/Definition 2

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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