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## Definition: Definition of Complex Numbers

A complex number $$x$$ is an ordered pair of real numbers $$a$$ and $$b$$:

$x:=(a,b),\quad\quad a,b\in\mathbb R$

$$\Re(x):=a$$ is called the real part and $$\Im(x):=b$$ is called the imaginary part of the complex number $$z$$.

This means that two complex numbers $z$ and $z’$ are equal, if and only if $\Re(z)=\Re(z’)$ and $\Im(z)=\Im(z’).$

The set of complex numbers is denoted by $$\mathbb C$$.

The set of complex numbers can be interpreted as a complex plane, in which every complex number $$z$$ is a single point. In the following figure, you can drag the point $$z$$ to see, how it moves through the complex plane and how the real and imaginary parts of the complex number change:

| | | | | created: 2014-02-20 23:57:48 | modified: 2020-06-14 12:28:43 | by: bookofproofs