# 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:

Chapters: 1 2 3 4 5
Corollaries: 6
Definitions: 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
Examples: 23 24
Lemmas: 25
Parts: 26 27
Proofs: 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
Propositions: 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78
Sections: 79
Theorems: 80 81 82 83

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