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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

1.Definition: Addition of Complex Numbers

2.Definition: Subtraction of Complex Numbers

3.Definition: Multiplication of Complex Numbers

4.Proposition: Distributivity Law for Complex Numbers

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Bibliography (further reading)