In the complex vector space of complex numbers over the field or real numbers, the set of vectors \(B:=\{1,i\}\) forms a basis, i.e. any complex number \(x\in\mathbb C\) can be represented by a linear combination of the vectors:

- complex number one \(1\) and the
- imaginary unit \(i\)

\[x=a \cdot 1+b \cdot i\]

for some real numbers \(a,b\in\mathbb R\). In particular, every complex number \(x\) can be written as \(a+bi\), where \(a\) is the **real part** of \(x\), also denoted by \(\Re (x)\), and \(b\) is the **imaginary part** of \(x\), also denoted by \(\Im (x)\).

In the following interactive figure, a complex number \(x=a+bi\) is visually represented as a pair of numbers \(a\) and \(b\) forming a vector (green) \(x\) in the complex plane.

| | | | | created: 2016-02-21 10:37:38 | modified: 2020-06-14 12:04:19 | by: *bookofproofs* | references: [696], [1038], [1692]

[696] **Kramer Jürg, von Pippich, Anna-Maria**: “Von den natürlichen Zahlen zu den Quaternionen”, Springer-Spektrum, 2013

[1692] **Timmann, Steffen**: “Repetitorium der Funktionentheorie”, Binomi-Verlag, 2003

[1038] **Wille, D; Holz., M **: “Repetitorium der Linearen Algebra”, Binomi Verlag, 1994