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## Lemma: Complex Numbers are Two-Dimensional and the Complex Numbers $$1$$ and Imaginary Unit $$i$$ Form Their Basis

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:

$x=a \cdot 1+b \cdot i$

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]

## 1.Proof: (related to "Complex Numbers are Two-Dimensional and the Complex Numbers $$1$$ and Imaginary Unit $$i$$ Form Their Basis")

### CC BY-SA 3.0

[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