# 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.

Proofs: 1

Definitions: 1 2 3 4 5 6 7 8
Lemmas: 9
Proofs: 10 11 12 13
Propositions: 14 15 16 17

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### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013
2. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994
3. Timmann, Steffen: "Repetitorium der Funktionentheorie", Binomi-Verlag, 2003