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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")

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)

[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