Definition: Addition of Complex Numbers

According to the definition of complex numbers, we can consider the complex numbers \(x,y\in\mathbb C\) as ordered pairs of some real numbers \(a,b,c,d\in\mathbb R\), i.e. \(x=(a,b)\) and \(y=(c,d)\).

The addition of complex numbers "\(+\)" is defined based on the addition of the real numbers of the corresponding ordered pairs, as follows:


Note that this kind of addition operation always produces a new ordered pair of real numbers \((a+c,b+d)\), which is a new complex number, called the sum of the complex numbers \(x\) and \(y\). Thus, the set of complex numbers \(\mathbb C\) is closed under this kind of addition operation.

The addition of two complex numbers can be interpreted as the addition of two vectors in the complex plain, which can be constructed using a parallelogram. In the following figure, you can drag the complex numbers \(x\) and \(y\) and see, how their position changes the position of the respective sum \(x+y\):

  1. Proposition: Addition of Complex Numbers Is Associative
  2. Proposition: Addition of Complex Numbers Is Commutative
  3. Proposition: Existence of Complex Zero (Neutral Element of Addition of Complex Numbers)
  4. Proposition: Existence of Inverse Complex Numbers With Respect to Addition
  5. Proposition: Uniqueness of Complex Zero

Chapters: 1 2
Definitions: 3
Parts: 4
Proofs: 5 6 7 8 9 10 11 12 13
Propositions: 14 15 16 17 18 19 20

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983