Definition: Multiplication 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 multiplication of complex numbers "\(\cdot\)" is defined based on the addition, subtraction, and multiplication of real numbers in the corresponding ordered pairs:

\[x\cdot y:=(a,b)\cdot (c,d):=(ac-bd,ad+bc),\]

Note that this kind of multiplication operation always produces a new ordered pair of real numbers \((ac-bd,ad+bc)\), which is a new complex number, called the product of the complex numbers \(x\) and \(y\). Thus, the set of complex numbers \(\mathbb C\) is closed under this kind of multiplication operation.

The multiplication of two complex numbers (points in the complex plane) \(x\) and \(y\) can be interpreted/constructed as follows:

  1. Connect \(x\) and \(y\) with the origin of the complex plane.
  2. Measure the angles of the segments with respect to the positive horizontal axis.
  3. Measure the lengths of the segments.
  4. Add the two angles.
  5. Multiply the two lengths.
  6. The point \(x\cdot y\) lies on the segment with the length gained in step 5 with an angle to the positive horizontal axis gained in step 4.

In the following figure, you can drag the complex numbers \(x\) and \(y\) and see, how the position of \(x\cdot y\) changes with the position of the respective factors \(x\) and \(y\):

Fun questions:

  1. Proposition: Multiplication of Complex Numbers Is Associative
  2. Proposition: Multiplication of Complex Numbers Is Commutative
  3. Proposition: Existence of Complex One (Neutral Element of Multiplication of Complex Numbers)
  4. Proposition: Existence of Inverse Complex Numbers With Respect to Multiplication

Chapters: 1
Parts: 2
Proofs: 3 4 5 6 7 8 9 10 11 12 13
Propositions: 14 15 16 17 18 19 20 21 22

Thank you to the contributors under CC BY-SA 4.0!




  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013