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:
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 \(y\):
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
