Where Do Complex Numbers Come From?

Let’s find out how complex numbers arose in the past. This will help us understand the use of complex signals in signal processing applications. DSP algorithms extensively use complex numbers to exploit the signal phase information in addition to the amplitude.

X-axis

Initial applications of mathematics included taxation and trade. This led to an evolution of the number system as follows.

Natural and whole numbers

The idea of countable objects comes naturally to humans, e.g., counting objects like fruits and houses. So the numbers $1, 2, 3, \cdots$ were called natural numbers. Whole numbers are natural numbers, which includes zero.

Rational and irrational numbers

Since many quantities need to be divided into multiple parts, (e.g., a field), this led to the discovery of rational numbers that can be written in fractional form. The remaining gaps on the real line are filled by irrational numbers such as $\pi$ or $\sqrt{2}$ ...