Contrary to what one might expect, complex numbers are not complicated
numbers. The term "complex" just means that complex numbers
have two components, one "real" and one "imaginary".

The set of numbers that people (including scientists and engineers)
use has gradually expanded. The first people who used numbers probably
only knew "1, 2, 3, and many", more or less. For a long time,
positive integers were sufficient. The Indians are generally credited
with having recognized "0" as a number. The acceptance of
negative integers may have arisen from the need to make a distinction
between a positive and a negative "bank account". The early
agricultural societies (Egypt? Mesopotamia?) invented fractions - very
useful - but even the ancient Greeks realised that that was not the
end of the story. After all, a square might very well have an area twice
as large as another square. Its side would then be the-square-root-of-two
times as large as the side of the smaller square. It was easy
to prove that this number could not be a fraction.

So the Greeks invented "irrational numbers" to complement
the "rational numbers" that had been identified up to that
point. But even that was not enough. It was discovered in the 18th and
19th centuries an additional extension to "transcendental numbers"
was needed, the distinction being that all numbers up to that point
could be solutions to an algebraic equation, while transcendental numbers
could not. (*e* and *π* are transcendental.)

"Irrational", "transcendental", "imaginary"
- all of these terms suggest that each expansion of the set of numbers
was greeted with some skepticism. Imaginary numbers, in particular,
trigger hostility. They are all based on the number *i*, the square
root of minus one. "There is no such number", was my first
reaction. But in fact, if you leave your prejudices aside and just accept
the definition, it turns out that "imaginary" numbers are
very useful. Their acceptance is easy, once you learn to interpret them
geometrically as numbers on the y axis in a two-dimensional plane. Complex
numbers then have the form *z* = *x* + *iy*, and all
the usual algebraic rules apply. Multiplication by *i* is equivalent
to a counterclockwise rotation of 90 degrees in the plane of complex
numbers.