Complex numbers

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.

  Last edited or checked October 11, 2005.

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