Complex Numbers

When we are first taught complex numbers, we are told that is it a "natural" completion of the real numbers. Something like "wouldn't we like x^2 + 1 = 0 to have a solution?". And then we continue with this baggage assuming this is a general conception that works for all Mathematical objects. Some modern Mathematics indeed works this way. But let's go back to see how complex numbers were first discussed. Here's an excerpt from wikipedia: (History of complex numbers)

Complex numbers became more prominent in the 16th century, when closed formulas for the roots of cubic and quarticpolynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. For example, Tartaglia's cubic formula gives the following solution to the equation x³ − x = 0:

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z3 = i has solutions –i, and . Substituting these in turn for in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x3 – x = 0.

This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in 1637 and was meant to be derogatory [citation needed] (see imaginary number for a discussion of the "reality" of complex numbers). A further source of confusion was that the equation

seemed to be capriciously inconsistent with the algebraic identity

, which is valid for positive real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity (and the related identity

) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of

to guard against this mistake.

The important thing here is Cardano didn't care about the complex solutions to the quadratic equation or the cubic equation, but even when solving a cubic equation all of whose roots are real you must manipulate complex numbers on the path to the real solutions. This is the true motivation. Later on these numbers started appearing in math other places in Mathematics. This is not surprising, giving their necessary, natural and unbidden appearance -- we might say that we didn't wan't them, but rather were forced onto us.