Themes > Science > Mathematics > Calculus > Real Number > What's "real" about the real numbers?

Probably the simplest way to understand "complex numbers" is to start with points in the plane, as I have done in the preceding paragraphs. However, by a historical accident, the simplest explanation was not the first explanation discovered. Indeed, the geometric, points-in-the-plane viewpoint wasn't discovered until the 19th century, long after the algebraic computations had been investigated. As early as the 16th century, mathematicians were devising new "numbers" as a way of solving polynomial equations; they were thinking in terms of algebraic formulas rather than pictures. They were particularly interested in the third and fourth degree equations at that time, but they even had new insights into the quadratic equation. The attitude that they took was something like this:

 We all know that there isn't really any "number" p that can satisfy the equation p2 = -1. Such a "number" can only exist in our imagination. But if it somehow did exist, what kind of arithmetic rules would it have to follow?

You have to admire the genius of the 16th century mathematicians: They correctly worked out the arithmetic rules of the complex numbers despite their lack of the simple geometric model; they calculated with "numbers" whose existence they didn't even believe in!

Their terminology was unfortunate, however. There is nothing fictitious or dreamlike about rotations of engines, but the name stuck. The points on the vertical axis are now called imaginary numbers, despite the fact that they have very tangible applications. The points on the horizontal axis are (by contrast) called real numbers. All the points in the plane are called complex numbers, because they are more complicated -- they have both a real part and an imaginary part.

Thus ends our tale about where the name "real number" comes from. But we have barely begun investigating the mathematical properties associated with that name.

Information provided by Eric Schechter