Imaginary Numbers

Numbers are broken up into a couple of different sets, as talked about in the previous post. There are two distinct groups: Real and Imaginary. But what does this mean? Wouldn’t a number have to “exist“ for it to be real? Real numbers are any number that you can think of that can be placed somewhere on a number line. Even a nonterminating number would still have a physical location on a counting line. An Imaginary number, however, is represented a little bit differently. An imaginary number uses a real number and the imaginary unit: I (the square root of -1).

That’s great and all, but what possible use could a number that doesn’t even exist have? Let’s first look at the history of i. In the 16th century, mathematicians were discovering that sometimes the answer to a quadratic equation or higher was the result of the square root of a negative number. Since negative numbers previously didn’t have square roots, this was a problem. Visually on a graph, an equation with no real roots would be a parabola with no x-intercepts. See the picture below.

Despite containing the square root of negative numbers, the rest of the solution would contain real numbers. This is where i came in handy. Representing the square of negative 1, any square root could now be represented. rad(-64) would be represented at rad(-1 * 64) = 8i. This gives a clear way to represent numbers that don’t exist. As pertaining to graphing, the imaginary number opens the door for a new set of axis: the imaginary axis. If a parabola doesn’t have roots on the real graph, its complex roots can be represented on the imaginary axis. If the solution to an equation is 2 + 7i (a complex conjugate), on the imaginary axis, there would be a point at (2, 7) where the usual x coordinate would be real and the usual y coordinate would be 7.

Rafael Bombelli was also a pioneer for this imaginary number in the late 16th century. He proved that complex numbers were excluded from the normal rules of addition. He discovered new rules when multiplying, adding, and subtracting imaginary numbers became the basis for calculations. He was also the first mathematician to firmly believed in the usefulness of imaginary numbers and values despite limited applications in his him.

“Imaginary“ was originally used by Descarte because he believed that it was a useless number; however, things have changed from seventeenth-century mathematics. Imaginary numbers are used in electricity and radio frequencies when doing calculations. The parabola problem also becomes useful in higher-level calculus along with relationships with i. Relatively new, the foundation of imaginary numbers leads to beautiful equations and solutions to equations that would otherwise be impossible.