Wednesday, September 24, 2008
Wednesday Math, Vol. 40: i, the imaginary number
Teaching math is tough enough, just dealing with the concepts, but sometimes even the language gets in the way. People grasp the concept of positive whole numbers fairly easily, and even zero, which is a tricky idea having a symbol expressing nothing, is now nearly universally accepted. But then we get negative numbers, and some people begin to have problems, and fractions cause even more trouble. Worse still, we have numbers that can't be expressed as fractions, like pi and the square root of 2, and it was decided that they would be called irrational. Then some clever guy decided we needed the square root of -1, and since it wasn't on the number line, it would best be called an imaginary number.
Many centuries ago, before the time of Isaac Newton, finding roots of polynomials was considered the most important thing in math, and the nationality of the best finders of roots was Italian. Instead of calling a person who excelled at algebra an algebraist, they used the word cosista, which translates directly from Italian as "thing-ist". Their goal was to find the thing, la cosa, so this was the word they coined.
The simplest polynomial that resisted having its roots found was x^2 + 1 = 0, and you have to admit it's pretty simple. To have a solution, you would need a square root of -1, and any real number when squared will be greater than or equal to zero. Okay, said one of the Italians, let's say there is a number that when squared equals -1, even though no real number works. Let's see if we can define such a number and work with it.
The square root of -1 was called the imaginary number. Some mathematicians used it, but many mocked it. Descartes thought it was idiotic. But the guy who really made an effort to understand it and integrate it into a consistent mathematical system was My Favorite Lenny, Leonhard Euler. If you have a number that is part real and part imaginary, it is called a complex number, and Lenny figured out how the math worked. If the imaginary number line was at 90 degrees from the real number line and the two lines met at zero, that would create the complex plane, and addition, subtraction, multiplication and division can be described by (relatively) simple geometric operations on vectors. This geometry is absolutely essential to the understanding of many scientific fields, most notably electrical engineering, but electrical engineers call the square root of -1 j instead of i, because there is nothing imaginary about it in their world.
After the excellent Euler, it is the great Gauss who makes the next most important steps in understanding the complex number system, including his many attempts to produce a good proof of the Fundamental Theorem of Algebra. Gauss did not give Euler much credit, or at least did not put him near the top of mathematicians, which is now considered an oversight on Gauss' part. He may have been miffed at Lenny for giving the root of -1 the official name of i, which stands for imaginary in a lot of languages. If Gauss had his way, positive numbers would be called direct, negative numbers would be called inverse and imaginary numbers would be called lateral or inverse lateral. (Yes, you can have negative imaginary numbers.) While I'm not thrilled with how little love Gauss showed for My Favorite Lenny, he was a fracking genius and he did have a very good idea here.
Of course, we have the problem that there are centuries of textbooks that use the words negative and imaginary, but maybe all we need to do is stay "on message". After all, now all swamps are "wetlands" and sludge is a "biosolid" and bad debts are "non-performing assets", why can't all mathematicians just start calling numbers direct, inverse and lateral? It's not actually a lie, just a re-naming, and it would make things a little easier for future generations of students and teachers alike.
Sometimes, even lying weasels can perform a public service with their example, if an honest person is open to a teachable moment.