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© Copyright 1999, Jim Loy
In mathematics, we sometimes deal with "imaginary numbers." They are not really imaginary, in the normal usage of that word. They are just rarely of any use to people, just as fractions are of no use to someone counting unbroken marbles. In some branches of mathematics, they are very useful. In my article on Irrational Numbers, I said that irrational numbers are important in order to solve certain equations. The same is true of imaginary numbers. There are a few practical uses for imaginary numbers. But, the main use for them is to make certain equations have solutions.
An imaginary number is a real number times the positive square root of -1. The positive square root of -1 is usually called i. It is not an integer, it is not real. It is the answer to the question, "What number squared is -1?" Well, no normal number can ever be the answer to that question. But, there are good reasons for defining a new kind of number, imaginary numbers, just to answer that question. 3i or -7i are imaginary numbers. Actually, if we define imaginary numbers, we also have to define complex numbers. A complex number is a real number plus an imaginary number. An imaginary number is really a complex number, like 0+3i.
If we don't allow imaginary numbers, then the equation x^2+1=0 has no solution. This should cause you no great difficulty. If you graph y=x^2+1, you will find that, indeed it has no solution, when y=0. The graph does not cross the x axis. There are sound theoretical reasons for saying that such an equation always has two solutions. These two solutions may be the same. And they may be complex. The two solutions to x^2+1=0 are i and -i. They are complex (imaginary, in this case).
Using sqr() as the square root function (as I don't have a square root symbol on the WWW), the equation sqr(x-1)=1-sqr(x+4) seems to have two solutions. Just solve it using traditional algebraic methods. Go ahead and try it. Now, before I tell you about the answer, go ahead and check your work, by substituting your two solutions back into the original equation. It didn't work, did it? This equation has no solution. It seems to have the solutions x=0 and -3. But neither of these works. That is a good reason to check your work.
Many equations have no solutions. 2x=2x+3 has no solution. But, when we try to solve that one, we get obvious nonsense, like 0=3. In our more complicated equation, we didn't get obvious nonsense, until we checked our work. What is going on there? Well, our solutions were not imaginary, but imaginary numbers were encountered during the attempted solution, and during the checking of our work. Does that matter? Well yes, in a way. On the left side of our equation we have sqr(x-1), which just may be an imaginary number. On the right side, we have 1-sqr(x+4), which just may be one more than another imaginary number. No imaginary number is one more than any imaginary number.
So, where did we go wrong, when we solved it. Well, we probably squared both sides. What if we have the obvious nonsense -2=2? Well, if we square both sides, we get the obvious truth 4=4. So, squaring both sides can be dangerous. We can start with an equation with no solution, square it, and end with an equation with a solution. We are starting with nonsense (maybe well disguised), and ending with something that is not nonsense. Here is an example: x-2=x+2, which has no solution. Square both sides (not something that you would normally do in this case), and you get x^2-2x+4=x^2+2x+4. Simplify, and you get -2x=2x. Solve for x and you get x=0, which does not work in the original equation (x-2=x+2). That did not involve imaginary numbers. I put it in this article, about imaginary numbers, because similar equations, but with imaginary numbers in them, demand that you square both sides in order to progress toward a possible solution. They encourage you to fall into that trap.
So, check your work.
One of the practical uses of complex numbers is in engineering where they are used as vectors in two dimensions. A vector like (3,2) becomes 3+2i. Vector addition and subtraction is identical to addition and subtraction of complex numbers. This is the reason that complex numbers were included in the computer programming language Fortran. But, these vectors are not imaginary numbers. They do not follow the rules of complex multiplication or division (which were supported by Fortran). So, I consider this a very strange use of complex numbers. Besides, there is no way to use complex numbers to represent a 3-dimensional vector (or one of more dimensions), except by analogy (like 3+2i+4j).
Note: I have made a couple corrections to the above article, based on email from Marco Coletti. One of the things that I had said (after correcting it) was, "y=x^2+1 crosses the x axis at i and -i." This was meant to be thought provoking. Just what would a graph of two complex variables look like. Well, you need four dimensions for such a graph. So, it is very difficult to visualize.