Square roots are confusing. I think this is the basis of our confusion. Variables make it worse. Let me show you what I mean. We say that r is the square root of a if r2=a. It take a long time to go from that definition to the plain fact that 2 is the square root of 4 or that 3 is the square root of 9, and these are concrete things. Then we math teachers, gleeful sadists that we are, spend a lot of time convincing our tortured students (I couldn't masochistic students because they would enjoy it) that r2 can never, ever be negative.
The argument goes like this. On one hand, suppose r is positive. A positive times a positive is positive. So r2, which is really just r times r, is positive. On the other hand, suppose r is negative. A negative times a negative is positive too. Really. You've just got to BUH-leeve! (Actually, can explain it to you, but it would slow us down.) So, in this case r2 is positive too. The only case left--if you trust me--is r=0. In this case r2=0.
There for there is no square root of -1, or of any other negative number.
Except there is.
What I've done in my argument, what all math teachers do when they make that argument, is to restrict the number r to what we call the set of real numbers. Here's the thing. When mathematicians say "real" we don't really mean it. It's just like when we say the homework assignment this weekend is going to be "short" or the test next Friday is going to be "easy." We are using these words in a different way than anybody else. While "real" originally did mean real, when we apply it to numbers, it has some to mean those numbers we thought were the totality of all numbers before we discovered we didn't know much.
Mathematicians now talk about a number i that is equal to the square root of -1 and refer to it as imaginary. We use it in expressions like x+yi that we manipulate formally like polynomials, but whenever i2 occurs, we replace it with -1. All of the numbers x+yi are refered to as the complex number and we can model the complex numbers geometrically on the plane. Indeed, one way mathematicians came to accept the complex numbers as being as "real" as any other is by creating an arithmetic of the ordered pairs of numbers which represent points on the Cartesian coordinate plane.
This impinges on algebra students in the following way. One of the most evil mathematical formulas that the bulk of the human race has to face is the general quadratic formula. (Did anyone else here evil laughter in the background?) The solution of the general quadratic equation, ax2+bx+c=0, is given as
It is a common practice to tell students at a certain level that, when b2-4ac is negative, there is no solution. This is what I was told. You and I, however, now know there is a solution, but that solution is a complex number. I suppose the reason some withhold this information is that it is an opening for other questions. So what's it good for? What about the blue lines in hockey? What's a pasty? Where do babies come from? Where is the money for the bail-out coming from?
While I can't answer some of these at all, and the ones I can answer, I can't deal with in the complexity I'd like, I will say this. When physics and engineering students are solving certain equations dealing with electrical circuits, it's nice to have these complex numbers around.
The end of it all
While I did say that complex numbers can be made precise in terms of real numbers and putting an arithmetical structure on the plane (and there is at least one other way to define them in terms of ring theory), the genesis of it all was simply pronouncing the square root of -1 to exist and running with it. This brings us to one of those other mysterious entities: ∞ (pronounced infinity).
Here one will get into a variety of arguments with a variety of personality types all falling toward the persnickety end of the scale. When we talk about infinity, what sort of infinity would we mean. We can talk about the cardinal numbers or the ordinal numbers. We can say that infinity means "unending." We can talk about about the Indian views and the Greek views. There are all sorts of ways that we can twist this puppy. To those let me say. "Ptttttttttppp." Let me rephrase, "Pttt, pttt, pttt."
No, I am talking about coming at this like a fifth grader would. I want a number that is larger than any other numbers. And I can hear you already, "But if you have a biggest number, you just add 1 to it, and you've got a bigger one. Therefore, there is no biggest number." Thinking like that is going to get us nowhere. We've got to do what was done when we stumbled upon the square root of negative one, i.e. we pronouce it to exist and run with it.
You are saying, "Okay, now, Mister Smart E. Pants, what is ∞+1 then?" Well, it's ∞. So is ∞+2 and ∞+3. And before you get there ∞+∞=∞ and ∞x∞=∞. However, it wasn't by accident or laziness that I left out ∞/∞ and ∞-∞ because division and subtraction are undefined for ∞, and this is probably why teachers don't go much into it. That and the fact that getting kids to learn the multiplication tables through 12 x 12 is hard enough; the ∞ x ∞ tables would be inhuman.
All of this having been said, ∞ can be made precise just like i can. We can define ∞ in terms of sequences that diverge to infinity. For those of you who've not used your calculus in a while or never had it to begin with, a sequence is an infinite list of numbers. Here I mean infinite in the sense of being unending. (I take back one of those earlier Pttt's.) A sequence is said to converge to a number L if its terms can eventually be made to stay as close as we desire to L. (I could make that more precise, but you wouldn't understand it any better.) We say that a sequence diverges to inifinity if its terms can eventually be made to stay as large as we desire. We can define ∞ to be the set of all of these sequences.
I've got to admit that defining a number to be a set of sequences is really, really weird. Really, really. However, one way the real numbers (you remember the real numbers) are defined is in terms of sequences of rational numbers. The gentle reader is at this point asking "Whut?" And I, dedicated scholar that I am, will simply pat him somewhat condescendingly on the back and answer, "It's technical; let's move on."
In adding ∞ to the real numbers in this manner, we are doing what we topologists call adding an end to the real number line. If we also add -∞ we are adding another end. This is all useful for very abstract mathematical purposes. What strikes me, after all these years, is how an idea that many smart-alecky fifth-graders have come up with on their own only to be shouted down by an over-worked teacher who has 40 other little smart-alecks to deal with, is actually legitimate mathematically.
