A brief interchange on Facebook yesterday and today with three of my former students prompted this discussion of the two questions stated in the title of this post. It all started with this implausible claim from a reliable math website:
I posted this with no comment other than “Context is important. This is from a reliable website.” Comments from two former students (A, just a year out of high school, now attending some college in Princeton, NJ; and J, having been a high school student in the ’70s) gave justifications. But then came a question from L (one year out of high school, just like A):
me and my friends were discussing why do negative numbers exist? They have no place in the real world, everything is in the positive dimension i.e. You can count 6 buildings or 1000 fields or 3 dogs, but you will never have/count a negative amount of anything. In addition to why do they exist, I am also wondering how come a negative number times a negative number equals a positive number. It’s something we learn very early on, but even as a math major I don’t understand the logic behind this.
We can respond to this from several points of view, on a spectrum from most theoretical to most applied. The most theoretical answer, from A, points out that
the purpose of negative numbers is to serve as inverses over the operation of addition.
looking at a polar diagram of the complex plane (and using r, theta coordinate multiplication rules) made it feel very logical to me (though I think this is the result, not the reason why).
OK, all that is true — but neither statement is going to convince anybody other than a real mathematician (which A is). And then B gives an overdrawn bank account as a “real world” example of negative numbers — an appropriate response too — only to be shot down by C, who makes this observation:
debt is a man-made concept… still holds no validity in what I call the real world
So where does that leave us? First of all, I want to point out that C’s original concern reflects exactly the POV of the website where I found the initial image at the top of this page: does number specifically mean counting number? In some ways it does, but in other ways the counting numbers form a subset of all numbers. In common usage, if we’re not mathematicians, when we say things like “I know a number of ways to answer that question,” we certainly don’t mean negative 3 ways or √2 ways or zero ways — or even one way, for that matter, but that’s another story. So the examples that C cites are all counting numbers. As we go through history, or as we go through the educational life of an individual student, we expand the idea of number from counting numbers (positive integers) to fractions, negative numbers, irrational numbers, and complex numbers — all of which we choose to designate as numbers. But we don’t usually call matrices or quaternions “numbers,” although we certainly could. We could formulate a precise definition of “number” that includes the sets we want and excludes the sets that we don’t want, but again that feels very abstract and theoretical. All this is why Leopold Kronecker said
Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.
(You can look it up, if your German is rusty.)
So the answer to why negative numbers exist is that they are very useful, both to theoreticians in order to be able to solve equations like x + 3 = 0, and to applied mathematicians in order to represent overdrawn bank accounts and the like. Similarly, complex numbers exist in order to be able to solve equations like x2 + 4 = 0 and in order to represent situations in electrical engineering.
Next, why do two negatives make a positive (only when multiplying, of course)? It all comes down to one’s definition of multiplication. If you think of multiplication as repeated addition, you won’t get anywhere, although many math teachers have attempted to create contorted explanations of how that definition leads to the correct conclusion about multiplying two negatives. You have to use the definition of multiplication that B refers to. Instead of being points on a number line, you think of numbers as being points in a number plane. (Note that the number line works nicely for much more than counting numbers, as it allows negatives, fractions, and irrationals. And the number plane then works nicely for complex numbers as well.) Every number is then identified by a pair of polar coordinates: r, the distance from the origin; and θ, the angle you have to turn through from 0° to reach that number. It then turns out — for reasons that would be far too long to explain even in this over-long post — that we multiply two numbers by multiplying their r values in the traditional way and adding their θ values. Since negative numbers are on the left side of the number line, their θ is 180°; multiply two of them and you get 360°, which of course is a positive number.
I’m not sure that that helps — especially without the details that I had to omit — but it really does give the explanation for why two negatives make a positive, and it has some added benefits that we won’t go into here, like how you find all three cube roots of 8. (It you’ve never studied this stuff, you probably think that 2 is the only one, but “there are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.”)