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## The Enemy of My Enemy

It’s traditional to teach kids subtraction right after addition.  That makes sense — the same facts about numbers get used in both, though in reverse.  And the black art of “borrowing,” so crucial to successful subtraction, is only a little more baroque than that of “carrying,” its counterpart for addition.  If you can cope with calculating 23 + 9, you’ll be ready for 23 – 9 soon enough.

Subtraction forces us to expand our conception of what numbers are.  Negative numbers are a lot more abstract than positive numbers — you can’t see negative 4 cookies and certainly can’t eat them — but you can think about them, and you have to, in all aspects of daily life, from debts and overdrafts to contending with freezing temperatures and parking garages.At a deeper level, however, subtraction raises a much more disturbing issue, one that never arises with addition.  Subtraction can generate negative numbers.  If I try to take 6 cookies away from you but you only have 2, I can’t do it — except in my mind, where you now have negative 4 cookies, whatever that means.

Still, many of us haven’t quite made peace with negative numbers.  As my colleague Andy Ruina has pointed out, people have concocted all sorts of funny little mental strategies to sidestep the dreaded negative sign.  On mutual fund statements, losses (negative numbers) are printed in red or nestled in parentheses, without a negative sign to be found.   The history books tell us that Julius Caesar was born in 100 B.C., not –100.  The subterranean levels in a parking garage often have names like B1 and B2.  Temperatures are one of the few exceptions: folks do say, especially here in Ithaca, that it’s –5 degrees outside, though even then, many prefer to say 5 below zero.  There’s something about that negative sign that just looks so unpleasant, so … negative.

Perhaps the most unsettling thing is that a negative times a negative is a positive.  So let me try to explain the thinking behind it.How should we define something like –1 × 3, where we’re multiplying a negative number by a positive number?  Well, just as 1 × 3 means 1 + 1 + 1, the natural definition for –1 × 3 is (–1) + (–1) + (–1), which equals –3.  This should be obvious in terms of money: if you owe me \$1 a week, after three weeks you’re \$3 in the hole.

From there it’s a short hop to see why a negative times a negative should be a positive.  Take a look at the following string of equations:
–1 × 3 = –3
–1 × 2 = –2
–1 × 1 = –1
–1 × 0 = 0
–1 × –1 = ?

Now look at the numbers on the far right and notice their orderly progression:
–3, –2, –1, 0, ?
At each step, we’re adding 1 to the number before it.  So wouldn’t you agree the next number should logically be 1?
That’s one argument for why (–1) × (–1) = 1.  The appeal of this definition is that it preserves the rules of ordinary arithmetic; what works for positive numbers also works for negative numbers.
But if you’re a hard-boiled pragmatist, you may be wondering if these abstractions have any parallels in the real world.  Admittedly, life sometimes seems to play by different rules.  In conventional morality, two wrongs don’t make a right.  Likewise, double negatives don’t always amount to positives; they can make negatives more intense, as in “I can’t get no satisfaction.”  (Actually, languages can be very tricky in this respect.  The eminent linguistic philosopher J. L. Austin of Oxford once gave a lecture in which he asserted that there are many languages in which a double negative makes a positive, but none in which a double positive makes a negative — to which the Columbia philosopher Sidney Morgenbesser, sitting in the audience, sarcastically replied, “Yeah, yeah.”)