Accentuate the Negative

Last week we pondered the rules of multiplication, why 3 x 4 equals 4 x 3, and so on. It was tricky. Indeed, it was not even clear what “multiplication” means. Concrete notions such as “repeated addition” seem natural, but we saw that they can only take us so far.

We also left two questions unanswered: first, why does 3 x (-4) equal (-4) x 3; second, why does a negative times a negative result in a positive? We hope now to provide some answers. However, these questions are even trickier: negative numbers are from a different world.

Natural numbers (positive whole numbers) are, well, very natural. They’ve been around for ages, at least since cavemen were counting rocks, and maybe forever. Mathematician Leopold Kronecker famously declared: God made the natural numbers, all else is the work of Man.

Whether or not we can thank God for the natural numbers, the rest has certainly been a lot of work. For instance, it took a very long time to recognize zero, the absence of quantity, as being a number in its own right. About 2500 years ago, the Babylonians employed zero as a sort of placeholder, as we do today. For example, using our modern (Hindu-Arabic) symbolism, we employ 0 to distinguish the numbers 34 and 304; the 0 in the second number indicates “no tens”, and that the 3 should be interpreted as “three hundreds” rather than “three tens”. 

The Babylonian use of 0 in so-called positional notation was very sophisticated, superior to the numeral systems of the later Greek and Roman civilizations. However, using 0 as a placeholder is not the same as considering 0 as a bona fide number, with which we can perform arithmetic. That required a second conceptual leap.

It was around 600 AD that Indian mathematicians first tackled the arithmetic of 0. So, they declared that 3 x 0 = 0, and so on.

That may not seem so difficult, simply a vacuous version of repeated addition. However, hindsight makes it easy to be smug, to overlook the conceptual difficulties. For example, if 0 is just another number then we should be able to make sense of 3/0. The fact that one cannot (at least not easily) caused the Indian mathematicians, and subsequent centuries of schoolchildren, plenty of headaches.

These same Indian mathematicians (and earlier Chinese mathematicians) also contemplated negative numbers. The mathematician Brahmagupta wrote of “fortune” and “debt” as numerical quantities of equal stature, and he deduced the arithmetic rules for these quantities.

Of course, the number line makes much of this beautifully clear. All such numbers then appear on one scale, with increasing fortune (positive numbers) to the right, and increasing debt (negative numbers) to the left.

The number line makes the addition of positive and negative numbers very intuitive. It is not mysterious why 4 + (-4) should equal 0. But what about multiplication?

Last week we interpreted 3 x (-4) as (-4) + (-4) + (-4), and so amounting to -12. But what can we do with (-4) x 3? For this, we need a leap of faith.

Any time we conceive of new numbers, we have to decide or determine what arithmetic rules the new numbers will obey. One approach is to simply assume that the familiar rules still apply, and cross our fingers that nothing goes wrong.

For example, since a x b always equals b x a for natural numbers, we can hope to apply this rule for negative numbers as well. In effect, that means we simply declare (-4) x 3 to be equal to 3 x (-4), and so equal to -12 (with our fingers still crossed).

What about (-3) x (-4)? We first recall another of our rules for natural numbers: the distributive law. This law says, for example, that 3 x (2 + 2) equals (3 x 2) + (3 x 2). Why should this be true? Well, as we did last week, such quantities can be interpreted as rectangles of balls:

Does the distribute law also hold for negative numbers? We cross our fingers again, declare that it does, and hope nothing goes wrong.  Having done so, let’s consider the product

 (-3) x (4 + (-4))

On the one hand, this product must equal (-3) x 0, which in turn (by our previous finger-crossing) is the same as 0 x (-3), and so equals 0. On the other hand, we can apply the distributive law. Doing so, we find that

 [ (-3) x 4 ] + [ (-3) x (-4) ] = 0

However, we have already declared that (-3) x 4 = -12. So, the only way the above equation can hold true is if (-3) x (-4) = 12: a negative times a negative must equal a positive.

But does this all work? Perhaps if we go further, we’ll find that our rules in this new context lead us to 3 = 4, or some other such absurdity. In fact, such finger-crossing in mathematics is not always successful.

However, it turns out that for negative numbers it all works perfectly. It can be checked that the familiar rules of arithmetic continue to function exactly as one would wish, and no absurdities result. In mathematics, at least sometimes, it seems there is a God.

 

Puzzles to ponder: Did God make the natural numbers? Is everything else the work of Mankind?

 

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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