I was talking with my elementary-school-aged child this evening about arithmetic.
"What is 2 x 2?" I asked.
"4," he replied.
"What is 2 x -2," I asked.
"Negative 4," he said, reasoning correctly that it was adding up two "negative 2's".
Hoping to stump him, I followed up with "What is -2 x -2?"
"Negative 4", he repeated.
"Wait, this is different from what I asked you before!", I said, "It's not obvious what it means, right?"
"Oh yeah!" he agreed.
"So what is it?" he asked. We were walking outside, feeling quite chilly in the evening air.
"Well, I'll tell you, but you're not going to believe it", I replied, "it's 4!"
He thought about it for a bit, and smiled at the incongruity of the answer. "4?!"
I could sense the puzzlement lingering in him. I knew I had to say more.
An awful feeling grew slowly in my stomach as I struggled with what to say next. "Well, it's not easy for me to explain why though..." my voice trailed off as I started to think about it more carefully.
"... I mean, I mean, I mean, I can't just say consider adding -2, -2 times", I stammered out, as we entered the warm indoors.
I thought back to how I myself had learned this. A shudder went through me as it occurred to me that I was probably simply told by the teacher that "negative times negative is positive" and made to absorb it simply as a convenient fact about numbers.
Up to this point, over months of conversations, I have been explaining arithmetic to my son using familiar concepts and ideas. Natural numbers correspond to putting up fingers one by one. Adding them is an extension of the counting process. It is easy to illustrate with one's hands why 2+3 is the same as 3+2. The positive numbers line up nicely on a line. Subtracting a smaller number from a bigger number corresponds to removing the smaller number of objects from a set of the larger number of objects and counting what remains. It also corresponds to going backwards on the count, or on the number line.
These explanations pile nicely, building on each other. From the above, it is intuitive and therefore easy to explain to a child why subtracting a bigger number from a smaller number requires us to put some more numbers on the line before 0, and that we call these negative numbers. It is easy to explain also on intuitive grounds that though we don't encounter them when counting objects, this could sometimes be a useful way of thinking of numbers (such as when we consider scenarios involving borrow and lending money). Building on the analogy of debt, it's possible to explain intuitively why negative 2 plus negative 2 is negative 4 and then note that this is the same as negative 2 minus 2, so that adding a negative number is the same as subtracting the positive counterpart of that number. It can also be pointed thereafter that 3 - 2 is not the same as 2 - 3, so that in this respect subtraction is quite asymmetric. Multiplication can be introduced as a shorthand for iterative addition: that when we say "3 times 4," what we mean is to add the total number of objects in 3 collections of 4 objects each. Pictorially, through rows and columns of objects, it can then be explained why "3 times 4" is the same as "4 times 3". And then, building on the prior understanding of negative numbers, it is not a stretch to explain to a child what it means to talk about "4 times -2", and how this is the same as "-4 times 2".
But "-2 times -2" is a qualitatively different animal. It does not follow directly from the above explanations. Because iterating a process a negative number of times is not meaningful, the earlier intuitive understanding of multiplication does not readily generalize*.
Determined to keep things intuitive, I put down my bag, took off my jacket and sat right down in front of the computer. I searched the web for an explanation suitable for a child. I found one immediately at
|("common sense" explanation from http://www.mathsisfun.com/multiplying-negatives.html)|
Reading this worthless bit of chicanery made me feel physically sick. It is insulting the intelligence of a child to even try and explain a mathematical concept through unrelated verbal shenanigans.
To be fair to that site, this junk is soon followed up by an alternative explanation, that is not bad. This second explanation equates -2 x 3 to walking backwards two steps on the number line three times, and 2 x -3 to turning around first (so that one is facing backwards) and then walking forward two steps three times. Finally, -2 x -3 is explained as a combination of these, turning around first, then walking backwards two steps three times, ending up at the right answer of 6.
As explanations go, this is more plausible. At least it builds a clean map between the operations in question and a physical process in such a way that the outcome in the physical process is consistent with the arithmetic operation. In this, it is in keeping with all the previous intuitive explanations I myself had been offering my child. But I couldn't help feeling that the analogy is still rather forced and somewhat lacking in elegance. Why should the negative in the first number correspond to walking backwards and the negative sign on the second number correspond to turning around?
One problem is that all of the above "intuitive explanations" are not rigorous. They are simply suggestive analogies. Because they are grounded in experience, they offer a scaffolding, helping children slowly build familiarity with numbers and what can be done with them. But this approach has its limitations.
Mathematics in its pure form is in fact not a description of our world as it really is. It is fundamentally about abstractions, axioms and deductive inference. We are fortunate that pure mathematics can in fact be applied; that elegant logical, axiomatic reasoning about abstract objects turn out to be useful in solving real-world problems in science, engineering and economics, once a connection can be established through the art of mathematical modeling.
I want to give my child some feel for this, pure mathematics, so I outlined for him a solution based on the following.
We use three statements that he currently understands and accepts on intuitive grounds:
A. a + (-a) = 0.
B. a x (b + c) = (a x b) + (a x c)
C. -a x b = - (a x b)
The explanation proceeds as follows:
- Start with the expression -2 x (-2 + 2). Since what is in the paranthesis is 0 (from A), we have that
- -2 x (-2 + 2) = -2 x 0 = 0 ---- (1)
- But also, from B, we have that
- -2 x (-2 + 2) = (-2 x -2) + (-2 x 2)
- And from C, we have that (-2 x 2) = -4. This yields that
- -2 x (-2 + 2) = (-2 x -2) + (-4) = 0, the last equality following from (1)
- From A, we then get that the term (-2 x -2) must be 4.
When we finished, I could tell him that "pure mathematics is all about axioms that you start with and the things you can prove from them." Though the hour was late, I think, I hope, it left some impression.
Graeme McRae has a nice page with a very accessible discussion of the axioms of integer arithmetic, that presents a minimal set of axioms from which all the familiar rules of arithmetic with integers can be derived. I am thinking this could be a good resource for a deeper discussion about integer arithmetic and axiomatic reasoning in the future.
Technically, the matter is actually a bit more complicated and interesting, it turns out. The axioms by McRae presented in the above link define what is called an Integral Domain, which is in modern algebra terms a commutative ring without zero divisors. Treating them as an Integral Domain is sufficient to derive most familiar rules of arithmetic with integers, but in fact it is not enough to characterize them completely and uniquely; what is needed in addition are some axioms pertaining to the strict ordering of integers (less-than), and one pertaining to Induction. A nice explanation of these technicalities is provided in these excellent notes on Axiomatizing Integer Arithmetic from a senior-level Applied Logic computer science course at Cornell.
We are so close to the heart of one the greatest (albeit negative) findings in pure mathematics and logic that I cannot resist mentioning here it at least in passing. Kurt Goedel in 1931 proved that in fact there are no set of axioms that can be used to prove all the properties of natural numbers (known as the Incompleteness Theorem). This doesn't mean that the axiomatic approach is totally useless, but shows that it has its limitations. His finding famously undermined Bertrand Russell and Alfred North Whitehead's ambitious joint attempt to put on firm axiomatic grounding all of mathematics in their Principia Mathematica.
Though I was exposed to Euclid's axioms of geometry and the idea of proving theorems as early as middle school, I think I only truly got what it's all about for the first time when I first took a senior-level undergraduate math course providing an introduction to Real Analysis, at Cornell. I remember one day early in the course working on a homework problem, struggling to deduce some seemingly trivial property of elementary integer arithmetic, and saying to myself, "how could I have gotten this far without understanding that all these 'facts' I took for granted can in fact be proved from much simpler axioms?!" One reason it took me so long to figure this out is that virtually all of the mathematics taught to engineers is from the perspective of "here are some useful facts and techniques that you can use to model real systems and analyze those models," and axiomatic development is consistently under-emphasized. This is perhaps as it should be, since abstract axiomatic development tends to be slow, painful, and of relatively little utility when it comes to applied mathematics, but it's a pity nonetheless.
*Update: Marc in a comment below points out a Khan academy video that gives a great intuitive explanation of the multiplication of negative numbers. In fact, if one models multiplying by a negative number as iterative subtraction, one can build an understanding of why -2 x -2 is 4 using intuition alone. I wish I had seen this sooner, but then I would have missed out on the opportunity to discuss with my child how deductive reasoning works.