Multiple Problems

Today’s topic is really simple: arithmetic! Well, …

OK, the times tables and mental arithmetic may not be as familiar as they once were. (And what could possibly be the causes?)  Still, we’re all reasonably comfortable with the basic rules, right? We all know that 3 + 4 = 4 + 3, and so forth.  

Perhaps we’re actually a little too comfortable with such rules, and in this column we want to take a closer look.  Even the rules for addition are well worth pondering, but we’ll focus upon multiplication.

Multiplication of natural numbers (that is, positive whole numbers) is usually introduced and thought of as repeated addition. For example, 3 x 4 is commonly expressed as “three fours”, and is then understood as the number of objects in three groups of 4, or 4 + 4 + 4. This explanation of multiplication may seem very natural, but it raises some puzzling questions.

Notice that in the product 3 x 4, the 3 and the 4 play fundamentally different roles. That prompts our first question: why is it that the order of the factors doesn’t matter, that a x b will always equal b x a? Is it really so obvious that three groups of 4 will amount to exactly the same number of objects as four groups of 3? 

To pose our second question, we’ll take a further step and consider products of three numbers, such as 3 x 4 x 5. What do we mean by such a product? Well, we can make sense of a triple product by inserting brackets: so, 3 x (4 x 5) = 3 x 20 = 60, or alternatively (3 x 4) x 5 = 12 x 5 = 60. But why can we be so blasé about where to place the brackets? How could we be sure that the two answers were going to be equal?

For our third question, we delve into the world of negative numbers. It seems that we can still use repeated addition to make sense of a product such as 3 x (-4): it’s presumably just (-4) + (-4) + (-4), summing to -12. But what about (-4) x 3? What is that even supposed to mean?

Our fourth and final question is concerned with the classic mantra: “A negative times a negative is a positive.” But why? Why, for example, is (-3) x (-4) = 12?

So much for the rules of arithmetic being easy. How do we explain all this? Clearly, it is insufficient to simply chant “repeated addition”.

Below, we ponder the question of what “multiplication” might actually be, but we’ll first offer some intuitive answers to our first two questions. To begin, to see why 3 x 4 should equal 4 x 3, we first shepherd our objects into a rectangular formation:

Next, we slice the rectangle in two different ways: 

So, since three rows of four balls is the same as four columns of three, it follows that 3 x 4 equals 4 x 3. Easy!

Next, we consider why 3 x (4 x 5) should equal (3 x 4) x 5. Building upon the rectangle above, we can stack layers of balls into a box formation:

Now, 3 x (4 x 5) can be thought of as three layers of (4 x 5) balls each. This is exactly our box as pictured on the left below.

What about (3 x 4) x 5? Well, we already know that we can reverse the order of multiplication, so that (3 x 4) x 5 will equal 5 x (3 x 4). Then, this latter product can be thought of as five layers of (3 x 4) balls; this is again our box, as pictured in the middle diagram above. So, since our box represents both 3 x (4 x 5) and (3 x 4) x 5, the two products must be equal.

We can of course multiply 3, 4 and 5 together in many different orders. The fact that all such products will be equal can be demonstrated by repeatedly applying our two rules above. Alternatively, all the products must be equal because they can all be represented by the same box. For example, the product 4 x (5 x 3) is represented above by the rightmost slicing of the box.

The geometric approach above is beautifully intuitive. However, it is difficult to see how our diagrams could be adapted for negative numbers. And, the fact that we can indeed multiply negative numbers suggests that there is much more to multiplication than the geometry above. In any case, we’ll return to the pondering of negative numbers next week, when we’ll also provide answers to our third and fourth questions above.

But let’s return to the underlying question: what is multiplication? One person to have given a very firm answer to this question is mathematician and maths populariser Keith Devlin. In a forthright column in 2008, Devlin declared that, whatever multiplication is, “It ain’t no repeated addition”. Following the large, mostly angry response, Devlin vigorously defended his stance (here, then here and here.)

One of Devlin’s main points is, even if for natural numbers repeated addition gives the same answers as multiplication, that does not mean that multiplication is repeated addition. And it definitely seems that we need more than repeated addition to understand how multiplication really works. Moreover, once we leave the familiar world of natural numbers, the concept of repeated addition is totally inadequate: “repeating” is, in its very nature, a natural number process.

So, your maths masters agree with Devlin: multiplication is not repeated addition. Not even for natural numbers.

Of course, how multiplication is ultimately defined is a very different question from deciding how to introduce multiplication to young students. Perhaps it’s fine to begin with the notion of repeated addition, with the view that the notion will later be corrected. That’s a little worrying: it’s seldom a good idea to begin with untruths. However, and though Devlin suggests otherwise, we’re not convinced there’s a better alternative.

But if multiplication is not actually repeated addition, and it’s not geometry, what then is it? Well, we feel like the computer Deep Thought in The Hitchiker’s Guide to the Galaxy. We have an answer for you, but we don’t think you’re going to like it.

The answer, to the fundamental question, as to what multiplication really is, is .................. 7 x 6 equals .................. 7 x 6.

We told you you weren’t going to like it.

 
Puzzles to Ponder: What is addition? Why is 3 + 4 = 4 + 3? What does 1/2 x 4 mean? What does √2 x 4 mean?

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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