What multiplication is … the central role of the distributive property

Posted by: Gary Ernest Davis on: January 7, 2011

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Tad Wattanabe raised some interesting points in his comments on the post Multiplication of whole numbers, and how we think aboutÂ it.

Part of what he said is that in Japanese texts, authors distinguish what multiplication IS as distinct from HOW TO multiply.

The issue of – psychological – models of whole number multiplication and repeated addition seem to raise the ire, if not the blood pressure, of many mathematics educators.

In this post I want to do something a little different: I want to demonstrate that there is one, and only one, candidate for multiplication of whole numbers.

As part of that demonstration it will become patently clear that multiplication is indeed repeated addition, by which I mean, as do most school teachers of mathematics, addition of a fixed number several times over, (as in $4+4+4+4$ but NOT as in $1+2+3+4$ ). See more about this in the postscript at the bottom of this post.

This is a very elementary demonstration, of a sort that most mathematics undergraduates experience in their degree studies.

In this demonstration, not all details will be filled in completely, but all can be, relatively easily.

The mathematics

First some notation.

The symbol $\mathbb{Z}$ will denote the set of all integers – positive or negative – and $\mathbb{Z}\times\mathbb{Z}$ will denote the set of pairs $(a,b)$ of integers $a \textrm{ and } b$ .

A function $M: \mathbb{Z}\times\mathbb{Z}\to \mathbb{Z}$ is called bilinear if $M(a,b+c)=M(a,b)+M(a,c) \textrm{ and } M(a+b,c)=M(a,c)+M(b,c)$ for all integers $a,b,c$ .

I claim: there is one and only one bilinear function $M: \mathbb{Z}\times\mathbb{Z}\to \mathbb{Z}$ for which $M(1,1)=1$ .

Here’s how we can establish this claim:

(1) First we will establish there is only one such bilinear function, by showing how it is determined. So suppose that

$M: \mathbb{Z}\times\mathbb{Z}\to \mathbb{Z}$

is a bilinear function with $M(1,1)=1$ .

(2) For all integers $a$ we have

$M(a,0)=M(a,0+0)=M(a,0)+M(a,0)$

(by bilinearity) so $M(a,0) = 0$ .

Ditto, $M(0,a)=0$ for all integers $a$ .

(3) Let $a$ be a positive integer. Then

$M(a,1)=M((a-1)+1,1)=M(a-1)+M(1,1)=M(a-1,1)+1$

Similarly, if $a\geq 2$ , we have

$M(a,1)=M(a-1,1)+1=M((a-2)+1,1) +1 =$

$M(a-2,1)+M(1,1)+1=M(a-2,1)+2$

More generally (by induction) $M(a,1)=\overbrace{1+\dots +1 }^{a \textrm{ times }}=a$

(4) Similarly, for $a\geq 1 , M(1,a)=a$ .

(5) For $a,b\geq 1$ we have

$M(a,b)=M(a,(b-1)+1)=$

$M(a,b-1)+M(a,1)=M(a,b-1)+a$

If $b\geq 2$ we have

$M(a,b)=M(a,b-1)+a=M(a,(b-2)+1)+a=$

$M(a,b-2)+M(a,1)+a=M(a,b-2)+a+a$

More generally, (by induction) $M(a,b)=\overbrace {a+\ldots +a}^{b\textrm{ times }}$ .

(6) For all integers $a$ we have

$0=M(a,0)=M(a,1+(-1)) = M(a,1)+M(a,-1)$

$M(a,-1)=-M(a,1)$

Ditto, $M(-1,a)=-M(a,1)$ for all integers $a$ .

(7) Using (6) we can see readily that for $a,b\geq 1$

$M(-a,b)=-M(a,b)$

$M(a,-b)=-M(a,b)$

$M(-a,-b)=M(a,b)$ .

Hence $M(a,b)$ is uniquely determined by its values for positive $a,b$ as

$M(a,b)=\overbrace {a+\ldots +a }^{b\textrm { times }}$

(8) This establishes that there is at most one bilinear function

$M: \mathbb{Z}\times\mathbb{Z}\to \mathbb{Z}$

with $M(1,1)=1$ .

(9) To see that there is a bilinear function

$M: \mathbb{Z}\times\mathbb{Z}\to \mathbb{Z}$

with $M(1,1)=1$ we define $M(a,1):=a \textrm{ and } M(a,b+1):=M(a,b)+a$ for positive integers $a,b$ .

We extend the definition to general integers by defining

$M(a,0)=0=M(0,a)$

$M(a,-b)=M(-a,b)=-M(a,b)$

and

$M(-a,-b)=M(a,b)$

for positive integers $a,b$ .

It is usual to write $M(a,b)=a\times b$ .

This function is bilinear and $1\times 1 = 1$ .

(10) So there is indeed one and only one bilinear function

$M: \mathbb{Z}\times\mathbb{Z}\to \mathbb{Z}$

with $M(1,1)=1$ .

That’s what multiplication IS; it is a a bilinear function $M: \mathbb{Z}\times\mathbb{Z}\to \mathbb{Z}$ Â with $M(1,1)=1$ .

Moreover, we have shown that this unique bilinear function (normalized to give the value 1 at the point $(1,1)$ ) is just repeated addition, in the sense explained at the beginning of the post.

Some people have suggested that this inductively defined multiplication is not “really” multiplication; it is, they say, just a model of multiplication. This reminds me of Karl Gauss’s assertion that the complex numbers as defined in the Argand diagram did not capture his more general understanding of a complex manifold. That may well be so, since Gauss was a supremely gifted mathematician, yet no one since has ever found a need to, or indeed been able to, express an understanding of complex numbers other than that which is now standard in textbooks.

I suspect a similar situation holds for multiplication of integers.

Mental models

No one in their right minds imagines that is a way to tell school students what multiplication of whole numbers is.

So what do we tell them?

That, I feel, is at the heart of the arguments over multiplication as repeated addition (which it clearly is).

No one that I am aware of has a good cognitive root for multiplication of integers … with the possible and outstanding example of Peter Braunfeld and his ideas of stretchers and shrinkers.

But that’s a topic for another post (to follow…)

The distributive property

The bilinearity of multiplication is just the distributive property: $a\times (b+c)=a\times b+a\times c$ .

This, together with the property $1\times 1=1$ , uniquely determines multiplication for integers.

So the demonstration above tells us that the most central, critical, and indeed defining, property of multiplication for integers is the distributive property.

This is the point that teachers need to emphasize when dealing with multiplication of whole numbers, because it is what tells us, together with $1\times 1=1$ , both what multiplication is, and how to calculate with it.

There are traps for an unwary teacher in using the most common form of modeling the distributive property through rectangular arrays. The danger is that adult teachers imagine young children see arrays in the same way they do. There is strong evidence that this is not so.

I discussed the distributive property and problems with rectangular arrays at length here.

Postscript on repeated addition

The issue of repeated addition is a constant source of irritation for some people. What seems to be the source of that irritation, so far as I can tell, is that the bilinear operation of multiplication is not an operation of repeated addition: rather it corresponds to an infinite number of repeated additions, one for each positive integer.

That, I would have thought, is obvious from the definition of multiplication. Multiplication by a number $m$ is determined inductively – via the distributive property (that is, bilinearity) – as $m\times 1 =m, m\times(n+1)=m\times n+m$ . This is repeated addition of the number $m$ .

The common school statement of “multiplication is repeated addition” refers exactly to this situation: multiplication by a given integer corresponds to repeated addition by that integer. At worst, this is an abus de langage. One does not hear the phrase “muliplication is repeated addition” outside of school, at least not in my experience, and it is, as anyone who reflects on the matter will appreciate, a shorthand for what I wrote above.

School teachers, particularly elementary teachers, will often refer to subtraction as the inverse of addition.

Strictly speaking this is not correct: addition is a binary operation, giving a result $m+n$ for every pair of integers $m, n$ .

Subtraction, as a binary operation, or partial operation on positive integers, is not the inverse of addition as a binary operation.

Nonetheless, for every integer $m$ addition of $m$ does have an inverse which is subtraction of $m$ .

To put it another way, every integer $m$ determinesÂ a unary, not binary, operation $n\to m+n$ . This unary operation has as an inverse the unary operation $n\to n-m$ .

Similarly, every integer $m$ determines a unary, not binary, operation $n\to m\times n$ . That unary operation is indeed repeated addition of the number $m$

If this is the essence of the long standing feud in the mathematics education community over multiplication as repeated additionÂ then I think it is a rather silly feud, and could easily have been resolved through careful listening and a little flexibility of thought.

Elementary teachers, and a majority of middle and high school teachers, do not have the language of unary and binary operations, so they do not have that vehicle to express their thoughts. Mathematics educators, at least those with a degree in mathematics, should have such language to hand, and should be able to explain, in a productive and non-hectoring way, the connection between the two.