# What multiplication is … the central role of the distributive property

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

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

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

so

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

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

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 $m$ determines  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 addition  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.

### 6 Responses to "What multiplication is … the central role of the distributive property"

So, is multiplication of whole numbers students study different from multiplication of (non-negative) rationals they study in Grades 4-6?

I think a major reason Japanese textbooks introduce the multiplicative comparison (scaling?) soon after they introduce multiplication of whole numbers is that they want to use it to extend multiplication to rational numbers (non-positive first in elementary, then to negative in Grade 7). They really want students to understand (by the end of Grade 6 when they complete the study of multiplication/division of decimals and fractions) from a proportional perspective. Suppose we have 3 quantities such that A is to 1 (unit of B), C is to B. If we want to calculate C given A and B, we use multiplication, C = A x B. If we want to calculate B, given A and C, we use quotitive division, B = C div A. Finally, if we want to calculate A, given B and C, we use partitive division, A = C div B.

Clearly, my exposition here is anywhere near rigorous mathematically as you wrote. But, I just feel very uneasy thinking about whole number multiplication “is” repeated addition because that might mean that multiplication with rational number is a distinct operation from whole number multiplication.

Tad, I am so glad you made that last point. One way to unify the operations of addition and multiplication of whole numbers and rational numbers more generally is through viewing numbers as operators. I am hoping to have a post on this topic up shortly.

When I write that multiplication is repeated addition- which it is (ask a computer scientist!) – I do not mean that’s necessarily, or even, how we should think about it.

One of the issues with arithmetical operations is that while they are defined for natural numbers we use them in a particular notation system – the base 10 system. A lot of what we call addition and multiplication facts are just short cuts for operating in this base 10 system.