Defining Multiplication

Date: 08/16/2002 at 19:22:52
From: Cindy Miller
Subject: Defining multiplication

I just read an article in Phi Delta Kappan (Feb. 2001) by Deirdre 
Dempsey and John Marshall titled "Dear Verity: Why Are All the 
Dictionaries Wrong?" They state that Euclid and some dictionaries 
define multiplication equations like 3x4 to mean "the number 3 four 
times." I was taught, I teach, and math textbooks say that 3x4 means 
3 lots/groups of 4. An array would look like 

   ****
   ****
   ****

Because multiplication is commutative, it may not matter symbolically; 
however, as the authors mentioned in the article, taking 4 pills a day 
for 21 days is a lot different from taking 21 pills a day for 4 days. 
If I want my students to know what multiplication "is," what is the 
correct explanation?

Date: 08/16/2002 at 23:41:38
From: Doctor Peterson
Subject: Re: Defining multiplication

Hi, Cindy.

Here is an answer I gave to a related question:

   Multiplicand, Multiplier
   http://mathforum.org/library/drmath/view/58567.html 

Your question is really not a math question, but a linguistic 
question, even though it involves symbols. Since, as you recognize, 
3*4 and 4*3 have the same value, there is no mathematical value in 
the distinction. You are only asking about the underlying image when 
we say "three times four" in English. (It may have been different in 
Euclid's Greek.) And I submit that it can be reasonably interpreted 
either way.

My first thought when analyzing the phrase is that originally '3 
times' meant to repeat what follows three times; that is, it would 
mean the same as '4, taken 3 times'. So 3 times 4 means 4+4+4.

But that seems awkward, considering the way we tend to say the phrase 
now. We might read through a calculation, saying "3, times 4 is 12, 
plus 2 is 14, ..." where each operation acts on the first number. 
we're really saying "3 multiplied by 4." Taken that way, we start 
with 3 and multiply it by 4, meaning that we repeat it 4 times. So 
3*4 = 3+3+3+3.

The difference here is entirely in the grammar: is '3 times' an 
adjective phrase modifying '4', or is 'times' a preposition 
equivalent to 'multiplied by', and 'times 4' a modifier of '3'?

When we introduce children to multiplication, it's reasonable to 
start with one specific meaning, just so they have a concrete image 
to start with. But we can almost immediately point out that

    ****
    ****
    ****

can be seen equally well as 3 rows of 4, or as 4 columns of 3. Once 
you've drawn the figure, or once you've written 3*4, no one can 
really tell which interpretation you meant. The distinction has been 
abstracted out of the problem.

And that is a good thing, not bad: in mathematics the ability to work 
abstractly and forget what the concrete problem was is a major 
advantage, because we can ignore details that don't affect the 
result, and rearrange the work to make it easier. If I took 4 pills a 
day for 21 days, and now I want ONLY to know how many pills I took, I 
can call it 4*21 (I just put the numbers in the order I saw them, 
without thinking about which is the multiplicand) and re-model the 
problem as 4 sets of 21, allowing me to add 21+21+21+21, which is 
easier than adding 21 4's. I can switch freely among different 
models, because I know that the outcome is equivalent. So I never 
bother to define whether 4*21 means 4 groups of 21 or 21 groups of 4, 
because that distinction would only be a hindrance. And if I were 
doing a calculation for which it made a difference whether I took 4 
or 21 pills a day, such as finding the concentration of medication in 
my blood, I wouldn't be multiplying, because that would be the wrong 
abstraction!

So "what is multiplication?" It is a commutative operation that can 
be modeled in two symmetrical ways as repeated addition (when applied 
to whole numbers).

Does that make sense?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 08/16/2002 at 23:14:46
From: Doctor Tom
Subject: Re: Defining multiplication

My understanding is that multiplication of natural numbers is defined 
as follows (and some stuff has to be proven as we go along - I'll just 
state the results).

0 = {} (zero is the empty set)
1 = {0} (the set containing zero)
2 = {0, 1}
3 = {0, 1, 2}
4 = {0, 1, 2, 3}
et cetera. This set is defined by the "axiom of infinity" in the 
Zarmelo-Frankel set theory.

Then you need to show that if m and n are two of the natural numbers 
above and there is a 1-1 mapping between them, they are identical.

Then define m x n (multiplication) as follows:

The ordered pair (a, b) is {{a}, {a, b}}

If S and T are two sets, then:

S X T = {(s, t) : s is in S and t is in T}

(I used upper-case X for the cross product above.)

To calculate m x n, construct the set m X n (this is just a set), and 
then find the unique natural number that can be mapped 1-1 onto this 
set.

Thus, if m = 2 and n = 3

m = {0, 1}, n = {0, 1, 2}

m X n = {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2)}

Define f mapping m X n to 6  = {0, 1, 2, 3, 4, 5} as follows:

f: (0, 0) -> 0
   (0, 1) -> 1
   (0, 2) -> 2
   (1, 0) -> 3
   (1, 1) -> 4
   (1, 2) -> 5

Show f is 1-1 and onto and you've proved that 2x3 = 6.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/