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Multiplication's Multiple Meanings: Unified by the Distributive

Date: 01/01/2015 at 16:11:08
From: Alonzo
Subject: What are the unique and common threads for Multiplication

Hello, 

I am a father of two young boys and I look forward to exploring
mathematics with them for as long as they will let me :-).

I would really like for them to have a deeper understanding of mathematics
than what I had when I was a young student; and as I think about how I
might approach some of the topics, there is one that remains particularly
unclear to me to this day: the multiplication operation. Now, I do not
have a strong background in mathematics (e.g., never had a course in
abstract algebra), so please forgive me if some things that I say are 
off -- maybe even way off.

I have seen debates online as to what multiplication is, and how to teach
it to students. From my perspective, I am most confused by the many
definitions for the multiplication operation depending on the type of
objects in question (real numbers, complex numbers, matrices, etc). I
always think to myself, "Why would mathematics allow the same name to be
associated with multiple definitions?" It seems like there must be
something that all the definitions must have in common. Surely, not just
any binary operation on a set of objects can be labeled multiplication on
a whim ... or can it?

There does not seem to be a solid answer that is agreed upon. Often,
debates turn into interpretations of multiplication (e.g., repeated
addition, scaling, etc.), but the discussion from this approach seems to
be fruitless. Other times, properties of multiplication are discussed, but
often the properties are the same as those found under different types of
operations. Integer multiplication may be associative, but so is integer
addition -- leaving me no more informed about the unique and universal
thread for the concept of multiplication.

So this is my question: is there a characterization of the multiplication
operation that 1) holds true for all operations labeled multiplication; 
2) is agreed on within the academic community; and 3) is unique enough to
be able to distinguish it from other operations (namely addition)? If so,
please do share. And if not, how would you explain why the same term has
various definitions in mathematics to students learning about operations
like multiplication?

From my limited mathematical knowledge, it appears that the only thing in
common among different definitions of multiplication on different objects
is that they all rely on the use of the addition operation in their
construction. So perhaps the term addition is used to reference an
operation for a set of objects that is considered to be the simplest
method for combining/connecting two objects in a set, and multiplication
is a more complex method for doing so (perhaps based on the use of simpler
operations, like addition, already defined for the set). But I would
prefer that my discussion with my sons not rely on such experience. Hence,
the reason for this question.

Many thanks for taking the time to review my write up, and I look forward
to any insight that may be offered.



Date: 01/02/2015 at 18:51:08
From: Doctor Peterson
Subject: Re: What are the unique and common threads for Multiplication

Hi, Alonzo.

As I see it, most of the definitions you found ARE related.

Multiplication is (almost) always the second operation, and it is its
relation to the first operation of addition that makes it what it is.

There is also one key feature that all "multiplications" have, and which
may even be thought of as a defining feature: the distributive property.

It all starts with the definition of multiplication for natural numbers.
In that context, multiplication is repeated addition. We can choose to
define it either as ...

   m * n = m + m + ... + m
           \_____________/
               n times

... or as

   m * n = n + n + ... + n
           \_____________/
               m times

We quickly learn that both definitions give the same results, so it
doesn't matter which we start with. This is the commutative property; and
we go on to discover the associative and distributive properties. We can
also develop additional models of multiplication, such as rectangular
areas and scaling. These will be useful in the next step, and help keep us
from being too dependent on that initial definition, which we will soon be
leaving behind.

Now, as we extend the meaning of "number" beyond the natural numbers, we
systematically generalize the definition of multiplication to apply to
each new kind of numbers. First, we include 0:

   m * n = 0 + m + m + ... + m
               \_____________/
                   n times

Here, m*0 and 0*n are defined (as 0). Then we include negative integers in
such a way as to retain the commutative, associative, and distributive
properties, by defining -1 * x = -x and -1 * -1 = 1, and applying
properties as needed, so that for example,

   x * -y = x * (-1 * y) 
          = (-1 * x) * y 
          = -1 * (x * y) 
          = -(xy)

We can also make some sense of this as a broader kind of "repeated
addition," but that starts to get fuzzy, and the other models of what
multiplication "means" become more important.

Then we extend to rational numbers by defining

    w     y    w * y
   --- * --- = -----
    x     z    x * z

This again is the only extension that will retain the properties we've
become accustomed to.

Next we extend this to all real numbers, including the irrationals, by
more elaborate methods I won't go into.

Finally, we extend to all complex numbers, by defining

   (a + bi)*(c + di) = (ac - bd) + (ad + bc)i

This is, once again, required if we are to retain the basic properties.

I'm not certain of this, but I think it is possible to say that
multiplication as we have defined it through this whole process is the
only operation that would distribute over addition. For example, if all we
know is that x(y + z) = xy + xz, we can show that it must be equivalent to
repeated addition when x is a natural number, because

   x * n = x(1 + ... + 1) = x*1 + ... + x*1 = x + ... + x
             \_________/    \_____________/   \_________/
               n times          n times         n times

So we could define multiplication as "the operation that distributes over
addition." (That's not a good way to start out in kindergarten, though!)

The other characterizations of multiplication (area, scaling, ...) all
continue to apply to any kind of number for which they make sense -- that
is, all real numbers. When we move on to complex numbers, those models
cease to be meaningful, so it is the properties themselves that make it
meaningful. We can say, for example, that complex numbers are solutions of
polynomials. Multiplication takes on a new wrinkle here, involving both
scaling and rotation!

For matrices, there are several reasons why the way we define both scalar
and matrix multiplication are useful; but quite likely one reason we call
them multiplication is that they, too, distribute over addition. The same
is true of scalar and vector multiplication of vectors.

Then we can move on to abstract algebra. Abstract algebra is essentially
about taking operations, reducing them to their basic properties, and then
asking "what if" questions, such as 

   What if the things being added are not numbers but ...? 
   What could we prove if we only used these properties and nothing else? 
   Is there anything else that has these properties?

In group theory, the operation (there is only one) can be called either
addition or multiplication, largely based on what notation we choose to
use, which may derive from which operation inspired our thinking. But when
we go to two operations (in rings and fields, for example), it is again
the distributive property that takes center stage. We are generalizing the
idea of multiplication beyond real numbers, looking only at the properties
(though by this time -- in fact ever since we multiplied matrices -- we
have found that the commutative property had to be left optional).

Bottom line: these two things (systematic extension, and retaining the
distributive property) are what tie all different kinds of
"multiplication" together, so that they all deserve the same name.

And for just starting out: while ultimately you want to explore other
models and think about how multiplication works in connection with the
other operations, I see nothing wrong with starting with the idea of
repeated addition. If you have the idea of generalization from the start,
and can see, for example, why scaling is really the same idea as repeated
addition but more general, you'll be headed in the right direction.

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


Date: 01/03/2015 at 00:03:25
From: Alonzo
Subject: What are the unique and common threads for Multiplication

Doctor Peterson, many thanks for the very thorough response. 

It sounds like there are a set of properties that have evolved into the
standard preferences, presumably due to their usefulness, for an algebraic
structure's multiplication operation (if one is claimed). And while some
of these properties may not be required for a multiplication operation, it
appears that the pattern of multiplication's distribution over addition is
inherently tied to the nature of those observations that are labeled as
"multiplication." Is this a fair understanding of your response?

There is something you stated that I would like to unpack a little:

> We are generalizing the idea of multiplication beyond real numbers,
> looking only at the properties (though by this time -- in fact ever since
> we multiplied matrices -- we have found that the commutative property had
> to be left optional).

I am curious as to why it is that the commutative property had to be left
optional. Is it because the distributive property alone is sufficient for
being considered multiplication, regardless of what other properties it
may or may not possess? So in the case of matrices, although it is not
always commutative, it can be labeled multiplication because there is a
matrix operation that demonstrates the distributive property?

I look forward to your thoughts. And again, I appreciate your time.



Date: 01/03/2015 at 10:54:17
From: Doctor Peterson
Subject: Re: What are the unique and common threads for Multiplication

Hi, Alonzo.

Yes, I think your interpretation of my response captures its main idea.

> I am curious as to why it is that the commutative property had to be left
> optional. Is it because the distributive property alone is sufficient for
> being considered multiplication, regardless of what other properties it
> may or may not possess? So in the case of matrices, although it is not
> always commutative, it can be labeled multiplication because there is a
> matrix operation that demonstrates the distributive property?
> 
> I look forward to your thoughts. And again, I appreciate your time.

Keep in mind that my discussion of this is just an attempt to analyze
after the fact something that has not necessarily been done consciously. I
don't know that anyone has ever specifically stated "We will call an
operation multiplication if and only if it distributes over addition." We
have simply recognized that it made sense to call something
"multiplication" for a variety of reasons, and then adjusted our thinking
about what that means to accommodate it.

When matrix multiplication was defined, it just made sense to call it that
even though it was recognized that the result is not commutative. Other
operations that have something in common with multiplication also turn out
to be non-commutative -- rotations in space, for example, and composition
of functions (both of which are related directly to matrix multiplication
in particular). So it has simply been recognized that commutativity is an
optional property of an operation, and the fact that an operation is not
commutative is not a major problem. This is not specific to
multiplication.

I'm not an expert in the history of either abstract algebra or matrices,
but a quick check suggests that matrix multiplication was first identified
as an abstract operation, and called multiplication, in the 1840's and
1850's, when Eisenstein and Cayley recognized that these operations had a
set of properties analogous to those of arithmetic in every way except for
commutativity. With these definitions, it was possible to perform
algebraic manipulations; and in fact the matrix then satisfies its own
characteristic equation, which is initially thought of as an equation
satisfied by real numbers (the eigenvalues). This makes the parallel
between matrix and real number operations quite secure, so that the
operation fully deserves the name "multiplication."

This illustrates the idea that various kinds of "multiplication" arise
either by direct extension of natural number arithmetic to larger sets
(and I should mention that the real numbers can be thought of as a subset
of matrices, by taking the real number k to correspond to the matrix kI,
where I is the identity matrix), or by parallelism in terms of properties.
When, in either sense or both, a new operation on a new kind of entity
"looks like" the multiplication we are familiar with, we will call it that
-- and, though we may not be aware of it, my sense is that the
distributive property is the most important way in which such a
parallelism is seen.

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




Date: 01/03/2015 at 21:37:26
From: Alonzo
Subject: Thank you (What are the unique and common threads for Multiplication)

Doctor Peterson, your patience with my questions has been much
appreciated. You have given me some useful data points that I can
reference as I take another look at the literature.

I find this type of support from the academic community to be a wonderful
service that I am hoping will still be accessible to my children as they
grow older and have questions of their own.

Another positive is that the automated information in the e-mail thread
made me aware that I can make donations to this service. Well, for
discussions like what you have shared with me, I believe it to be a small
exchange for a lifetime of enlightenment.

Thanks again for your insight and time.
Associated Topics:
Elementary Definitions
Elementary Multiplication
Elementary Number Sense/About Numbers
Middle School About Math
Middle School Definitions
Middle School Number Sense/About Numbers

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