What is Multiplication … Really?

I want to close out this series on multiplication with a very different kind of question. We have seen that multiplication of natural numbers can be modeled as a repeated sum of the multiplicand, taken the number of times indicated by the multiplier; and that the terms “multiplier” and “multiplicand” reflect only this model, not the operation itself or how it is written (that is, which comes first). But that’s just how it is first introduced to children. When you get to higher levels, you find that you can multiply not only natural numbers, fractions, and radicals, but also complex numbers, vectors, and matrices. What makes all those more advanced operations “multiplication” as well? It’s a lot bigger than it looked from the outside.

Why are so many things called multiplication?

Here is the question, from 2015, which I will break into two parts so I can catch my breath:

Multiplication's Multiple Meanings: Unified by the Distributive

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.

He is referring to debates starting in 2008 as to whether “multiplication is not repeated addition” — that is, whether it should be introduced as we have been discussing so far, or in an entirely different way. As we’ll see, I (as someone who is not an elementary teacher) lean toward sticking with the repeated addition idea as an initial model, but using other models as well. I don’t find anything in our archives responding to that debate, though we did touch on it in a few unarchived discussions; and quite often a student has introduced a question about, say, multiplication of negative numbers, by saying “I know multiplication is just repeated addition”, which we then have to correct!

But Alonzo’s question is not really about that specifically, but about what makes all the higher versions of multiplication worthy of having the same name. What makes multiplication, multiplication?

Alonzo continues:

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.

I think his last paragraph is pretty close to a reasonable explanation, but his question triggered enough thoughts that I said more in my answer. First, the basics:

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.

I think I said almost always because there are cases, like “groups”, where there is only one operation, but it is traditionally thought of as a multiplication, though it is sometimes written as addition.

Next, I quickly ran through how the meaning of multiplication is extended from natural numbers, all the way up to the complex numbers, step by step. At each point, it is the same operation on the “old” numbers, which are a subset of the new numbers, but now applies to “new” numbers as well. It makes perfect sense to still call such an extension by the same name:

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.

Now I make a conjecture:

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!)

What I’m suggesting here is that, for any of the sets of numbers we’ve considered, we could define multiplication entirely by the distributive property (assuming addition has already been defined), and that might be a sufficient definition. If I am right, then this is, as I suggest, the defining property of multiplication throughout the complex numbers.

On the other hand, although we can wave our hands enough to claim that “repeated addition” in some stretched sense still works even for real numbers, other ways of describing it fit better until we reach the complex numbers:

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!

I like to point out that multiplication of a complex number by a real number scales it, while multiplication by an imaginary number rotates it (as, in fact, multiplication by a negative number also does). The models we use have broadened, but still apply so far.

When we move beyond numbers of any kind, none of those definitions make any sense, but we still have the distributive property:

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).

I’m not sure I could prove that the only operations we could define in each case that distributes over addition are the multiplications that have been defined; but it seems likely.

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.

Alonzo responded with a couple more questions:

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?

My reply:

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

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.

My aside on eigenvalues is particularly interesting. If you know something about matrices, but not about eigenvalues, the brief idea is that for a matrix A we can form an equation like \(\lambda^2 – 5\lambda – 2 = 0\), whose solutions are called eigenvalues of A; and according to the Cayley-Hamilton theorem, A itself satisfies this equation: \(\mathbf{A}^2 – 5\mathbf{A} – 2 \mathbf{I} = \mathbf{0}\). This makes a direct connection between multiplication of matrices and multiplication of numbers. If we didn’t call the former multiplication, we could never say this! Whether this is tied to my hypothesis about the distributive property might be interesting to explore.