We have received many questions over the years about the meaning of multiplication. When we multiply \(2\times 3\), what are we really doing? This can confuse not only students and their parents, but also teachers. The next couple posts will deal with various aspects of this question.

## Is the multiplier on the left or on the right?

First we have questions like this one, from 2001:

Multiplicand, Multiplier Greetings from the Philippines! Do you agree with me that in the following mathematical sentence 456 x 10, 456 is called themultiplicandand 10 is called themultiplier? Why does the Encyclopaedia Britannica define them in the other way - 456 x 10, (456 is the multiplier and 10 is the multiplicand)? Some parents insist the one written in the Encyclopaedia Britannica is the correct one; no matter how much I explain to them, they refuse to accept. Since there is what we know ascommutative property of multiplication,why are they insisting all the textbooks and the teachers are committing a mistake?

It is amazing what issues can inflame people. I agreed with Christopher in part, but took an even more “relativist” stance:

To my mind,it makes no difference at allwhich is which. In fact, today it is more common to call them both "factors" and not make such a distinction. I wouldn't fight over this, on either side. I recently saw a facsimile of a 19th-century text that defined the multiplier as the SMALLER of the two numbers, regardless of the order. So there's yet a third definition to use.

So now we have three ways to take it:

Christopher: Multiplier is first: \(\text{multiplicand} \times \text{multiplier}\)

Britannica: Multiplier is second: \(\text{multiplier} \times \text{multiplicand}\)

Old text: Multiplier is smaller, multiplicand is larger

As we’ll eventually see, there is a sense in which I favor that last one, yet I disagree with all of them in another sense.

Really, the only distinction that can be made relates to the meaning in a givenapplication: thenumber you start with(say, the size of each of several groups) is the multiplicand ("thing to be multiplied" in Latin), and the one being thought of as thenumber of groups, by which the original number is multiplied, is the multiplier. I would tend to read 456 x 10 as "456 ten's," giving me Britannica's definition; but I can also see it as "456, multiplied by 10," giving me your definition. If I write it as 456 x 10 ---- I see 10 as the multiplier, because in the usual process of multiplying, I multiply each digit of 456 BY a digit of 10. I'm operating on the 456, using the 10. Even then, I'm not sure that means anything. But I suspect this is the reason for calling the smaller number the multiplier, because it is easier to use the smaller number on the bottom (or to add that many of the larger number).

This is my key idea: the distinction really only applies to the application of multiplication, and possibly to the technique of multiplication, not to the meaning of the expression we write.

I went on to look up what Britannica actually says (always a good idea when you are told what someone says), and found that what they said was not stated as a rule, but arose from their context:

You can see that their usage depends on their definition of multiplication: From the above laws, it is evident that a repeated sum such as 5 + 5 + 5 is independent of the way in which the summands are grouped and is written 3 x 5. Thus, a second binary operation called multiplication is defined. The number 5 is called the multiplicand; the number 3, which denotes the number of summands, is called the multiplier; and the result 3 x 5 is called the product. Since they take the 3 as the number of 5's, it must be the multiplier. Again, the distinction lies only in the assumed meaning of the multiplication. When a multiplication problem is given abstractly, there is no such distinction, so we prefer to use the symmetrical term "factor."

I don’t recall what the old book was that I had seen; just now I did a search of Google books and found none that say *only* that the multiplier is the smaller number, but many that say something like what follows. A particularly nice example is White’s A New Complete Arithmetic: Uniting Oral and Written Exercises (1897), which says the following on page 22:

The

Multiplicandis the number taken or multiplied.

TheMultiplieris the number denoting how many times the multiplicand is taken. …The product may be obtained by adding the multiplicand to itself

as many times less oneas there are units in the multiplier, and hence multiplication is a short method of finding the sum of several equal numbers. …When the sign [×] is placed between two numbers, it shows that they are to be multiplied together; and, since the order of the factors does not affect the product,

either number may be made the multiplier. The multiplier is usually written after the sign, which is then read; when the multiplier is written before the sign, it is readmultiplied by. …timesThe multiplier must always be regarded as an abstract number.

The multiplicand and product are like numbers, and may be either concrete or abstract.

When one of the factors is concrete, the concrete number is the

truemultiplicand, butwhen it is the smaller number, it may be used.abstractlyas the multiplier

I love this! First, multiplicand and multiplier are defined only in terms of the application or (later) the algorithm. Second, where today people often describe \(2+2+2\) as “adding 2 to itself 3 times”, this author sees as I do that there are only *two* additions involved there, so he says “as many times less one”. (I prefer to say either that we are adding together 3 copies of the 2, or that we start with 0 and add 2, 3 times.) Third, he emphasizes that *either* number in the written form may be thought of as the multiplier; he distinguishes the two by the way it is read (“multiplied by” or “times”), as I suggested above and will be discussing later. Finally, he brings in the idea of concrete numbers (numbers with units), which we will look at below. In essence, this determines which number is the multiplier in the meaning of the problem; but he points out that we can carry out the multiplication abstractly, and take the most convenient number (the smaller one) as the multiplier.

I don’t know that the pedagogy of this presentation is the best, and the mathematics differs from what I am used to, but the handling of words is excellent!

## Is the multiplier on the top or on the bottom?

The next question is from 2002:

Groups in Multiplication This may seem like a ridiculous question, but the resources I've used to determine the answer contradict each other. I would like to knowwhen multiplying factors vertically, which factor represents the number of groups?Is it the top factor or the bottom factor next to the multiplication symbol? Some resources show it as being read from the top down, and others from the bottom up. For example, when multiplying factors where the groups are nine and the amount in each is eight, would the nine (groups) be placed on the top or bottom next to the sign? It is simple when the direction is horizontal - 9x8. The factor for groups is the first one (on the left). I would sincerely appreciate your help as it makes a big difference if you have to draw out the multiplication facts.

Since the “number of groups” is the multiplier and the “amount in each” is the multiplicand, Leslee clearly assumes the multiplier goes on the left, but finds disagreement when it is written vertically. I responded by first referring to the answers above and next time, and then dealing with the vertical case:

When we write something like 835 * 24 ---- in order to multiply two numbers, we are not representing a physical problem on paper. Rather, we have already determined that in order to solve a problem, we must multiply these two numbers together; and we know certain techniques for doing so. One of the things we know is that theorder doesn't affect the result; we can either multiply 835 by 24, or multiply 24 by 835. So when we come to doing the actual multiplication,we can choose whichever way is easier. In this case, I would always put the 24 on the bottom, because it is smaller. So the fact that 24 is on the bottom has absolutely nothing to do with whether I have 24 groups, or groups of 24. Even when written as 835 * 24 (or 24 * 835) the order does not really reflect the problem. As you'll see in one of the answers cited, I myself vacillate on how to interpret it. That's because, again, it makes no difference at all. It may be of use to choose one meaning when you first introduce multiplication to children, and stick with that yourself for the sake of consistency; but you must not insist that they learn that as the only correct order, but rather should emphasize from the start that the order is unimportant, and that this gives them the freedom to see the same multiplication in two ways, and to change views at will if it makes anything easier.

As I already said above, in the vertical form, which is properly thought of a part of the algorithm for multiplying rather than as just stating a problem, it is appropriate to think of the bottom number as the multiplier as far as that algorithm is concerned; but this will typically be the smaller number, with no necessary connection to the original horizontal order, or to the application.

## Can the multiplier have units?

In 2011, we got a question from a very different perspective:

Further Musings on "Multiplicand" and "Multiplier" I have seen Dr. Math's answer to the definition of multiplicand and multiplier, and would like to share my thoughts. Consider the possible multiplicand and multiplier in (9 x 4) = 36 I believethese designations become clearer when the objective is writtenor spoken, such as "What is your age times 4?" If your age is 4, then four is the multiplicand and the multiplier is 9. If on the other hand your age is 9, then the multiplicand is nine and the factor is 4. The distinction between multiplicand and multiplier is less clear with questions about the total of contributions if, to continue the example, four individuals each give nine dollars. In my opinion,the multiplicand is the number that has the same units as the product. For example, I would say that the multiplicand is the dollar amount, because it is a four dollar contribution that is magnified by the number of contributors.

Carter may have read any of the discussions we looked at above (or the one we’ll see next time); but the question is rightly on the application rather than the written form. And he is almost right; you will see echoes here of the 1897 book I quoted, “When one of the factors is concrete, the concrete number is the true multiplicand”. A concrete number relates to objects, either as a mere count or in terms of units (a denominate number), as opposed to an abstract number (which is purely number). When the multiplier is abstract, the product has the same units as the multiplicand, as Carter says. So, if we calculate 4 times 9 years, the product is 36 years, and 4 is being used as the multiplier; the same is true if we multiply $9 by 4. But there’s more to it than that.

I started by asking which page he had seen, and emphasizing that the way the multiplication is written is irrelevant:

In any APPLICATION of multiplication, the multiplicand is the number to be multiplied (or scaled up, or repeated, or whatever), and the multiplier is the number by which it is to be multiplied (aka, the scale factor, repeat count, etc.). As you say, that is really unrelated to the way it happens to be written. The equation 9 x 4 = 36 need not represent "What is your age times 4?" It might just as well be what you'd write for "What is 9 times your age?" In either case, it is clear that the age (9 or 4 years, respectively) is the multiplicand, because it is the number you start with and modify. But it is written as the first number in one case, and the second in the other.

As we said before, either the multiplier or the multiplicand may be written first, or said first. The distinction is entirely in the application.

But what about the question of units?

I agree with you that dollar amounts (unit prices) are multiplicands, while numbers or quantities are multipliers. ButI would not consider it a good general principle to say that the multiplicand has the same units as the product. In the case of 9 pounds at 4 dollars per pound, the product is in dollars;no two numbers have the same unit! What you say would apply only when the multiplier is a dimensionless quantity (a mere number of times, or items). So in simple problems that require multiplication, it's fairly easy to identify the multiplier and multiplicand based on the application. The distinction, however, becomes less and less meaningful as you do more complex things. (For example, when calculating the force of gravity using F = GMm/d^2, which of the two masses is the multiplicand?) In the abstract, however, just given as A x B with no connection to an application,they are both just "factors" and play an equal role.

White, the 1897 author, might not say that 4 dollars per pound is a multiplier, though it still seems somewhat reasonable to call it that. The reality is that at this point we are multiplying two denominate numbers, but verging on the abstract. The distinction hardly matters now; we are in the realm where the numbers being multiplied are mere factors.

## More on denominate numbers

Let’s take a short side trip into a related issue associated with units:

Defining Counts Why can I not intuitively resolve this equivalency of dollars and cents?!100D = 1CI know that the ratio of dollars to cents can be expressed 1:100 and 1/100. This makes sense in numbers alone: 100 x NUMBER of dollars = 1 x NUMBER of cents And that goes ahead and plots on an x-y graph just fine. But the fact that1 dollar is equal to 100 centsin VALUE keeps messing me up. Surely, that would suggest that1D = 100C? I know the discrepancy has to do with the value not being the number. But as cents HAVE a dollar value, is there any way to resolve this intuitively? It works fine in my head for things that don't have an equivalent value, but are still directly proportional -- things like ingredients. But how can 100 DOLLARS = 1 CENT?! 100 dollars equals 10,000 cents! Almost everything that I find online about ratios explains it just fine, but this little detail keeps confusing my intuition.

I myself have occasionally fallen into the same trap, so I knew what to say:

You're confusing the idea of aunitwith the idea of avariable representing a number of units. When you say 100D = 1C, you are implicitly defining D = number of dollars C = number of cents Evidently, you obtained this equation from the proportion D dollars 1 --------- = --- C cents 100 On the other hand, you have this: 1D = 100C Here, you are using "D" to represent the unit "dollars," and "C" to represent the unit "cents." That is, you mean $1 = 100¢ or 1 dollar = 100 cents This is entirely different.Unit names are not variables. (In what follows, I will NEVER use a single letter to represent a unit, but will write out the unit name. In practice, you can use symbols or abbreviations for units, but must not think of them as if they were variables.)

The tricky part comes when we use dimensional analysis to convert units, where we include the units in an expression, and in fact treat the units as if they were variables! This is what lies behind Dominic’s equation \(100D = 1C\), where *D* and *C* are variables and units are not stated.

Now, can we put these two perspectives together? In the equation, which represents a unit conversion, we are implicitly using units in definitions of the variables D and C. So let's reveal that meaning: 100 cents/dollar * D dollars = C cents Multiplication of cents/dollar by dollars "cancels" units, leaving cents; so this is correct. When we work with the equation, we drop the units and just work with the numbers, 100D = CWhen we do include units in an equation, they don't represent numbers, but quantities. The quantity "D dollars" can be thought of as the number D times the unit quantity "1 dollar," which is a specific amount of money. Units transform "pure numbers" into "concrete numbers" (or, specifically, "denominate numbers") that represent physical quantities. We can express our equation in this way, multiplying quantities rather than just bare numbers: (100 cents/dollar) * D*(1 dollar) = C*(1 cent) Now, go back to your "equation" of units, which I rewrote as 1 dollar = 100 cents We can write that as (1 dollar) = 100*(1 cent) So we can replace (1 dollar) with 100*(1 cent) to obtain an equivalent quantity: D*(1 dollar) = D*100*(1 cent) But since we have defined our variable C as the number of cents, this says that 100*D*(1 cent) = C*(1 cent) Removing units, 100D = C The usual way we obtain this equation is to multiply D*(1 dollar) by 1, in the form of the fraction (100 cents)/(1 dollar), obtaining 100*D*(1 cent). Now we are back at the start: this is the equation relating amounts of dollars and cents. Again, note thatI used units in my definitions of the variablesD and C. For the sake of clarity, I recommend always doing this (and always writing out such definitions, rather than keeping them unstated). This has bothered me a few times, too, when I tried to do the math mechanically rather than thinking about what it means. So the main answer I can give is that you just have to be clear on what the parts mean.

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Vishaal KSome examples from algebra or vector spaces should be sufficient to convince anyone who doubts the order is multiplier x multiplicand, depsite the oddities in speech:

8 x 9 is 8 times 9, and 9 multiplied by 8, and 9 over 8 (when written as a multiplication vertically).

Example 1. Polynomials are the central object of study in algebra (non-abstract). We’ve arranged our notation to be able to write them easily (multiplication has higher precedence than addition). Further, if p(x) = 3x^2 + 2x + 1, most would agree that 3x^2 is more naturally thought of as 3 groups of x^2, or x^2 three times, rather than x^2 groups of 3, or 3, x^2 times. In this example, the multiplier is on the left, not the right.

Example 2. If you’re familiar with vector spaces, you’d agree that we do the same with vector notation. For a vector v in V over R, we write the scalar multiplier on the left, e.g. 3v rather than v3. Since there is generally no concept of adding 3, v times, 3v can only be interpreted as v 3 times, meaning the multiplier is on the left.

I think this concept is complex because English is a subject-verb-object language, it would be natural to see the first number as the one being multiplied, and in fact with addition or division, the first number is the main object to which something is “done” (added, divided).

Dave PetersonThanks for the comment, Vishaal. This is an interesting additional perspective.

In both of your examples, there is a

conventionto write the numerical coefficient, or the scalar multiplier, on the left. This does support the suggestion that we most naturally think of a multiplication as “multiplier times multiplicand”.On the other hand, it is not

illegalto write “x^2 3” or “v3”; for evidence of the latter, see Wikipedia, which talks about both left and right scalar multiplication (which are equivalent when the scalar comes from a commutative ring). So it is only a convention, not a requirement, that the “multiplier” is written first.As for your grammatical point, I have pointed out that “times” in “a times b” serves not as a verb, but as an adjective or a preposition (my dictionary says the latter); and “multiplied by” is a participle, not a finite verb that would be followed by an object but a phrase followed by an agent that does the action. The latter reading definitely puts the multiplier on the right. I think the main grammatical point to make is ambiguity: we have different ways to read it in English.

Considering only symbolic forms and ignoring language, consistency with addition, subtraction, and division supports thinking of the multiplier as the second number as you suggest.

Ultimately, at best your thoughts suggest a tendency; but this relates only to the specific applications you are using, which was my main point anyway. It is not the symbols written, but the

applicationfor which they are used, that determines which is the multiplier.