Multiplicand, Multiplier

Date: 07/05/2001 at 07:01:25
From: Christiopher Dungca (Elementary Math Teacher)
Subject: Definition of Terms in Multiplication

Dear Dr. Math:

Greetings from the Philippines!

Do you agree with me that in the following mathematical sentence
  456 x 10, 
456 is called the multiplicand and 10 is called the multiplier?

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 as commutative property of multiplication, 
why are they insisting all the textbooks and the teachers are 
committing a mistake?

Thank you in advance for the advice that you may give me.

Christopher Dungca 

Date: 07/05/2001 at 08:58:50
From: Doctor Peterson
Subject: Re: Definition of Terms in Multiplication

Hi, Christopher.

To my mind, it makes no difference at all which 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.

Really, the only distinction that can be made relates to the meaning 
in a given application: the number 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 the number 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).

I found the Britannica definition online at

   http://208.154.71.60/bcom/eb/article/7/0,5716,117157+2+109384,00.html   

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."

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