Order of Digits in Multiplication

Date: 03/27/2001 at 09:33:35
From: Hannah Jones
Subject: Multiplication place value order

Why do we learn to multiply starting with the least significant digits 
in the ones place value instead of starting with the larger place 
values? If I had 325 pieces of gum and I had 300 in one hand and 25 in 
the other, I would choose the 300. I don't understand why we start 
with the smaller place value when multiplying. Wouldn't it be easier 
when we have a problem such as 325 x 54 to start multiplying by the 
hundred's to get 1500?

If you can answer this I would greatly appreciate it.

Thank you
Hannah Jones

Date: 03/27/2001 at 12:37:22
From: Doctor Peterson
Subject: Re: Multiplication place value order

Hi, Hannah.

I've just been teaching my daughter how to multiply, and she's found 
that you can do it in any order you want, as long as you put 
everything in the right place. There are two reasons for always 
starting at the right: it's a good idea to do it the same way every 
time so you don't get confused; and starting at the right allows you 
to do a bit less writing and avoid erasing. There are some older 
methods that were just right for use on an abacus, where there's no 
writing or erasing, or on a sand table or chalk board, where erasing 
was easy. What we teach now seems to work best with pencil and paper.

The basic idea behind all these methods doesn't depend on order. 
What's happening is that you multiply each part of one number by each 
part of the other. In your example, we're really multiplying:

     300 + 20 + 5  by  50 + 4

and we do so by multiplying 300 by both 50 and 4, then 20 by 50 and 4, 
and so on. (For example, if I had 5 bags, each containing 50 red 
marbles and 4 blue marbles, I would have 5x50 red marbles and 5x4 blue 
marbles, which is the same as 5x(50+4) marbles.) We can see it this 
way:

       (300 + 20 + 5) x (50 + 4)

     = (300 + 20 + 5) x 50 + (300 + 20 + 5) x 4

     = 300 x 50 + 20 x 50 + 5 x 50 + 300 x 4 + 20 x 4 + 5 x 4

     = 15000 + 1000 + 250 + 1200 + 80 + 20

I can add these together in any order I want. But since the ones 
column will carry into the tens column, I usually add starting at the 
right so I don't have to erase the number I already wrote for the 
tens, after I add the ones. That makes me tend to do everything from 
the right.

Here's one way you can write this down:

       325
     x  54
     -----
        20 <-- 5 x 4 = 20
        8  <-- 20 x 4 = 80
      12   <-- 300 x 4 = 1200
       25  <-- 5 x 50 = 250
      10   <-- 20 x 50 = 100
     15    <-- 300 x 50 = 15000
     -----
     17550

The usual way "telescopes" the first three lines into one, by writing 
down a digit and carrying the rest:

       325
     x  54
     -----
      1300 <-- 5 x 4 + 20 x 4 + 300 x 4 = 20 + 80 + 1200
     1625  <-- 5 x 50 + 20 x 50 + 300 x 50 = 250 + 1000 + 15000
     -----
     17550

Within each line, we started from the right so the carries would work 
neatly; but there's no particular reason why we have to multiply the 
4 first, then the 5. We could have done this:

      325
    x  54
    -----
    1625  <-- 5 x 50 + 20 x 50 + 300 x 50 = 250 + 1000 + 15000
     1300 <-- 5 x 4 + 20 x 4 + 300 x 4 = 20 + 80 + 1200
    -----
    17550

The only problem with this is that it can be hard to keep everything 
lined up right; it's easier to remember that the first line starts at 
the right, and each line after that moves one place to the left.

Here are some answers in our archives that deal with the same issues a 
little more completely, including some about an old method called 
lattice multiplication where you multiply all the digits in any order 
you want:

   Learning to Multiply Three-Digit Numbers
   http://mathforum.org/dr.math/problems/jonathan7.3.98.html   

   Lattice Multiplication
   http://mathforum.org/dr.math/problems/susan.8.340.96.html   

   Lattice Multiplication Explained
   http://mathforum.org/dr.math/problems/durham.10.20.99.html   

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