Lining Up Decimal Points

Date: 03/03/2003 at 18:16:11
From: Rashonda
Subject: Decimals

Why do addition and subtraction have to be lined up by the decimals 
but division and multiplication don't have to be?

Date: 03/04/2003 at 14:48:48
From: Doctor Ian
Subject: Re: Decimals

Hi Rashonda,

That's a really good question! It's easy to just take it for granted,
without ever stopping to think about it.  

Suppose I have 

  3 cherries, 2 apples, 3 pears

and you have 

  4 apples, 1 pear, 3 watermelons

and we want to know what we have together.  Could we do this?

  3 cherries   2 apples   3 pears
  4 apples     1 pear     3 watermelons
  ----------   --------   -------------
  7            3          6

That doesn't make any sense at all, does it?  We'd have to make sure
that we're adding the same kinds of things:

  3 cherries   2 apples   3 pears
               4 apples   1 pear     3 watermelons
  ----------   --------   -------    -------------
  3 cherries   6 apples   4 pears    3 watermelons

This is also what's going on when we add decimals.  When we write a
number like '123.4', that's really just a very compact way of writing
the sum

   1 * 100   +   2 * 10   +    3 * 1   +   4 * 1/10    

And when we write a number like 5.432, we're just writing the sum

   5 * 1   +   4 * 1/10   +   3 * 1/100   +   2 * 1/1000

Now, following the example of the fruit, can we do this?

   1 * 100       2 * 10       3 * 1       4 * 1/10    
   5 * 1         4 * 1/10     3 * 1/100   2 * 1/1000
   -------       --------     ---------   --------
   6             6            6           6

Does that make any more sense than adding cherries to apples?  No!  So
we need to line up terms where the digits have the same meaning:

   1 * 100   2 * 10   3 * 1   4 * 1/10    
                      5 * 1   4 * 1/10   3 * 1/100   2 * 1/1000
   -------   ------   -----   --------   ---------   ----------
   1 * 100   2 * 10   8 * 1   8 * 1/10   3 * 1/100   2 * 1/100

But how do we do this?  By lining up the decimal points!  So lining up
the decimal points is just an easy way of making sure that we're not
adding apples to pears, so to speak.  

Does that make sense? 

Okay, so much for addition and subtraction.  What about multiplication
and division?  Let's consider the product

     3.21
  *  23.4
  -------

Now, what we'd _really_ like is to get rid of those decimal points
entirely, or at least temporarily.  We can use a trick to do that. 
Note that

  3.21 = 32.1 * 1/10 = 321 * 1/100

and

  23.4 = 234 * 1/10

So 

    3.21 * 23.4 

  = (321 * 1/100) * 23.4             'Save' two decimal places

  = (321 * 1/100) * (234 * 1/10)     'Save' one deimal place

  = (321 * 234) * (1/100 * 1/10)        

  = (321 * 234) * 1/1000                

  = 75114 * 1/1000                      

  = 75.114                           'Restore' three decimal places

So we can just forget about the decimal points while we're doing the
multiplication, so long as we know where to put the decimal point when
we're done.  In practice, we do this:

     3.21    <-- 'save' two decimal places
  *  23.4    <-- 'save' one decimal place
  -------

      321    
  *   234    
  -------
    75114 * 1/1000   <-- 'restore' three decimal places
                          to get 75.114  

So in a sense, the answer to the question "Why don't we have to line
up the decimal points when we multiply?" is that we remove them until
we're done. You can't line up what isn't there.

What about division?  
        _____
  3.21 ) 23.4

We do a slightly different trick here, which is essentially the same
thing we do when we find equivalent fractions:

  23.4   23.4   100
  ---- = ---- * ---
  3.21   3.21   100

         2340
       = ----
          321

So now we have
      ________
  321 ) 2340.0

Once again, a little extra work up front lets us get rid of the
decimal point in the divisor. So once again, there's nothing to line
up.    

I hope this helps!  Write back if you'd like to talk more about this,
or anything else. 

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