Number of Places in Products of Decimals

Date: 09/19/2002 at 13:47:34
From: Nicholas
Subject: Multiplying decimals


When you multiply decimals, why is there always the same number of 
decimal places in the answer as in the problem? Why couldn't there be 
more or fewer than there are in the problem?  

For example, if you have a number with two decimal places (like 35.47) 
times another one with two decimal place (like 57.91), why do there 
HAVE to be four decimal places in the answer?

Date: 09/19/2002 at 15:46:11
From: Doctor Rick
Subject: Re: Multiplying decimals

Hi, Nicholas.

I could give several different answers to your question.

First: You can regard your example as multiplying 35 47/100 by 
59 91/100, or 3547/100 by 5991/100. The denominator will be 100*100, 
or 10,000. A number of 10,000ths is a number with 4 decimal places. 
When you're doing the multiplication in the usual way, you ignore the 
decimal points until the end. Then you count off as many decimal 
places as there are in the factors combined, and put the decimal 
point there.

         3 5.4 7
       * 5 9.9 1
       ---------
         3 5 4 7
     3 1 9 2 3
   3 1 9 2 3
 1 7 7 3 5
 ---------------
 2 1 2 5.0 0 7 7

What you've done is to multiply 3547 by 5991, then divide by 10,000 
(when you put in the decimal point).

Second: In a sense, there don't HAVE to be 4 decimal places in the 
answer. Think again about multiplying decimals: the first answer we 
get, by multiplying the numerators and multiplying the denominators, 
may not be in lowest terms; if not, you can reduce the denominator: 
it might end up being 1000 or even 100 instead of 10,000, which means 
only 3 or 2 decimals in the answer. Here's an example:

  35.25 * 59.92 = 2112.1800

I wrote that with four decimal places in the answer, but the number 
is the same as 2112.18, which has only two decimal places.

Third: When you do this multiplication the usual way, you'll see the 
zeros at the end, and they must be counted among the 4 decimal 
places. You can't drop them before you place the decimal point.

         3 5.2 5
       * 5 9.9 2
       ---------
         7 0 5 0
     3 1 7 2 5
   3 1 7 2 5
 1 7 6 2 5
 ---------------
 2 1 1 2.1 8 0 0

You can't ignore the zeros because you're doing the same thing as 
before: multiplying the numerators and dividing by 10,000. When you 
ignore the decimal places, you get 3525 * 5992 = 21121800 - those 
zeros make a big difference! You're doing this:

  35.25 * 69.92 = 3525/100 * 5992/100

    3525 * 5992   21121800   211218
  = ----------- = -------- = ------ = 2112.18
     100 * 100      10000      100

Fourth: In my second answer I said that the answer might not need four 
decimal places, because 2112.1800 is the same as 2112.18. But for 
some purposes that isn't true. Sometimes the number of digits we write 
in an answer tells people how accurate we believe the answer to be. 
For instance, if I say a line is 34.12 meters long, it implies that 
the measurement is accurate to the nearest hundredth of a meter. If I 
only knew the distance to the nearest tenth of a meter, then that 
final 2 would be only a wild guess and I shouldn't say it. So, if I 
say the distance is 34.10 meters, this is different from saying that 
it is 34.1 meters: the extra zero tells people that I believe the 
answer is correct to the nearest hundredth of a meter, not just the 
nearest tenth of a meter.

If I know the sides of a rectangle are 35.25 meters and 59.92 meters 
to the nearest hundredth of a meter, then I know the area of that 
rectangle is 2112.1800 to the nearest 10,000th of a meter. If I said 
the product was 2112.18, I'd be claiming less precision than I 
actually have. So in this sense, if I get zeros at the end of my 
answer, I should leave them there and stick with four decimal 
places.

I hope you can see that there is a good reason why a product of 
decimals has as many decimal places as the two factors combined.

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