Counting Decimal Places

Date: 05/22/2001 at 23:46:49
From: Craig Nadeau
Subject: Decimals

When multiplying decimals why does placing the decimal in the answer 
by counting the number of decimal places in the problem and counting 
from the right the same number of places work?

I have searched the Internet and even looked in my math book, but I 
still can't find anything. I asked one of the teachers at my school 
and he didn't know. My own math teacher won't help because she says I 
need to learn it on my own whether I get it or not. I would really 
like to show her up and prove I'm not a complete idiot. All the 
answers I've tried are apparently not good enough for her.

Date: 05/23/2001 at 09:16:22
From: Doctor Peterson
Subject: Re: Decimals

Hi, Craig.

I tried searching our archives for this, hoping I could help you find 
the answer for yourself, but I didn't find a good complete answer. 
Here's how it works:

Suppose we are multiplying 1.23 by 4.5.

First, we can convert each number to a decimal by using all its digits 
as the numerator (with no decimal point), and using for a denominator 
a 1 followed by as many zeros as there were decimal places, so that 
1.23 is 123/100. You may want to think of it this way, where I 
repeatedly move the decimal point to the right to multiply by 10, 
until everything is a whole number:

            1.23   12.3   123.
     1.23 = ---- = ---- = ----
            1.00   10.0   100.

(Remember, if you multiply the numerator and denominator of a fraction 
by the same number, you haven't changed the value.)

Writing both decimals as fractions, we have:

                  123   45   123*45   5535
     1.23 * 4.5 = --- * -- = ------ = ---- = 5.535
                  100   10   100*10   1000

Now, when I multiplied the fractions, the denominator was the product 
of two powers of ten, and turned out to be a 1 followed by the total 
number of zeros in the two numbers, which is the total number of 
decimal places in the two decimals! When I convert back to a decimal 
at the end, that's how many decimal places I need.

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