Why Decimal Division Works

Date: 04/24/2001 at 03:14:31
From: Marc
Subject: WHY division by decimals

For my college math education class I have to write a paper on why 
(not how) the procedure of division with decimals works. The question 
is, "All elementary school students learn how to divide with decimals 
such as in the problem 551.2/1.06. Explain as if talking to 5th and 
6th graders why this procedure works." If you could help me it would 
help very much!

Thanks,
Marc

Date: 04/24/2001 at 08:51:51
From: Doctor Peterson
Subject: Re: WHY division by decimals

Hi, Marc.

If you'd like to look over our shoulders as we actually explain 
division of decimals to a 5th or 6th grader, visit our archives:

   Fractions and Decimals - Elementary School Level
   http://mathforum.org/dr.math/tocs/fractions.elem.html   

   Division - Elementary School Level
   http://mathforum.org/dr.math/tocs/division.elem.html   

Here's a sample:

   How to Divide Decimals by Decimals
   http://mathforum.org/dr.math/problems/wright3.7.98.html   

That should give you some good ideas for your own explanation.

The basic idea is that we first learn a method for dividing whole 
numbers. When we introduce decimals, we find that the work can be 
divided into two parts, just as in multiplication: we can first ignore 
the decimal point entirely, and get the DIGITS of the answer using the 
methods we've already learned; then we can determine where in that 
answer to put the decimal point. This is because, for example, we can 
rewrite a division of decimals as a division of whole numbers times a 
division of powers of ten:

     1.95   195 * 0.01   195   0.01   195
     ---- = ---------- = --- * ---- = --- * 0.01 * 10 = 3 * 0.01 * 10
     6.5    65 * 0.1      65   0.1     65

This says that, after dividing 195 by 65, we put the decimal point in 
the same place it was in the dividend, 1.95 (multiplying 3 by 0.01 to 
get 0.03), and then shift it one place to the right (multiplying by 
10, which is the same as dividing by 1/10, to get 0.3).

The traditional way to make this memorable is to first move the 
decimal points in the divisor and dividend until the dividend is 
whole, and then put the decimal point in the quotient directly above 
the new position in the dividend. This effectively shifts the decimal 
point in the quotient first to the left, accounting for the decimal in 
the dividend, and then the to right, accounting for the divisor.

So you see, it's not that we MUST divide only by a whole number; 
rather, this is just one easily remembered way to decide where to put 
the decimal point in the quotient. It's not much different from what 
we do in multiplication of decimals, where we first ignore the decimal 
point, and then add the number of decimal places in the two numbers. 
In division, we are really SUBTRACTING the number of decimal places in 
the two numbers; but explaining it in terms of moving the decimal 
point ensures that we won't accidentally subtract in the wrong order.

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