Zeros after the Decimal Point

Date: 09/17/2001 at 22:19:53
From: Carolyn Bronk
Subject: Decimals

I have a group of students that just can't comprehend my explanation 
of how 3.70 is equivalent to 3.7 or 3.700 or 3.7000... especially 
when you say 3.70 is 3 and 70 hundredths. Can you give me some 
insight into how you might go about the process of explaining the 
equivalent aspect of this concept? Thanks.

P.S. I have noted that adding the extra zeros does not change the 
value.

Date: 09/18/2001 at 08:59:47
From: Doctor Peterson
Subject: Re: Decimals

Hi, Carolyn.

There are two main ways to explain this:

By place value:

    3.7  = 3(1) + 7(1/10)

    3.70 = 3(1) + 7(1/10) + 0(1/100)

    The only difference between them is adding an expression that is
    equal to zero.

    This differs from the case for whole numbers, where adding a zero
    on the right changes the place value of all the other digits (e.g.
    37 -> 370); but it is the same as adding a zero on the LEFT of a
    whole number (37 -> 037), where nothing is changed.

By equivalent fractions:

    3.7 = 3 7/10

    3.70 = 3 70/100

    The fractional parts are equivalent, because the latter can be
    obtained from the former by multiplying numerator and denominator
    by 10:

     7     7 * 10    70
    --- = ------- = ---
     10   10 * 10   100

    When we add a zero on the end, we are multiplying both the 
    numerator and the denominator by 10, so the value doesn't change.

I don't know whether this helps. If you still face resistance, you 
might want to make it more concrete (say, ask whether they want you to 
give them three dollars and seventy cents, or three dollars and seven 
dimes). Or, ask them to convince you that the numbers are different, 
and then tell me what they said so I can see their reasoning.

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