Why the Decimal Point Goes Where It Does

Date: 09/21/2003 at 12:07:28
From: Michelle
Subject: Decimal Place Value

I am trying to help my daughter understand decimal place value.  Could 
you help me with a chart for her to study with?

I'm having trouble getting her to understand that the first place
before a decimal is "ones", and the first number after the decimal is
"tenths".

I've been trying to explain it using money. I'm using dollars as
resemblance to whole numbers and change as resemblance to fraction.

Date: 10/11/2003 at 19:12:49
From: Doctor Ian
Subject: Re: Decimal Place Value

Hi Michelle,

I'm somewhat hesitant to provide a chart.  The thing about charts is
that you either understand enough to make one yourself, or you don't.
In the latter case, it means that the most you can do with the chart
is try to memorize what's on it, and that's usually a bad idea. 

Money can be good, but for this to make sense, she needs to understand
the basic idea of how decimal notation works.  

Let's look at a number like 

  123.45

We can write this as a sum:

   100
    20
     3
      .4
 +    .05
 --------
   123.45

Does this make sense?  When we see it broken up like this, we can see
that each digit tells us how many groups of a certain size we have. 
In this case, we have

  1 group of  100
  2 groups of  10
  3 groups of   1
  4 groups of    .1
  5 groups of    .01

First, let's look at the groups to the left of the decimal point. 
Each one of them is a power of 10, that is, 10 raised to an exponent:

  1000 = 10^3              10*10*10 = 1000
   100 = 10^2                 10*10 = 100
    10 = 10^1
     1 = 10^0             

The last one is a little weird, but it's important to know about it,
or the rest of the system doesn't really make sense. 

So when we write a number like 3047, that's the same as

  3047 = 3*1000 + 0*100  + 4*10   + 7*1

       = 3*10^3 + 0*10^2 + 4*10^1 + 7*10^0

Note that we have to have the zero in there (between the 3 and the 4).
Do you see why? 

So in this way, we cover all the group sizes where the exponent is
zero or larger.  What about when it's less than zero? 

Before reading any further, you should probably take a look at this:

  Properties of Exponents
    http://mathforum.org/library/drmath/view/57293.html 

Ready to go on? 

We can have negative powers of 10, as well as positive ones:

  1000     = 10^3              10*10*10 = 1000
   100     = 10^2                 10*10 = 100
    10     = 10^1
     1     = 10^0             
      .1   = 10^-1               1/10^1 = 1/10
      .01  = 10^-2               1/10^2 = 1/100
      .001 = 10^-3               1/10^3 = 1/1000

So as you can see, we're using _all_ the powers of 10.  The only
question remaining is:  Where should we put the decimal point?  

In theory, we could put it anywhere.  For example, we could put it
between 10^1 and 10^0.  Then we'd have something like

  ********** + ****  = 1.4

instead of 

  ********** + ****  = 14

This wouldn't help us with anything, and it would make things more
complicated.  Well, what if we put it between 10^-1 and 10^-2?  Then
we'd have to be putting in extra zeros all the time.  The number we
now write as 6 would have to be '60', to indicate that there are no
tenths.  

In the end, we want numbers like 

    *     = 1
    **    = 2
    ***   = 3

and so on to be as simple as possible to write, since they're the ones
we write most often.   Of all the places where we might put the
decimal point, between 10^0 and 10^-1 is the one that causes the least
trouble. 

So if you want to figure out the value corresponding to a place,
here's how to do it without a chart.  Suppose we have a number like

  3175634.442586

What role is the 7 playing?  Let's write out just that part of the sum,

  3175634.442586
    70000

and let's write the group size by changing the 7 to 1:

  3175634.442586
    70000
    10000

How big is this?  Adding commas can help:

  3,175,634.442586
     70,000
     10,000          <-- ten thousand

What about the 8?  Let's start the same way:

  3175634.442586
         .00008
         .00001

Now we can count up the exponents:

  3175634.442586
         .00008
         .00001
          |||||
          1||||
           2|||
            3||
             4|
              5     So the group size is 1/10^5, 
                    or 1/100,000 (1 with 5 zeros)

I know this seems like a lot to take in, but believe me, it's worth
understanding, rather than trying to memorize a chart.  

Does this help?  Write back if you'd like to talk more about this, or
anything else. 
  
- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/