Which Place?

Date: 07/28/2003 at 03:20:15
From: Kelly Mirelle
Subject: Place value

In which place is the digit 6 in the number 3164297 ?

In my book there is an answer that says it is in the 100000, and when 
I asked my mom she said 10000. I dont know which one is correct!

Date: 07/28/2003 at 09:25:47
From: Doctor Ian
Subject: Re: Place value

Hi Kelly, 

One way to solve a problem like this is to write down all the possible
place values.  We do that by starting with 1, and multiplying by 10.  

   10,000   1000   100    1

Now, these quickly get pretty big!  So to make things a little more
compact, we use exponents. If you're not familiar with exponents, the
basic idea is to use one number to represent a bunch of 
multiplications:

   1
  2  = 1 * 2

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

   4
  2  = 1 * 2 * 2 * 2 * 2

Do you see the pattern?  When we write something like 

    9
  10

we mean 

  start with 1, and multiply it by 10, 9 times

that is,

    9
  10  = 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 

      = 1,000,000,000

Now, when we use exponents with 10's, there is a nice pattern:

    1
  10  = 10

    2
  10  = 100

    3
  10  = 1000

Do you see the pattern?  The exponent is the same as the number of
zeros.  So for a number like 

    24
  10

we don't have to do 24 multiplications. We can just write a 1 with
24 zeros after it:

    24
  10   = 1,000,000,000,000,000,000,000,000

There is one other thing you need to know, one that will eventually 
seem pretty natural. Anything raised to the 0th power is 1.  That is,

   0
  2  = 1

    0 
  10  = 1

and so on.  One way to think of it is from the definition we started
with.  To compute 2^0 (which is another way to write the exponent,
when you want to fit everything on one line), we start with 1, and
multiply it by 2 zero times. That is, we start with 1, and do nothing 
to it, so we end up with 1. 

Why am I telling you all this? Because now we can find the place value  
of each digit in a number like 3164297 by using exponents:

    6    5    4    3    2    1    0
  10   10   10   10   10   10   10     <-  place values

   3    1    6    4    2    9    7     <-  digits

So to find the place value of a digit, we count over from the right,
starting at 0:

   3164297     <- digits
     ^
     |   0
     |  1
     | 2
     |3
     4         <- counting from zero

When we get to the digit 6, we're at place 4; so the place value of 6
in this number is 10^4, which is 1 with 4 zeros, or 10,000. 

So you and your mom are correct. 

By the way, here's a less complicated way to do the same thing.  For a
number like 

   32571

we can write it as a sum:

    30000
     2000
      500
       70
  +     1
  -------
    32571

Now we can see immediately what the place values are. For your
number, it would look like this:

   3000000
    100000
     60000    <--  60,000 is 6 times 10,000
      4000         so 10,000 is the place value
       200         for the digit 6 in this number
        90
  +      7
  --------
   3164297

Note that each of the items in the sum corresponds to one of the
exponents that we saw earlier:

   3000000     3 times 10^6
    100000     1 times 10^5
     60000     6 times 10^4
      4000     4 times 10^3
       200     2 times 10^2
        90     9 times 10^1
  +      7     7 times 10^0
  --------
   3164297

Does this make sense?  Write back if any of it wasn't clear, or if you
have any questions about it. 

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