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### 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

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

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Place Value

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