Explanation of Place Values

Date: 05/08/2000 at 19:16:17
From: Enzo
Subject: The increase of each place

Hi,

I guess this is a bit philosophical, but here it goes. It is written 
that as you progress from the units place to the tens place to the 
hundreds place, the value is increasing ten times the numeral on its 
right. For example, in 589 the 8 is ten times greater than the 9 and 
the 5 is ten times greater than the 8. How does this increase by ten 
times work out? If you count from 1 to 10 I imagine that you are ten 
greater than nothing or 0 (which is a place holder). Then you count 10 
ten times to reach 100 but you must also count from the number 10 up 
to 100. What that means is that 100, which is 10 times the number 10, 
also must include the 10 as the beginning of the counting. Can you 
explain to me on a deeper level what is going on?

Thanks,
Enzo

Date: 05/09/2000 at 11:59:59
From: Doctor Peterson
Subject: Re: The increase of each place by 10x the place on right

Hi, Enzo.

What increases at each digit is the granularity of the count - the 
size of the unit we are counting with. It's similar to measuring a 
distance as, say, 5 miles, 8 feet, and 9 inches. As you go several 
miles, each mile includes every foot you have gone along the way, 
EXCEPT for the last few, which were not enough to make a full mile. In 
the final measurement you give, the 8 feet are not part of the 5 miles 
you counted, but are the leftover feet that were not part of any full 
mile.

Similarly, each digit counts a different size group. If I say I have 
589 objects, it means I can count by 100's until I have 5 of them, 
with less than a hundred left; then count by 10's until I have 8 tens, 
with less than ten left; and then count what's left by 1's, finding 9 
of them. We can picture this as putting them into packages of 
decreasing sizes; there are ten tens in a hundred, and ten units in a 
ten, package inside package; when there are not enough to fill another 
package of one size, we start using smaller packages to use up the 
remainder:

500
     +-----------------------------------------+
     | +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ |
     +-----------------------------------------+

     +-----------------------------------------+
     | +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ |
     +-----------------------------------------+

     +-----------------------------------------+
     | +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ |
     +-----------------------------------------+

     +-----------------------------------------+
     | +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ |
     +-----------------------------------------+

     +-----------------------------------------+
     | +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | |*| |*| |*| |*| |*| |*| |*| |*| |*| |*| |
     | +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+ |
     +-----------------------------------------+
80
       +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+
       |*| |*| |*| |*| |*| |*| |*| |*|
       |*| |*| |*| |*| |*| |*| |*| |*|
       |*| |*| |*| |*| |*| |*| |*| |*|
       |*| |*| |*| |*| |*| |*| |*| |*|
       |*| |*| |*| |*| |*| |*| |*| |*|
       |*| |*| |*| |*| |*| |*| |*| |*|
       |*| |*| |*| |*| |*| |*| |*| |*|
       |*| |*| |*| |*| |*| |*| |*| |*|
       |*| |*| |*| |*| |*| |*| |*| |*|
       |*| |*| |*| |*| |*| |*| |*| |*|
       +-+ +-+ +-+ +-+ +-+ +-+ +-+ +-+
9
        *
        *
        *
        *
        *
        *
        *
        *
        *

When we count by ones up to 589, as you discussed, we have to 
constantly change smaller groups for larger ones. Each time we reach a 
multiple of ten we repackage them as a single ten, with no individual 
units left. This is what happens when we change from 9 to 10 or from 
289 to 290, setting the units place back to 0 and adding one to the 
tens place. Each time we reach a multiple of 100, we package our last 
ten 10's as a single hundred, clearing the other digits, as when we go 
from 99 to 100 or from 299 to 300. 

Each place thus incorporates whatever was counted before, taking over 
what had been in the lower digits. This process can be seen in an 
odometer, where each numbered wheel counts the number of times the 
wheel to its right passed 9 and returned to 0.

So yes, the 5, which represents 5 hundreds, includes the first ten you 
counted; it starts at zero, not after the first ten. Yet the 500 does 
not include the 80 or the 9. When you count starting with the larger 
units as I did above, you see more clearly that the smaller units are 
OUTSIDE the larger units, but it really works the same either way.

I hope this clarifies what place value is all about. Talking about 
counting and packaging doesn't sound like a "deeper level," but I 
think it shows what we mean more clearly than deeper math would. Let 
me know if you want a more advanced perspective on this.

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