The natural numbers are 1, 2, 3, 4, ..., up to infinity. They look simple, but they are not. One of the first interesting things about these numbers is that they confront us in a very natural way with the idea of infinity. The list of natural numbers never ends! (although the reader should realize that it does have a beginning).
Well, there is a very important symbol to add to this list: The symbol which represents "nothing". Yes, that is 0. Some people include it as a natural number, some others do not. If you pay close attention to the way you write numbers, you realize that all natural numbers are written by using just 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Why ten symbols? Why not 5 symbols, or 3, or just 2 symbols? I guess the number ten is beloved by us humans because the total number of fingers in our hands is 10.
Let us examine closely how we count, which is a very smart thing. I am going to use temporarily an extra symbol to represent "a package of ten", let us say T. Let us assume we count stones. We go like this:
0 stones
1 One stone
2 Two stones
3 Three stones
4 Four stones
5 Five stones
6 Six stones
7 Seven stones
8 Eight stones
9 Nine stones
1T= One package of ten stones
Now we can continue like this:
1T
1 Eleven stones
1T
2 Twelve stones
....
9T
9 Ninety nine stones,
and what do we do now? Well, we continue with...
1TT One package of ten packages of ten stones!!! (usually called "One hundred" to make it simpler... :)
Now we can go on and on and on... But observe that we can replace the T's just for empty spaces and, not only we save some ink, but also we can compress even more the way to write the numbers:
2T
3 (twenty three)
turns into
23
This trick leads to our decimal system of writing numbers. The position of each symbol tells us whether we are thinking of units, packages of ten, packages of packages of ten, etc. This is really nice, because this trick allows us to write BIG numbers with just 10 symbols, and in a very systematic and economic way.