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Decimal (Base 10)

Let's count. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ... Fun isn't it? When we count we're using the words ``one,'' ``two'' and so on (or the symbols ``1,'' ``2'' and so on) to represent a quantity of things. Two eyes, two ears, two teeth... The word ``two'' or the symbol ``2'' is just used to denote how many of something we have.

One of the great things about our numbering system is that we don't need a symbol for every quantity - we recycle. What I mean by this is, we have individual symbols that we can write down which indicate the quantities zero through to nine. Once we get past nine, however, we have to use two symbols to denote the number ten - we write ``one zero'' but we mean ``ten.'' This is very efficient for lots of reasons, not the least of which is the fact that, if we had to have a different symbol for every number, our laptop computers would have to be much bigger to accomodate the keyboard.

This raises a couple of issues:

1. why do we only have ten of those symbols?
2. how does the recycling system work?

We have ten symbols because most of us have ten fingers. When we learned to count, we started by counting our fingers. In fact, another word for ``finger'' is ``digit'' which is why we use the word digit for the symbols that we use for counting - the digit ``0'' represents the quantity (or the number) ``zero.''

How does the system work? This won't come as a surprise, but we'll go through it anyway... Let's look at the number 7354. What does this represent? Well, one way to write is to say ``seven thousand, three hundred and fifty-four.'' In fact, this tells us right away how the system works. Each digit represents a different ``power'' of ten... Take a look at Table 1.2.

Table 1.2: An illustration of how the location of a digit within a number (in this case, the number 7354) determines the power of ten by which it's multiplied.

7		3		5		4
Thousand's place		Hundred's place		Ten's place		One's place
$7 * 1000$	+	$3 * 100$	+	$5 * 10$	+	$4 * 1$
$7 * 10^{3}$	+	$3 * 10^{2}$	+	$5 * 10^{1}$	+	$4 * 10^{0}$
		= 7354

Following this table, we can see that if we write a string of digits together, each of the digits is multiplied by a power of ten where the placement of the digit in question determines the exponent. The right-most digit multiplied by the 0th power of ten, the next digit to the left is multiplied by the 1st power of ten, the next is multiplied by the 2nd power of ten and so on until you run out of digits. Also, we can see why we're taught phrases like ``the thousand's place'' - the digit 7 in the number above is multiplied by 1000 ($10^{3}$ because of its location in the written number - it's in the ``thousand's place'')

This is a very efficient method of writing down a number because each time you add an extra digit, you increase the number of possible numbers you can represent by a factor of ten. For example, if I have three digits, I can represent a total of one thousand different numbers (000 - 999). If I add one more digit and make it a four-digit number, I can represent ten thousand different numbers (0000 - 9999) - an increase by a factor of ten.

This particular property of the system makes some specific mathematical functions very easy. If you want to multiply a number by ten, you just stick a ``0'' on the right end of it. For example, 346 multiplied by ten is 3460. By adding a zero, you shift all the digits to the left and the multiplication is automatic. In fact, what you're doing here is using the way you write the number to do the multiplication for you - by shifting digits, you wind up multiplying the digits by new powers of ten in your head when you read the number aloud.

Similarly, if you don't mind a little inaccuracy, you can divide by ten by removing the right-most digit. This is a little less effective because it's not perfect - you are throwing some details away - but it's pretty close. For example, 346 divided by ten is pretty close to 34.

We typically call this system the decimal numbering system (beacuse the root ``dec'' means ``ten'' - therefore words like ``decimate'' - to reduce in number by a power of ten, and ``decathalon'' for a sporting contest with 10 different events). There are those among us, however, who like our lives to be a little more ordered - they use a different name for this system. They call it base 10 - indicating that there are a total of ten different digits at our disposal and that the location of the digits in a number correspond to some power of 10.