How We Write Numbers

You have probably heard at some point in your life that we use a "base ten" notation system for our numbers, without necessarily understanding what this actually means.

The point of a notation system for numbers is to keep track of how many items have been counted (as in a census, inventory report, etc.).  Therefore, the first question is how people count.  The answer is that people generally count on their fingers (you may be thinking only of small children at this point, but try listing all seven dwarves from Snow White without using your fingers to keep track of how many you've gotten (and writing them down is cheating)).  Counting on your fingers is a convenient way of keeping track of relatively small numbers, but what happens when they get bigger?  This is when you need to get creative.

One solution would be to gather several of your friends together and have someone new take over counting each time the previous person runs out of fingers, but then you have to add them all up at the end (essentially counting again), and it would take five people just to count to forty-one, which is not an outlandishly big number.  I'd like a better solution.

Suppose that instead of having a friend continue the count when I run out of fingers, I instead ask my friend to count how many times I run out of fingers.  Then forty-one is simply four of my friend's fingers and one of mine.  This essentially what we do in base ten notation.

There are ten symbols we use to write our numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.  Each of those symbols alone represents a number, but it helps to think of numbers in terms of "how many."  Here, the symbol "3" means "three fingers" (similarly for the other symbols).  The pair of symbols "41" (in that order) means "four sets of fingers and one leftover finger" (also known as forty-one, since we have ten fingers).  This shorthand notation is significantly easier than having to write down forty-one symbols to represent the forty-one items you counted.

If I have a second friend counting how many times my first friend runs out of fingers, we can make some even bigger numbers.  The string of symbols "123" means "one set of ten sets of ten fingers (one hundred fingers) and two sets of ten fingers (twenty) and three more fingers," or one hundred and twenty-three fingers.

You may have noticed that in general we are using the symbol "10" to mean "one set of all my fingers, with no leftovers" and begun to wonder why we should always count with ten fingers.  Suppose, for instance, that one hand is busy with something else and therefore can't help with our counting?  This would result in a base five system (which did develop in some cultures).

In a base five notation system, you would only need five symbols: 0, 1, 2, 3, and 4.  The symbol "10" would mean "one set of five fingers with no leftovers."  If you wanted to write ten, it would look like "20" because you would need two sets of five with no leftovers.  The number directly before this, nine, would look like "14."  And forty-one would have to be written as "131" (one set of five fives (twenty-five) plus three sets of fives (fifteen) plus one).  Your arithmetic would look a bit different, but it would work just as well (try a few calculations and see how it turns out).

For any whole number n bigger than or equal to two, you can create a base n notation system.  Currently, most of the human race uses base ten, most likely simply because we have ten fingers.  Most human cultures have used base five, ten, or twenty (I'm sure you can think of justifications for these).  One notable exception is the ancient Babylonians, who used base sixty.  As far as I know, no one is quite sure how they decided on this (though it has been pointed out that sixty is the smallest number divisible by 2, 3, 4, 5, and 6, which makes for fewer repeating decimals for fractions), but you can see its influence in how we measure time and angles.