Binary: Representing Numeric Data

You've probably heard that everything is stored inside a computer as binary, but what does this mean, and why was this done?

The Decimal System

As a general rule, humans count and express quantities with a decimal, or a base 10 numeric system. That is, any digit can be any of ten different values. The fact that this is so universal probably has to do with the number of fingers we have. When we count, instead of having a number after 9, we put a 1 in the "tens place", and start back at 0 in the "ones place".

The odometer in your car is a perfect example of this. When you pick up your car at the lot, there are only a handful of miles on it. The odometer reads 6. After you drive for a few miles, it says 9. The 9 is printed on a little wheel that has the digits 0 through 9 on it. As the wheel rotates from 9 back to the 0, it nudges the wheel next to it one place. The right-most wheel moves once every mile. The wheel next to it moves once ever full rotation of the right-most wheel, so it moves once every ten miles. The next wheel over moves once every 100 miles, and so on. With six little wheels, we can show almost a million miles.

Now, think of each little wheel as a storage unit. It stores a value, from 0 to 9. A set of six wheels has six different values, but each wheel is worth ten times the miles of the one next to it. We read the value of the entire display naturally because it is in the familiar decimal system.

Tasmanian Math, and other base systems

I will be forever indebted to my sixth grade math teacher (Mr. Smart, would you believe) for teaching me Tasmanian Math. It was a fictional base 3 numbering system, but he didn't tell us that at first. Instead, he told us this story, as if things really happened that way. (This story is completely fictional, and no offense is intended to anyone from Tasmania.)

Tasmania, as you should know, is a largish island off the southern coast of Australia. Obviously, the native peoples didn't own automobiles or live in fancy suburbs, so the status symbols of native Tasmanian society were quite different from ours. In Tasmania, it was a status symbol to own a wombat (also known as a Tasmanian Devil).

Lower class citizens could not afford a wombat. Middle class citizens prided themselves in owning a wombat and keeping it in their homes. The truly filthy rich upper class showed their frivolity by actually owning two wombats.

Somewhere along the line, Tasmanians realized the need to do some serious counting, and even some math. Unlike most civilizations which modeled their numbering system on the fingers they all counted with, the Tasmanians elected to base their numbering system on the symbol that best represented status to them: the wombat (obviously this was decided by someone with more wombats than anyone really needed).

This gave them three numbers to work with. The first number, Op, was representative of the lower class. It was drawn as an empty hut -- with no wombats. The next number was known as Diddle. It was drawn as a hut with one wombat, as was typical for most middle class Tasmanian families of the day. Finally, there was Doodle, an audacious hut with two wombats inside.

While this system had obvious merits to the upper class, the have-nots were quick to realize you couldn't count very many things this way. No one needed to count more than two wombats, but it became awkward and confusing to count beads or sheep or termites.

Out of necessity, this was quickly resolved. Since no one had ever heard of three wombats in a single hut, it was decided that the number to come after Doodle would be Diddle-Op. It was drawn as two huts side by side, one with one wombat and one empty. The hut to the left stood for more than one wombat (or termite). Instead, because of its placement, it represented one times three wombats (or sheep).

Soon a whole system of math evolved. Tasmanians learned about carrying Diddle to the next hut when they added two numbers together. They learned to multiply wombats (or beads) and do long division. Some scholarly mathematicion had even calculated pi to Doodle-Diddle-Doodle-Op-Doodle huts after the Tasmanian math point. Sadly, the native Tasmanian way of life was obliterated by European settlers before electronic calculators were invented to make all this easier.

The Binary System

Now, getting back to reality, imagine that the digits 2 through 9 didn't exist. All we have are 1's and 0's. How would we count with that? What comes after 1? Well, just as we do in a decimal system after 9, we go back to 0 and put a 1 in the place next to it. So after 1 is 10 (binary).

But don't let that confuse you.