How many ways can you express the number 12? Well, there's "one dozen," of course. But there's also 1100 (Base 2), 20 (Base 6), and more. You see, the values that numerals represent depend on the number system (or Base) you use.
We mostly use the decimal system (or Base 10) to do arithmetic. The decimal system has 10 numerals: 0 through 9. The position of each numeral is very important. There's the place for 1s, the place for 10s, 100s, 1,000s, and so on.
But why limit yourself to Base 10? Robert McGuigan is chairman of the mathematics department at Westfield (Mass.) State College. "Different bases make certain types of arithmetic easier," he says. Base 2 has just two numbers: 0 and 1. In binary arithmetic, the place values (from right to left) are: 1s, 2s, 4s, 8s, 16s, and so on.
Look at it this way: In decimal, the place values vary by multiples of 10 (10x10, 10x10x10, etc.). In binary, the place values are multiples of 2 (2x2, 2x2x2, etc.).
A binary number written as 11001 is 25 in our Base 10 (decimal) system. To understand why, McGuigan suggests imagining a balance scale. The scale has two dishes. One dish holds decimal values, the other holds binary. The sides must be equal to balance.
Numbered blocks representing various quantities are put on the scale. Let's put one block labeled 25 in the decimal dish. Now we have to balance the binary side. Our Base 2 blocks are labeled 1, 2, 4, 8, 16, and so on. What combination of binary blocks adds up to 25? Answer: one 16, one 8, (zero 4s, zero 2s), and one 1. To represent that as a binary number, we write: 11001 (Base 2).
What good is Base 2? It's perfect for computers, where "on" and "off" can represent 1 and 0, respectively.
Computers can also use Base 8 (octal) and Base 16 (hexadecimal). Numbers in the hexadecimal system are interesting: 0 through 9, followed by the letters A (10) through F (16). In hexadecimal, the decimal number 27 becomes "1B." (That is: one 16 and 11 ones. with 11 expressed as the letter B.)
Using different bases is as old as civilization itself. Our decimal system traces its roots to the ancient Chinese, Egyptians, and Sumerians. Early Babylonians often used Base 60 (sexagesimal). Parts of the ancient Mayan calendar incorporated Base 20.
Having lots of choices is nice. But just think of the problems we'd have if we didn't all agree on which base to use!
*To view an ancient number system, see:
To use an online calculator to convert numbers to different bases, go to:
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