Definition of Base - Numbering Systems

We use a numbering system everytime we do any mathematical operations. Balance a checkbook? Count the number of toothpicks, Rainman? We use the base 10 numbering system, or decimal, when using everyday numbers. The traditional numbers we use 0,1,2,3,4,5,6,7,8,9 make up the decimal system.

Let's dive right in and see what exactly a numbering system is and how it works. We'll pick a number, say 1984, and break it down. In English, we say this number as one thousand nine hundred eighty four. This number is actually equal to:

(1 * (10^3)) + (9 * (10^2)) + (8 * (10^1)) + (4 * (10^0))

Which is:

(1 * 1000) + (9 * 100) + (8 * 10) + (4 * 1) = 1000 + 900 + 80 + 4 = 1984

Very interesting. You will notice that the numbers we can put in each "place" total 10. That is we have 0,1,2,3,4,5,6,7,8, or 9 we can use in this numbering system of decimal, or base 10. Coincidence? I think not.

The same is true for every base. Another common base is called binary, or base2. This has two number you can put in a "place" - 0 or 1. Using the system above let's see if you can try to figure out the deicmal value of the binary number 1001

Well.....

You break it down just as we did for the decimal number 1984 above, except rather than having a BASE of 10, we now use a base of 2:

(1 * (2^3)) + (0 * (2^2)) + (0 * (2^1)) + (1 * (2^0))

This expression evaluated further is:

(1 * 8) + (0) + (0) + (1 * 1) = 8 + 0 + 0 + 1 = 9

Great. That is the foundation of numbering systems. Once you grasp that you can use any numbering system thrown your way. But wait? What do we do when the numbering system's base is above 10?

One of the most commonly used numbering systems in computer science is hexadecimal, or base 16. The way we figure numbers is no different than any decimal or binay, but just as with those systems there are 16 numbers we can put in any place. We only have 10 numbers, so we use letters above the initial 10:

0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, and F make up the unit numbers. The base is 16.

So to represent the number 11 in hexadecimal we simply write : B or 0B

which is nothing more than (B * (16^0)) or (11 * 1) = 11 in decimal.