## Number Systems"base" of the decimal system is 10 because there are 10 digits represented in it: 0 thru 9. We use the combination of these 10 digits to form all other numbers. The value of a digit in a number depends upon its position in the number. Take for example the decimal number 2448:
Some say we came up with the decimal number system because we have 10 fingers! Could be! But the decimal system isn't the only number system. Especially since computers came around, other number systems have come into widespread usage. Since a computer finds it easy to have an electronic circuit be either "on" or "off" (like a light switch), computers like a number system that has only 2 numbers: 0 (off) and 1 (on). So, each inary digb ("bit") in a computer is like a digit in the decimal system. Where the decimal system has the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, the itbinary system has only 0 and 1. Just as the decimal system has place representations for powers of 10 (1,10, 100, . . .) and for numbers less than 1 (1/10, 1/100, 1/1000 . . .) , the binary system has place representations for each power of 2: (1, 2, 4, 8, 16, . . .) and for numbers less than 1: (1/2, 1/4, 1/8, 1/16 . . .) A number in the binary system is rather boring. They look like this: "0 0 1 0 0 1 1". Each "bit", starting at the right, is a power of 2. For our 7-bit example, the powers of 2 are:
19.
Each 1 or 0 tells you whether that power of 2 is in the number. In our above example, the number is 7 'bits' long and tells you which of the powers of 2 up to 2
So, our hexadecimal number, 2FE, is, in decimals, 512 + 240 + 14 = 766. Think about it. Our 3-digit hexadecimal number, 2FE is, in binary, 1011111110. 2FE is a lot easier to read and to input. If you were inputting numbers like 1011111110 into the computer it would be easy to miss a 1 and mess that up! A computer doesn't actually do arithmetic in hexadecimal. It just uses it to communicate with the outside world! When you're dealing with more than one number system, you indicate the base of your number with a little subscript. Our example would become:2FE _{16} = 1011111110_{2} = 766_{10}You can see that without that little subscript, a number like 286 would be confusing. It is quite different in decimal or hexadecimal. 286 is a valid number in both systems! It isn't valid in binary because it contains digits other than 0 or 1. 286 _{10} is just 286, where 286_{16} is 646_{10} or, if you're interested, 1010000110_{2}
There are other number systems out there. Octal, based on the number 8, was used early in the computer world but is no longer used. You could even make up a system based on the number 7 where the number 166 _{7} would be x71^{2} + x76^{1} + x76^{0} = 49 + 42 + 6 = 97 _{10}, but it wouldn't be very useful! By the way, if you want to be cool, you call hexadecimal numbers "hex"! |