### Numbering Systems

People are used to dealing with the decimal number system, which has 10 different digits. But machines use schemes that have some power of 2 digits, such as 2 (21), 4 (22), 8 (23), 16 (24), 32 (25), 64 (26), and so on.

#### Decimal

The decimal number system is also called base 10 or radix 10. The set of possible digits is {0, 1, 2, 3,4, 5, 6, 7, 8, 9}. The first digit to the left of the radix or “decimal” point is multiplied by 100, or 1. The next digit to the left is multiplied by 101, or 10. The power of 10 increases as you move farther to the left. The first digit to the right of the decimal point is multiplied by a factor of 10−1, or 1⁄ 10. The next digit to the right is multiplied by 10−2, or 1⁄ 100. This continues as you go farther to the right. Once the process of multiplying each digit is completed, the resulting values are added up. For example:

2 × 103 + 7 × 102 + 0 × 101 + 4 × 100 + 5 × 10−1 + 3 × 10−2 + 8 × 10−3 + 1 × 10−4 + 6 × 10−5 = 2704.53816

#### Binary

The binary number system is a method of expressing numbers using only the digits 0 and 1. It is sometimes called base 2 or radix 2. The digit immediately to the left of the radix point is the “ones” digit. The next digit to the left is a “twos” digit; after that comes the “fours” digit. Moving farther to the left, the digits represent 8, 16, 32, 64, and so on, doubling every time. To the right of the radix point, the value of each digit is cut in half again and again, that is, 1⁄ 2, 1⁄ 4, 1⁄ 8, 1⁄ 16, 1⁄ 32, 1⁄ 64, and so on.
Consider the decimal number 94. In the binary number system, this number is written as 1011110. It breaks down as follows:
0 × 20 + 1 × 21 + 1 × 22 + 1 × 23 + 1 × 24 + 0 × 25 + 1 × 26 = 1011110
When you work with a computer or calculator, you give it a decimal number that is converted into binary form. The computer or calculator does its operations entirely using the digits 0 and 1. When the process is complete, the machine converts the result back into decimal form for display.
In a communications system, binary numbers can represent alphanumeric characters, shades of color, frequencies of sound, and other variable quantities.

#### Octal

Another scheme, sometimes used in computer programming, is the octal number system, so named because it has eight symbols (according to our way of hinking), or 23. Every digit is an element of the set {0, 1, 2, 3, 4, 5, 6, 7}. This system is also known as base 8 or radix 8. Hexadecimal
Another system used in computer work is the hexadecimal number system. It has 16 (24) symbols. These digits are the usual 0 through 9 plus six more, represented by A through F, the first six letters of the alphabet. The digit set is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}. This system is sometimes called base 16 or radix 16.