Number systems

A number system represents numbers with a set of symbols (digits). The number of symbols is its base (or radix). Four number systems are used with computers:

These are positional systems: the value of a digit depends on its position. Positions are numbered from the decimal point: the rightmost digit of the whole-number part is position 0, then 1, 2, 3… going left. After the point, positions are –1, –2, –3… going right. Each digit is worth digit × base^position, and the number is the sum of these.

For example, (345.6)₁₀ = 3 × 10² + 4 × 10¹ + 5 × 10⁰ + 6 × 10⁻¹ = 300 + 40 + 5 + 0.6.

• Binary uses only 0 and 1, which match the OFF and ON states of a computer's transistors.
• Octal uses 0-7. Since 8 = 2³, one octal digit stands for exactly 3 bits.
• Hexadecimal uses 0-9 and the letters A-F, where A = 10, B = 11, C = 12, D = 13, E = 14 and F = 15. Since 16 = 2⁴, one hexadecimal digit stands for exactly 4 bits.

Hexadecimal makes long binary numbers short and easier to read. A 16-bit memory address such as 1010111100001111 can be written as just AF0F. Colours on a screen use 24 bits (8 each for red, green and blue), giving about 16 million colours; they are written as three hexadecimal pairs: pure green is (00, FF, 00), white is (FF, FF, FF).