Number systems

A number system is a set of digits (signs) for writing numbers together with the rules for using them. The number of digits in the system is its base: in the decimal system the base is 10 (digits 0 to 9), in the binary system it is 2 (0 and 1). One-place signs are digits, many-place writings made of them are numbers: 5, 6, 8 are digits, 568 is a number. In the past various peoples used five-based, twelve-based, twenty-based (Maya) and sixty-based (Babylonian) systems; the 60 minutes of an hour and the 12 + 12 hours of a day remind us of this. In a positional system the value of a digit depends on its place (position) in the number: in 333 the three 3s mean 300, 30 and 3. The Roman system is not positional: in XXX each X always means 10. A system with base p has digits from 0 to p – 1; when the base is above 10, letters A, B, C, … are added (in base 16, A = 10, …, F = 15). The next number is made by “shifting” the last digit, while the largest digit turns into 0 and shifts its left neighbour. In every positional system its own base is written “10”, so the base is shown as a subscript: 101₂, 101₈. In expanded form a number is written as the sum of its digits times powers of the base: 4072 = 4 · 10³ + 0 · 10² + 7 · 10¹ + 2 · 10⁰. One of the scholars who spread today’s decimal positional system was Muhammad al-Khwarizmi (born in Khorezm, about 780–850): his treatise on Hindu arithmetic was translated into Latin, and its opening words “Dixit Algorizmi” gave rise to the word “algorithm”.