Positional notation

Positional notation, also known as place-value notation, positional numeral system, or simply place value, usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred (however, the values may be modified when combined). In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.

The Babylonian numeral system, base 60, was the first positional system to be developed, and its influence is present today in the way time and angles are counted in tallies related to 60, such as 60 minutes in an hour and 360 degrees in a circle. The Inca used knots tied in a decimal positional system to store numbers and other values in quipu cords.

Today, the Hindu–Arabic numeral system (base ten) is the most commonly used system globally. However, the binary numeral system (base two) is used in almost all computers and electronic devices because it is easier to implement efficiently in electronic circuits.

Systems with negative base, complex base or negative digits have been described. Most of them do not require a minus sign for designating negative numbers.

The use of a radix point (decimal point in base ten), extends to include fractions and allows the representation of any real number with arbitrary accuracy. With positional notation, arithmetical computations are much simpler than with any older numeral system; this led to the rapid spread of the notation when it was introduced in western Europe.

Today, the base-10 (decimal) system, which is presumably motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past, and some continue to be used today. For example, the Babylonian numeral system, credited as the first positional numeral system, was base-60. However, it lacked a real zero. Initially inferred only from context, later, by about 700 BC, zero came to be indicated by a "space" or a "punctuation symbol" (such as two slanted wedges) between numerals. It was a placeholder rather than a true zero because it was not used alone or at the end of a number. Numbers like 2 and 120 (2×60) looked the same because the larger number lacked a final placeholder. Only context could differentiate them.

The polymath Archimedes (ca. 287–212 BC) invented a decimal positional system based on 10⁸ in his Sand Reckoner; 19th century German mathematician Carl Gauss lamented how science might have progressed had Archimedes only made the leap to something akin to the modern decimal system. Hellenistic and Roman astronomers used a base-60 system based on the Babylonian model (see Greek numerals § Zero).

Before positional notation became standard, simple additive systems (sign-value notation) such as Roman numerals or Chinese numerals were used, and accountants in the past used the abacus or stone counters to do arithmetic until the introduction of positional notation.