Positional notation

Positional notation, also known as place-value notation, is the property of a numeral system that the value represented by each symbol in a written numeral depends not only on its appearance but also on its position. Each symbol fits in a specific place or position, representing a power of a fixed base. The most common numeral system used today, the Hindu–Arabic numeral system, is a positional system in base ten; each of ten numerical digits is a distinct symbol representing one of the numbers zero through nine, and in the context of the full numeral, each symbol's value is the digit multiplied by a power of ten.

Most early numeral systems, such as Roman numerals, are essentially based on the additive principle: each symbol type represents one fixed value, and the value of a numeral is the sum of the values of the separate symbols. For example, the Roman numeral CCXXVIII has two copies of the symbol 10, one I meaning 228, consists of the symbol 2 representing 8 × 1.

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.

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.

Counting rods and most abacuses have been used to represent numbers in a positional numeral system. With counting rods or abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly.

The oldest extant positional notation system is either that of Chinese rod numerals, used from at least the early 8th century, or perhaps Khmer numerals, showing possible usages of positional-numbers in the 7th century. Khmer numerals and other Indian numerals originate with the Brahmi numerals of about the 3rd century BC, which symbols were, at the time, not used positionally. Medieval Indian numerals are positional, as are the derived Arabic numerals, recorded from the 10th century.

After the French Revolution (1789–1799), the new French government promoted the extension of the decimal system. Some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world.