Positional Notation · aiwiki.page

Positional notation is a method of writing a number in which a digit’s contribution depends on both the digit itself and its position. In a conventional fixed-base system, successive places correspond to successive powers of a base. The familiar Hindu–Arabic numeral system uses base ten: in 555, the three identical symbols represent five hundreds, five tens, and five units. Positional notation also underlies binary and other systems used in computers. (openstax.org)

Mathematical structure

An ordinary positional system chooses an integer base $b\geq2$ , also called a radix, and digits with values $0,1,\ldots,b-1$ . A finite numeral has the interpretation

(d_n d_{n-1}\cdots d_0.d_{-1}\cdots d_{-m})_b =\sum_{k=-m}^{n}d_kb^k.

The radix point separates the units place from fractional places. Positions to its left have weights $1,b,b^2,\ldots$ ; those to its right have weights $b^{-1},b^{-2},\ldots$ . These weights use exponentiation, including negative exponents for fractions. (mathworld.wolfram.com)

For example,

(304.25)_{10} =3\times10^2+0\times10+4 +2\times10^{-1}+5\times10^{-2}.

In binary, the same principle gives

(101.01)_2=4+1+\frac14=5.25.

The base subscript identifies how the symbols should be interpreted; it does not change the represented quantity. A numeral is therefore distinct from the number it denotes. (openstax.org)

Digits, zero, and nonpositional systems

Zero preserves an unoccupied place. In 304, it indicates that no tens contribute to the value; removing it produces 34, a different number. Leading zeros ordinarily leave a numerical value unchanged, although they may serve formatting purposes. Thus 0034 and 34 denote the same integer. (openstax.org)

The shapes of the digits are separate from the place-value principle. A system may use individual characters or groups of marks to express each digit value. Babylonian notation, for example, combined unit and ten signs within a place while assigning powers of sixty to successive places. (mathshistory.st-andrews.ac.uk)

Positional notation contrasts with systems such as Roman numerals. Roman symbols have relatively fixed values, combined through additive and subtractive conventions. Their order matters, but their positions do not assign successive powers of a base. Merely making symbol order significant does not make a notation positional. (openstax.org)

Historical development

Babylonian scribes in Mesopotamia developed a sexagesimal, or base-sixty, positional system. Each place could express values up to fifty-nine. Early notation lacked a consistently used placeholder for empty positions, and the absolute scale of a numeral often depended on context. Later scribes introduced a placeholder, but it did not function in every respect like modern numerical zero. (mathshistory.st-andrews.ac.uk)

The Maya developed another positional tradition using base twenty, with dot-and-bar digit forms and a zero sign. Their calendrical notation included modified place weights, so not every Maya numeral should be interpreted as an unqualified sequence of powers of twenty. (openstax.org)

Decimal positional notation developed in India through a history distinct from the evolution of the digit shapes themselves. Earlier Brahmi numerals were not a fully positional decimal system. Indian place-value notation incorporating zero subsequently spread through Arabic-language scholarship and into Europe. Al-Khwarizmi helped transmit Indian calculation methods, while Fibonacci presented them in his Liber abaci of 1202. (mathshistory.st-andrews.ac.uk)

Arithmetic and base conversion

Place value provides the structure for written arithmetic. Addition aligns equal-weight positions. When a column total reaches the base, units are exchanged for a unit in the next position: in binary, $1+1=10_2$ . Subtraction reverses this exchange through borrowing. Multiplication combines digit products with the appropriate positional shifts. (openstax.org)

A base-conversion algorithm for a nonnegative integer repeatedly divides by the target base. The remainders give successive digits from right to left. Conversely, evaluating a numeral’s weighted sum converts it into a numerical value. For example, repeated division of thirteen by two gives remainders $1,0,1,1$ , read in reverse as $1101_2$ . (openstax.org)

Fractional expansions and computing

Positional notation also represents fractions and real numbers. An infinite fractional expansion is interpreted through the limit of its finite truncations. Rational numbers have terminating or eventually repeating expansions in an ordinary integer base, whereas irrational numbers have nonterminating expansions that are not eventually periodic. Whether a particular fraction terminates depends on the base. (maths.ucd.ie)

Binary notation uses two digit values, corresponding to the possible values of a bit. Hexadecimal notation uses sixteen values, conventionally written 0–9 and A–F, and offers a compact way to express binary quantities. Decimal 0.1 has an infinite binary expansion; consequently, finite binary floating-point arithmetic generally stores an approximation rather than that exact value. This is a limitation of the chosen representation, not a change in the underlying number. (openstax.org)

The broader positional principle also permits mixed-radix systems, negative bases, and noninteger bases. These generalizations alter the place weights or allowed representations and need not share all the properties of ordinary positive-integer-base notation. (mathworld.wolfram.com)