Decimal

The decimal numeral system (also called base-ten positional numeral system, and occasionally called denary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system.^[1] The way of denoting numbers in the decimal system is often referred to as decimal notation.^[2]

A decimal numeral, or just decimal, or casually decimal number, refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified for containing a decimal separator (for example the "." in 10.00 or 3.14159). "Decimal" may also refer specifically to the digits after the decimal separator, such as in "3.14 is the approximation of π to two decimals".

The numbers that may be represented in the decimal system are the decimal fractions, that is the fractions of the form a/10ⁿ, where a is an integer, and n is a non-negative integer.

The decimal system has been extended to infinite decimals, for representing any real number, by using an infinite sequence of digits after the decimal separator (see Decimal representation). In this context, the decimal numerals with a finite number of non–zero places after the decimal separator are sometimes called terminating decimals. A repeating decimal, is an infinite decimal, that, after some place repeats indefinitely the same sequence of digits (for example 5.123144144144144... = 5.123144).^[3] An infinite decimal represents a rational number if and only if it is a repeating decimal or has a finite number of nonzero digits.

OriginEdit

Ten fingers on two hands, the possible starting point of the decimal counting.

Many numeral systems of ancient civilisations use ten and its powers for representing numbers, probably because there are ten fingers on two hands and people started counting by using their fingers.^{[citation needed]}Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, and Chinese numerals. Very large numbers were difficult to represent in these old numeral systems, and, only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some non-integer numbers, called decimal fractions or decimal numbers for forming the decimal numeral system.

Decimal fractionsEdit

The numbers that are represented by decimal numeral are the decimal fractions (sometimes called decimal numbers), that is, the rational numbers that may be expressed as a fraction, the denominator of which is a power of ten.^[5] For example, the numerals ${\displaystyle 0.8,14.89,0.00024}$ represent the fractions 8/10, 1489/100, 24/100000. More generally, a decimal with n digits after the separator represents the fraction with denominator 10ⁿ, whose numerator is the integer obtained by removing the separator.

Expressed as a fully reduced fraction, the decimal numbers are those, whose denominator is a product of a power of 2 by a power of 5. Thus the smallest denominators of decimal numbers are

${\displaystyle 1=2^{0}\cdot 5^{0},2=2^{1}\cdot 5^{0},4=2^{2}\cdot 5^{0},5=2^{0}\cdot 5^{1},8=2^{3}\cdot 5^{0},10=2^{1}\cdot 5^{1},16=2^{4}\cdot 5^{0},25=2^{0}\cdot 5^{2},\ldots }$

The integer part, or integral part of a decimal is the integer written to the left of the decimal separator (see also truncation). For a non-negative decimal, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is the fractional part, which equals the difference between the numeral and its integer part.

When the integral part of a numeral is zero, it may occur, typically in computing, that the integer part is not written (for example .1234, instead of 0.1234). In normal writing, this is generally avoided because of the risk of confusion between the decimal mark and other punctuation.

HistoryEdit

The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the Warring States period in China.

Many ancient cultures calculated with numerals based on ten, sometimes argued due to human hands typically having ten digits. Standardized weights used in Indus Valley Civilization (c.3300-1300 BCE) were based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, while their standardized ruler – the Mohenjo-daro ruler – was divided into ten equal parts. Egyptian hieroglyphs, in evidence since around 3000 BC, used a purely decimal system, just as the Cretan hieroglyphs (ca. 1625−1500 BC) of the Minoans whose numerals are closely based on the Egyptian model. The decimal system was handed down to the consecutive Bronze Age cultures of Greece, including Linear A (ca. 18th century BC−1450 BC) and Linear B (ca. 1375−1200 BC) — the number system of classical Greece also used powers of ten, including, like the Roman numerals did, an intermediate base of 5. Notably, the polymath Archimedes (ca. 287–212 BC) invented a decimal positional system in his Sand Reckoner which was based on 10 and later led the German mathematician Carl Friedrich Gauss to lament what heights science would have already reached in his days if Archimedes had fully realized the potential of his ingenious discovery. The Hittites hieroglyphs (since 15th century BC), just like the Egyptian and early numerals in Greece, was strictly decimal.

Some non-mathematical ancient texts like the Vedas dating back to 1900–1700 BCE make use of decimals and mathematical decimal fractions.

The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20, to 90, 100, 200, to 900, 1000, 2000, 3000, 4000, to 10,000. The world's earliest positional decimal system was the Chinese rod calculus.

The world's earliest positional decimal system
Upper row vertical form
Lower row horizontal form

History of decimal fractionsEdit

counting rod decimal fraction 1/7

Decimal fractions were first developed and used by the Chinese in the end of 4th century BC, and then spread to the Middle East and from there to Europe. The written Chinese decimal fractions were non-positional. However, counting rod fractions were positional.

Qin Jiushao in his book Mathematical Treatise in Nine Sections (1247) denoted 0.96644 by

寸

     , meaning

寸

096644