Decimal

2008/9 Schools Wikipedia Selection. Related subjects: Mathematics

The decimal (base ten or occasionally denary) numeral system has ten as its base. It is the most widely used numeral system, perhaps because humans have four fingers and a thumb on each hand, giving a total of ten digits over both hands.

Decimal notation

Decimal notation is the writing of numbers in the base-ten numeral system, which uses various symbols (called digits) for no more than ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent any numbers, no matter how large. These digits are often used with a decimal separator which indicates the start of a fractional part, and with one of the sign symbols + (positive) or − (negative) in front of the numerals to indicate sign. There are only two truly positional decimal systems in ancient civilization, the Chinese counting rods system and Hindu-Arabic numeric system, both required no more than ten symbols. Other numeric systems require more or fewer symbols.

The decimal system is a positional numeral system; it has positions for units, tens, hundreds, etc. The position of each digit conveys the multiplier (a power of ten) to be used with that digit—each position has a value ten times that of the position to its right.

Ten is the number which is the count of fingers and thumbs on both hands (or toes on the feet). In many languages the word digit or its translation is also the anatomical term referring to fingers and toes. In English, decimal (decimus < Lat.) means tenth, decimate means reduce by a tenth, and denary (denarius < Lat.) means the unit of ten. The symbols for the digits in common use around the globe today are called Arabic numerals by Europeans and Indian numerals by Arabs, the two groups' terms both referring to the culture from which they learned the system. However, the symbols used in different areas are not identical; for instance, Western Arabic numerals (from which the European numerals are derived) differ from the forms used by other Arab cultures.

Alternative notations

Some cultures do, or used to, use other numeral systems, including pre-Columbian Mesoamerican cultures such as the Maya, who use a vigesimal system (using all twenty fingers and toes), some Nigerians who use several duodecimal (base 12) systems, the Babylonians, who used sexagesimal (base 60), and the Yuki, who reportedly used octal (base 8).

Computer hardware and software systems commonly use a binary representation, internally. For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems. For most purposes, however, binary values are converted to the equivalent decimal values for presentation to and manipulation by humans.

Both computer hardware and software also use internal representations which are effectively decimal for storing decimal values and doing arithmetic. Often this arithmetic is done on data which are encoded using binary-coded decimal, but there are other decimal representations in use (see IEEE 754r), especially in database implementations. Decimal arithmetic is used in computers so that decimal fractional results can be computed exactly, which is not possible using a binary fractional representation. This is often important for financial and other calculations .

Decimal fractions

A decimal fraction is a fraction where the denominator is a power of ten.

Decimal fractions are commonly expressed without a denominator, the decimal separator being inserted into the numerator (with leading zeros added if needed), at the position from the right corresponding to the power of ten of the denominator. e.g., 8/10, 833/100, 83/1000, and 8/10000 are expressed as: 0.8, 8.33, 0.083, and 0.0008. In English-speaking countries, a dot (·) or period (.) is used as the decimal separator; in most other languages a comma is used.

The integer part or integral part of a decimal number is the part to the left of the decimal separator (see also floor function). The part from the decimal separator to the right is the fractional part; if considered as a separate number, a zero is often written in front. Especially for negative numbers, we have to distinguish between the fractional part of the notation and the fractional part of the number itself, because the latter gets its own minus sign. It is usual for a decimal number which is less than one to have a leading zero.

Trailing zeros after the decimal point are not necessary, although in science, engineering and statistics they can be retained to indicate a required precision or to show a level of confidence in the accuracy of the number: Whereas 0.080 and 0.08 are numerically equal, in engineering 0.080 suggests a measurement with an error of up to 1 part in two thousand (±0.0005), while 0.08 suggests a measurement with an error of up to 1 in two hundred (see Significant figures).

Other rational numbers

Any rational number which cannot be expressed as a decimal fraction has a unique infinite decimal expansion ending with recurring decimals.

Ten is the product of the first and third prime numbers, is one greater than the square of the second prime number, and is one less than the fifth prime number. This leads to plenty of simple decimal fractions:

1/2 = 0.5

1/3 = 0.333333… (with 3 repeating)

1/4 = 0.25

1/5 = 0.2

1/6 = 0.166666… (with 6 repeating)

1/8 = 0.125

1/9 = 0.111111… (with 1 repeating)

1/10 = 0.1

1/11 = 0.090909… (with 09 repeating)

1/12 = 0.083333… (with 3 repeating)

1/81 = 0.012345679012… (with 012345679 repeating)

Other prime factors in the denominator will give longer recurring sequences, see for instance 7, 13.

That a rational number must have a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are only q-1 possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q. For instance to find 3/7 by long division:

      .4 2 8 5 7 1 4 ...
 7 ) 3.0 0 0 0 0 0 0 0 
     2 8                         30/7 = 4 r 2
       2 0
       1 4                       20/7 = 2 r 6
         6 0
         5 6                     60/7 = 8 r 4
           4 0
           3 5                   40/7 = 5 r 5
             5 0
             4 9                 50/7 = 7 r 1
               1 0
                 7               10/7 = 1 r 3
                 3 0
                 2 8             30/7 = 4 r 2  (again)
                   2 0
                        etc

The converse to this observation is that every recurring decimal represents a rational number p/q. This is a consequence of the fact the recurring part of a decimal representation is, in fact, an infinite geometric series which will sum to a rational number. For instance,

Real numbers

Every real number has a (possibly infinite) decimal representation, i.e., it can be written as

where

• sign() is the sign function,
• a_i ∈ { 0,1,…,9 } for all i ∈ Z, are its decimal digits, equal to zero for all i greater than some number (that number being the common logarithm of |x|).

Such a sum converges as i decreases, even if there are infinitely many nonzero a_i.

Rational numbers (e.g. p/q) with prime factors in the denominator other than 2 and 5 (when reduced to simplest terms) have a unique recurring decimal representation.

Consider those rational numbers which have only the factors 2 and 5 in the denominator, i.e. which can be written as p/(2^a5^b). In this case there is a terminating decimal representation. For instance 1/1=1, 1/2=0.5, 3/5=0.6, 3/25=0.12 and 1306/1250=1.0448. Such numbers are the only real numbers which don't have a unique decimal representation, as they can also be written as a representation that has a recurring 9, for instance 1=0.99999…, 1/2=0.499999…, etc.

This leaves the irrational numbers. They also have unique infinite decimal representation, and can be characterised as the numbers whose decimal representations neither terminate nor recur.

So in general the decimal representation is unique, if one excludes representations that end in a recurring 9.

Naturally, the same trichotomy holds for other base-n positional numeral systems:

• Terminating representation: rational where the denominator divides some n^k
• Recurring representation: other rational
• Non-terminating, non-recurring representation: irrational

and a version of this even holds for irrational-base numeration systems, such as golden mean base representation.

History

There follows a chronological list of recorded decimal writers.

Decimal writers

• c. 3500 - 2500 BC Elamites of Iran possibly used early forms of decimal system.
• c. 2900 BC Egyptian hieroglyphs show counting in powers of 10 (1 million + 400,000 goats, etc.) – see Ifrah, below
• c. 2600 BC Indus Valley Civilization, earliest known physical use of decimal fractions in ancient weight system: 1/20, 1/10, 1/5, 1/2. See Ancient Indus Valley weights and measures
• c. 1400 BC Chinese writers show familiarity with the concept: for example, 547 is written 'Five hundred plus four decades plus seven of days' in some manuscripts
• c. 1200 BC In ancient India, the Vedic text Yajur-Veda states the powers of 10, up to 10⁵⁵
• c. 400 BC Pingala – develops the binary number system for Sanskrit prosody, with a clear mapping to the base-10 decimal system
• c. 250 BC Archimedes writes the Sand Reckoner, which takes decimal calculation up to 10^{80,000,000,000,000,000}
• c. 100–200 The Satkhandagama written in India – earliest use of decimal logarithms
• c. 476–550 Aryabhata – uses an alphabetic cipher system for numbers that used zero
• c. 598–670 Brahmagupta – explains the Hindu-Arabic numerals (modern number system) which uses decimal integers, negative integers, and zero
• c. 780–850 Muḥammad ibn Mūsā al-Ḵwārizmī – first to expound on algorism outside India
• c. 920–980 Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi – earliest known direct mathematical treatment of decimal fractions.
• c. 1300–1500 The Kerala School in South India – decimal floating point numbers
• 1548/49–1620 Simon Stevin – author of De Thiende ('the tenth')
• 1561–1613 Bartholemaeus Pitiscus – (possibly) decimal point notation.
• 1550–1617 John Napier – use of decimal logarithms as a computational tool
• 1765 Johann Heinrich Lambert – discusses (with few if any proofs) patterns in decimal expansions of rational numbers and notes a connection with Fermat's little theorem in the case of prime denominators
• 1800 Karl Friedrich Gauss – uses number theory to systematically explain patterns in recurring decimal expansions of rational numbers (e.g., the relation between period length of the recurring part and the denominator, which fractions with the same denominator have recurring decimal parts which are shifts of each other, like 1/7 and 2/7) and also poses questions which remain open to this day (e.g., a special case of Artin's conjecture on primitive roots: is 10 a generator modulo p for infinitely many primes p?).
• 1925 Louis Charles Karpinski – The History of Arithmetic
• 1959 Werner Buchholz – Fingers or Fists? (The Choice of Decimal or Binary representation)
• 1974 Hermann Schmid – Decimal Computation
• 2000 Georges Ifrah – The Universal History of Numbers: From Prehistory to the Invention of the Computer
• 2003 Mike Cowlishaw – Decimal Floating-Point: Algorism for Computers.

Natural languages

A straightforward decimal system, in which 11 is expressed as ten-one and 23 as two-ten-three, is found in Chinese languages except Wu, and in Vietnamese with a few irregularities. Japanese, Korean, and Thai have imported the Chinese decimal system. Many other languages with a decimal system have special words for the numbers between 10 and 20, and decades.

Incan languages such as Quechua and Aymara have an almost straightforward decimal system, in which 11 is expressed as ten with one and 23 as two-ten with three.

Some psychologists suggest irregularities of numerals in a language may hinder children's counting ability.