Numerical digit

A numerical digit or numeral is a single symbol used alone, or in combinations, to represent numbers in positional notation, such as the common base 10. The name "digit" originates from the Latin digiti meaning fingers.
For any numeral system with an integer base, the number of different digits required is the absolute value of the base. For example, decimal requires ten digits, and binary requires only two digits. Bases greater than 10 require more than 10 digits, for instance hexadecimal requires 16 digits.

Overview

In a basic digital system, a numeral is a sequence of digits, which may be of arbitrary length. Each position in the sequence has a place value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its place value, and summing the results.

Digital values

Each digit in a number system represents an integer. For example, in decimal the digit "1" represents the integer one, and in the hexadecimal system, the letter "A" represents the number ten. A positional number system has one unique digit for each integer from zero up to, but not including, the radix of the number system.
Thus in the positional decimal system, the numbers 0 to 9 can be expressed using their respective numerals "0" to "9" in the rightmost "units" position. The number 12 is expressed with the numeral "2" in the units position, and with the numeral "1" in the "tens" position, to the left of the "2" while the number 312 is expressed with three numerals: "3" in the "hundreds" position, "1" in the "tens" position, and "2" in the "units" position.

Computation of place values

The decimal numeral system uses a decimal separator, commonly a period in English, or a comma in other European languages, to denote the "ones place" or "units place", which has a place value one. Each successive place to the left of this has a place value equal to the place value of the previous digit times the base. Similarly, each successive place to the right of the separator has a place value equal to the place value of the previous digit divided by the base. For example, in the numeral 10.34,
The total value of the number is 1 ten, 0 ones, 3 tenths, and 4 hundredths. The zero, which contributes no value to the number, indicates that the 1 is in the tens place rather than the ones place.
The place value of any given digit in a numeral can be given by a simple calculation, which in itself is a complement to the logic behind numeral systems. The calculation involves the multiplication of the given digit by the base raised by the exponent, where n represents the position of the digit from the separator; the value of n is positive, but this is only if the digit is to the left of the separator. And to the right, the digit is multiplied by the base raised by a negative n. For example, in the number 10.34,

History

Western Arabic	0	1	2	3	4	5	6	7	8	9
Eastern Arabic	٠	١	٢	٣	٤	٥	٦	٧	٨	٩
Persian	۰	۱	۲	۳	۴	۵	۶	۷	۸	۹
Devanagari	०	१	२	३	४	५	६	७	८	९
Kadamba	೦	೧	೨	೩	೪	೫	೬	೭	೮	೯

The first true written positional numeral system is considered to be the Hindu–Arabic numeral system. This system was established by the 7th century in India, but was not yet in its modern form because the use of the digit zero had not yet been widely accepted. Instead of a zero sometimes the digits were marked with dots to indicate their significance, or a space was used as a placeholder. The first widely acknowledged use of zero was in 876. The original numerals were very similar to the modern ones, even down to the glyphs used to represent digits.
Image:Maya.svg|thumb|left|150px|The digits of the Maya numeral system
By the 13th century, Western Arabic numerals were accepted in European mathematical circles. They began to enter common use in the 15th century. By the end of the 20th century virtually all non-computerized calculations in the world were done with Arabic numerals, which have replaced native numeral systems in most cultures.

Other historical numeral systems using digits

The exact age of the Maya numerals is unclear, but it is possible that it is older than the Hindu–Arabic system. The system was vigesimal, so it has twenty digits. The Mayas used a shell symbol to represent zero. Numerals were written vertically, with the ones place at the bottom. The Mayas had no equivalent of the modern decimal separator, so their system could not represent fractions.
The Thai numeral system is identical to the Hindu–Arabic numeral system except for the symbols used to represent digits. The use of these digits is less common in Thailand than it once was, but they are still used alongside Arabic numerals.
The rod numerals, the written forms of counting rods once used by Chinese and Japanese mathematicians, are a decimal positional system able to represent not only zero but also negative numbers. Counting rods themselves predate the Hindu–Arabic numeral system. The Suzhou numerals are variants of rod numerals.

0	1	2	3	4	5	6	7	8	9
Image:Counting rod 0.png	Image:Counting rod v1.png	Image:Counting rod v2.png	Image:Counting rod v3.png	Image:Counting rod v4.png	Image:Counting rod v5.png	Image:Counting rod v6.png	Image:Counting rod v7.png	Image:Counting rod v8.png	Image:Counting rod v9.png
−0	−1	−2	−3	−4	−5	−6	−7	−8	−9
Image:Counting rod -0.png	Image:Counting rod v-1.png	Image:Counting rod v-2.png	Image:Counting rod v-3.png	Image:Counting rod v-4.png	Image:Counting rod v-5.png	Image:Counting rod v-6.png	Image:Counting rod v-7.png	Image:Counting rod v-8.png	Image:Counting rod v-9.png

Modern digital systems

In computer science

The binary, octal, and hexadecimal systems, extensively used in computer science, all follow the conventions of the Hindu–Arabic numeral system. The binary system uses only the digits "0" and "1", while the octal system uses the digits from "0" through "7". The hexadecimal system uses all the digits from the decimal system, plus the letters "A" through "F", which represent the numbers 10 to 15 respectively. When the binary system is used, the term "bit" is typically used as an alternative for "digit", being a portmanteau of the term "binary digit".

Unusual systems

The ternary and balanced ternary systems have sometimes been used. They are both base 3 systems.
Balanced ternary is unusual in having the digit values 1, 0 and −1. Balanced ternary turns out to have some useful properties and the system has been used in the experimental Russian Setun computers.
Several authors in the last 300 years have noted a facility of positional notation that amounts to a modified decimal representation. Some advantages are cited for use of numerical digits that represent negative values. In 1840 Augustin-Louis Cauchy advocated use of signed-digit representation of numbers, and in 1928 Florian Cajori presented his collection of references for negative numerals. The concept of signed-digit representation has also been taken up in computer design.

Digits in mathematics

Despite the essential role of digits in describing numbers, they are relatively unimportant to modern mathematics. Nevertheless, there are a few important mathematical concepts that make use of the representation of a number as a sequence of digits.

Digital roots

The digital root is the single-digit number obtained by summing the digits of a given number, then summing the digits of the result, and so on until a single-digit number is obtained.

Casting out nines

is a procedure for checking arithmetic done by hand. To describe it, let represent the digital root of, as described above. Casting out nines makes use of the fact that if, then. In the process of casting out nines, both sides of the latter equation are computed, and if they are not equal, the original addition must have been faulty.

Repunits and repdigits

Repunits are integers that are represented with only the digit 1. For example, 1111 is a repunit. Repdigits are a generalization of repunits; they are integers represented by repeated instances of the same digit. For example, 333 is a repdigit. The primality of repunits is of interest to mathematicians.

Palindromic numbers and Lychrel numbers

Palindromic numbers are numbers that read the same when their digits are reversed. A Lychrel number is a positive integer that never yields a palindromic number when subjected to the iterative process of being added to itself with digits reversed. The question of whether there are any Lychrel numbers in base 10 is an open problem in recreational mathematics; the smallest candidate is 196.

History of ancient numbers

Counting aids, especially the use of body parts, were certainly used in prehistoric times as today. There are many variations. Besides counting ten fingers, some cultures have counted knuckles, the space between fingers, and toes as well as fingers. The Oksapmin culture of New Guinea uses a system of 27 upper body locations to represent numbers.
To preserve numerical information, tallies carved in wood, bone, and stone have been used since prehistoric times. Stone age cultures, including ancient indigenous American groups, used tallies for gambling, personal services, and trade-goods.
A method of preserving numeric information in clay was invented by the Sumerians between 8000 and 3500 BC. This was done with small clay tokens of various shapes that were strung like beads on a string. Beginning about 3500 BC, clay tokens were gradually replaced by number signs impressed with a round stylus at different angles in clay tablets which were then baked. About 3100 BC, written numbers were dissociated from the things being counted and became abstract numerals.
Between 2700 and 2000 BC, in Sumer, the round stylus was gradually replaced by a reed stylus that was used to press wedge-shaped cuneiform signs in clay. These cuneiform number signs resembled the round number signs they replaced and retained the additive sign-value notation of the round number signs. These systems gradually converged on a common sexagesimal number system; this was a place-value system consisting of only two impressed marks, the vertical wedge and the chevron, which could also represent fractions. This sexagesimal number system was fully developed at the beginning of the Old Babylonia period and became standard in Babylonia.
Sexagesimal numerals were a mixed radix system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons. By 1950 BC, this was a positional notation system. Sexagesimal numerals came to be widely used in commerce, but were also used in astronomical and other calculations. This system was exported from Babylonia and used throughout Mesopotamia, and by every Mediterranean nation that used standard Babylonian units of measure and counting, including the Greeks, Romans and Egyptians. Babylonian-style sexagesimal numeration is still used in modern societies to measure time and angles.