A numerical digit is a single symbol (such as "2" or "5") used alone, or in combinations (such as "25"), to represent (such as the number 25) according to some positional . The single digits –as one-digit-numerals– and their combinations (such as "25") are the numerals of the numeral system they belong to. The name "digit" comes from the fact that the ten digits (Latin digiti meaning fingers) of the hands correspond to the ten symbols of the common base 10 numeral system, i.e. the decimal (ancient Latin adjective decem meaning ten) digits.
For a given numeral system with an integer radix, the number of digits required to express arbitrary numbers is given by the absolute value of the base. For example, the decimal system (base 10) requires ten digits (0 through to 9), whereas the binary system (base 2) has two digits (e.g.: 0 and 1).
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 can be 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 can be expressed by three numerals: "3" in the "hundreds" position, "1" in the "tens" position, and "2" in the "units" position.
The total value of the number is 1 ten, 0 ones, 3 tenths, and 4 hundredths. Note that 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 compliment 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 (written in base 10),
By the 13th century, Hindu–Arabic numerals were accepted in European mathematical circles (Fibonacci used them in his Liber Abaci). 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.
The Thai numerals 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 Hindu–Arabic numerals.
The rod numerals, the written forms of counting rods once used by China and 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.
|+ Rod numerals (vertical)|
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.
To preserve numerical information, Tally marks 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 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 (originally containers for tokens) 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 (about 1950 BC) 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 (minutes per hour) and (degrees).
The oldest Greek system was that of the Attic numerals, but in the 4th century BC they began to use a quasidecimal alphabetic system (see Greek numerals). Jews began using a similar system (Hebrew numerals), with the oldest examples known being coins from around 100 BC.
The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers. The Roman numerals remained in common use in Europe until positional notation came into common use in the 16th century.
The Maya numerals of Central America used a mixed base 18 and base 20 system, possibly inherited from the Olmec, including advanced features such as positional notation and a zero.. They used this system to make advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit of Venus.
The Incan Empire ran a large command economy using quipu, tallies made by knotting colored fibers. Knowledge of the encodings of the knots and colors was suppressed by the Spain in the 16th century, and has not survived although simple quipu-like recording devices are still used in the Andes region.
Some authorities believe that positional arithmetic began with the wide use of counting rods in China. The earliest written positional records seem to be rod calculus results in China around 400. In particular, zero was correctly described by Chinese mathematicians around 932.
The modern positional Arabic numeral system was developed by mathematicians in India, and passed on to Muslim mathematicians, along with astronomical tables brought to Baghdad by an Indian ambassador around 773.
From India, the thriving trade between Islamic sultans and Africa carried the concept to Cairo. Arabic mathematicians extended the system to include Decimal, and Muḥammad ibn Mūsā al-Ḵwārizmī wrote an important work about it in the 9th century. The modern Arabic numerals were introduced to Europe with the translation of this work in the 12th century in Spain and Leonardo of Pisa's Liber Abaci of 1201. In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century.
The binary system (base 2), was propagated in the 17th century by Gottfried Leibniz. Leibniz had developed the concept early in his career, and had revisited it when he reviewed a copy of the I ching from China. Binary numbers came into common use in the 20th century because of computer applications.