Numerical digit

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The ten digits of the Hindu-Arabic numeral system, in order of value.

In mathematics and computer science, a digit is a symbol (a number symbol, e.g. "3" or "7") used in numerals (combinations of symbols, e.g. "37"), to represent numbers, (integers or real numbers) in positional numeral systems. The name "digit" comes from the fact that the 10 digits (ancient Latin digita meaning fingers) of the hands correspond to the 10 symbols of the common base 10 number system, i.e. the decimal (ancient Latin adjective dec. meaning ten) digits.

In a given number system, if the base is an integer, the number of digits required is always equal to the absolute value of the base.

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 total 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 the Hindu-Arabic numeral system the digit "1" represents the integer one, and in the hexadecimal system, the digit "A" represents the number ten. A positional number system must have a digit representing the integers from zero up to, but not including, the radix of the number system.

Computation of place values

The Arabic numeral system uses a separator, commonly a period in the United States or a comma in Europe, to denote the "ones 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 (written in base ten),

the 0 is immediately to the left of the separator, so it is in the ones place;
the 1 to the left of the zero has a place value of one, and is in the tens place;
the 3 is to the right of the ones place, so it is in the tenths place; and
the 4 to the right of the tenths place is in the hundredths place.

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.

History

Glyphs used to represent digits of the Hindu-Arabic numeral system.

The first true written positional numeral system is considered to be the Hindu-Arabic numeral system. This system was established by the 7th century, 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, a space was left in the numeral as a placeholder. The first widely acknowledged use of zero was in 876. Although the original Hindu-Arabic system was very similar to the modern one, even down to the glyphs used to represent digits, the direction of writing was reversed, so that place values increased to the right rather than to the left.

The digits of the Maya numeral system, with Hindu-Arabic equivalents

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.

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 (base twenty), 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 Hindu-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 Hindu-Arabic numeral system. The Suzhou nemerals are variants of rod numerals.

Rod numerals (vertical)
0 1 2 3 4 5 6 7 8 9
-0 -1 -2 -3 -4 -5 -6 -7 -8 -9

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 is base two and uses only the digits "0" and "1", while the octal system is base eight and 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 from ten to fifteen respectively.

Unusual systems

The ternary system is infrequently used; it is a simple base-three system.

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

Casting out nines is a procedure for checking arithmetic done by hand. To describe it, let $f(x)\,$ represent the digital root of $x\,$, as described above. Casting out nines makes use of the fact that if $A + B = C\,$, then $f(f(A) + f(B)) = f(C)\,$. 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 (one thousand, one hundred eleven) 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 primacy 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.