In mathematics and computer science, a digit is a symbol (a number symbol, e. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their g. "3" or "7") used in numerals (combinations of symbols, e. g. "37"), to represent numbers, (integers or real numbers) in positional numeral systems. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the real numbers may be described informally in several different ways A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a A numeral system (or system of numeration) is a Mathematical notation for representing numbers of a given set by symbols in a consistent manner 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. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. e. the decimal (ancient Latin adjective dec. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. 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. In mathematical numeral systems, the base or radix is usually the number of unique digits, including zero that a positional Numeral In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.
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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. A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a 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. The Hindu-Arabic numeral system is a Positional Decimal Numeral system first documented in the ninth century Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In Mathematics and Computer science, hexadecimal (also base -, hexa, or hex) is a Numeral system with a A positional number system must have a digit representing the integers from zero up to, but not including, the radix of the number system. A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a In mathematical numeral systems, the base or radix is usually the number of unique digits, including zero that a positional Numeral
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. In a positional Numeral system, the decimal separator is a Symbol used to mark the boundary between the integral and the fractional A full stop or period (sometimes stop, full point, decimal point, or dot) is the Punctuation mark commonly placed at the The United States of America —commonly referred to as the Each successive place to the left of this has a place value equal to the place value of the previous digit times the base. radix|basis (topologyIn Arithmetic, the base refers to the number b in an expression of the form b n. 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 decimal ( base ten or occasionally denary) Numeral system has ten as its base.
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
The first true written positional numeral system is considered to be the Hindu-Arabic numeral system. A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a The Hindu-Arabic numeral system is a Positional Decimal Numeral system first documented in the ninth century This system was established by the 7th century[1], but was not yet in its modern form because the use of the digit zero had not yet been widely accepted. The 7th century is the period from 601 to 700 in accordance with the Julian calendar in the Christian / Common Era. Instead of a zero, a space was left in the numeral as a placeholder. The first widely acknowledged use of zero was in 876. Events Births Deaths Louis the German, King of East Francia Map-bms876 Simple876 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. A glyph is an element of writing Two or more glyphs representing the same symbol whether interchangeable or context-dependent are called Allographs the abstract unit they [1]
By the 13th century, Hindu-Arabic numerals were accepted in European mathematical circles (Fibonacci used them in his Liber Abaci). The arabic numerals (often capitalized are the ten Digits (0 1 2 3 4 5 6 7 8 9 which—along with the system Leonardo of Pisa (c 1170 – c 1250 also known as Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or most commonly simply Fibonacci Liber Abaci (1202 also spelled as Liber Abbaci) is an historic book on Arithmetic by Leonardo of Pisa known later by his nickname Fibonacci 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 twentieth century of the Common Era began on
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 Pre-Columbian Maya civilization used a Vigesimal ( base - twenty) Numeral system. The system was vigesimal (base twenty), so it has twenty digits. The vigesimal or base - numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten "Twenty" redirects here For the village in England, see Twenty Lincolnshire. 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 Maya civilization is a Mesoamerican Civilization, noted for the only known fully developed written language of the Pre-Columbian Americas In a positional Numeral system, the decimal separator is a Symbol used to mark the boundary between the integral and the fractional
The Thai numeral system is identical to the Hindu-Arabic numeral system except for the symbols used to represent digits. Thai numerals (เลขไทย are a set of numerals traditionally used in Thailand, although the Arabic numerals are more common The Hindu-Arabic numeral system is a Positional Decimal Numeral system first documented in the ninth century The use of these digits is less common in Thailand than it once was, but they are still used alongside Hindu-Arabic numerals. The Kingdom of Thailand (ˈtaɪlænd ราชอาณาจักรไทย, râːtɕʰa-ʔaːnaːtɕɑ̀k-tʰɑj
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 ( Japanese: 算木 sangi are small bars typically 3-14 cm long used by mathematicians for calculation in China, Japan China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National For a topic outline on this subject see List of basic Japan topics. Counting rods themselves predate Hindu-Arabic numeral system. The Suzhou nemerals are variants of rod numerals. Chinese numerals are characters for writing Numbers in Chinese.
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Modern digital systems
In computer science
The binary (base 2), octal (base 8), and hexadecimal (base 16) systems, extensively used in computer science, all follow the conventions of the Hindu-Arabic numeral system. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. The octal Numeral system, or oct for short is the base -8 number system and uses the digits 0 to 7 In Mathematics and Computer science, hexadecimal (also base -, hexa, or hex) is a Numeral system with a The Hindu-Arabic numeral system is a Positional Decimal Numeral system first documented in the ninth century 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.
Unusual systems
The ternary system is infrequently used; it is a simple base-three system. Ternary or trinary is the base - Numeral system. Analogous to a " Bit " a ternary digit is known as a trit (
Digits in mathematics
Despite the essential role of digits in describing numbers, they are relatively unimportant to modern mathematics. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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. The digital root (also repeated digital sum) of a number is the number obtained by adding all the digits then adding the digits of that number and then continuing until a single-digit
Casting out nines
Casting out nines is a procedure for checking arithmetic done by hand. Casting out nines is a Sanity check to ensure that hand computations of sums differences products and quotients of Integers are correct Casting out nines is a Sanity check to ensure that hand computations of sums differences products and quotients of Integers are correct To describe it, let 
represent the digital root of 
, as described above. The digital root (also repeated digital sum) of a number is the number obtained by adding all the digits then adding the digits of that number and then continuing until a single-digit 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. An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent
Repunits and repdigits
Repunits are integers that are represented with only the digit 1. In Recreational mathematics, a repunit is a Number like 11, 111, or 1111 that contains 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. In Recreational mathematics, a repdigit is a Natural number composed of repeated instances of the same digit most often in the decimal numeral system. For example, 333 is a repdigit. The primacy of repunits is of interest to mathematicians[2]
Palindromic numbers and Lychrel numbers
Palindromic numbers are numbers that read the same when their digits are reversed. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 A palindromic number or numeral palindrome is a 'symmetrical' number like 16461 that remains the same when its 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. A Lychrel number is a Natural number which cannot form a Palindrome through the iterative process of repeatedly reversing its Base 10 digits The question of whether there are any Lychrel numbers in base 10 is an open problem in recreational mathematics; the smallest candidate is 196. A Lychrel number is a Natural number which cannot form a Palindrome through the iterative process of repeatedly reversing its Base 10 digits Recreational mathematics is an umbrella term referring to Mathematical puzzles and Mathematical games. 196 is a Natural number following 195 and preceding 197. It is the square of 14.
See also
References
- ^ a b O'Connor, J. A positional notation or place-value notation system is a Numeral system in which each position is related to the next by a Constant multiplier a A bit is a binary digit, taking a value of either 0 or 1 Binary digits are a basic unit of Information storage and communication The significant figures (also called significant digits and abbreviated sig figs) of a number are those digits that carry meaning contributing to its accuracy J. and Robertson, E. F. Arabic Numerals. January 2001. Retrieved on 2007-02-20.
- ^ Eric W. Weisstein. Repunit, from MathWorld ([1]). MathWorld is an online Mathematics reference work created and largely written by Eric W Retrieved on 2007-02-20.
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