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{{short description|Internal representation of numeric values in a digital computer}}
{{refimprove|date=October 2022}}
▲ }}</ref> Numerical values are stored as groupings of [[bit]]s, such as [[byte]]s and words. The encoding between numerical values and bit patterns is chosen for convenience of the operation of the computer;{{cn|date=June 2020}} the encoding used by the computer's instruction set generally requires conversion for external use, such as for printing and display. Different types of processors may have different internal representations of numerical values and different conventions are used for integer and real numbers. Most calculations are carried out with number formats that fit into a processor register, but some software systems allow representation of arbitrarily large numbers using multiple words of memory.
==Binary number representation==
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|+ Table 1: Binary to octal
|-
! Binary string !! Octal value
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| 000 || 0
|-
| 001 || 1
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| 010 || 2
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| 011 || 3
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| 100 || 4
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| 101 || 5
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| 110 || 6
|-
| 111 || 7
|}
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While a single bit, on its own, is able to represent only two values, a [[Bit string|string of bits]] may be used to represent larger values. For example, a string of three bits can represent up to eight distinct values as illustrated in Table 1.
As the number of bits composing a string increases, the number of possible ''0'' and ''1'' combinations increases [[Exponentiation|exponentially]]. A single bit allows only two value-combinations, two bits combined can make four separate values, three bits for eight, and so on, increasing with the formula 2
Groupings with a specific number of bits are used to represent varying things and have specific names.
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The number of bits needed for the precision and range desired must be chosen to store the fractional and integer parts of a number. For instance, using a 32-bit format, 16 bits may be used for the integer and 16 for the fraction.
The eight's bit is followed by the four's bit, then the two's bit, then the one's bit. The fractional bits continue the pattern set by the integer bits. The next bit is the half's bit, then the quarter's bit, then the
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* a sign bit, giving the sign of the number.
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The representation has a limited precision. For example, only 15 decimal digits can be represented with a 64-bit real. If a very small floating-point number is added to a large one, the result is just the large one. The small number was too small to even show up in 15 or 16 digits of resolution, and the computer effectively discards it. Analyzing the effect of limited precision is a well-studied problem. Estimates of the magnitude of round-off errors and methods to limit their effect on large calculations are part of any large computation project. The precision limit is different from the range limit, as it affects the significand, not the exponent.
The significand is a binary fraction that doesn't necessarily perfectly match a decimal fraction. In many cases a sum of reciprocal powers of 2 does not match a specific decimal fraction, and the results of computations will be slightly off. For example, the decimal fraction "0.1" is equivalent to an infinitely repeating binary fraction: 0.000110011 ...<ref>{{cite web|last=Goebel|first=Greg|title=Computer Numbering Format|url=http://www.vectorsite.net/tsfloat.html|archive-url=https://archive.today/20130222091425/http://www.vectorsite.net/tsfloat.html|url-status=usurped|archive-date=February 22, 2013|access-date=10 September 2012}}</ref>
==Numbers in programming languages==
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* [[Gray code]]
* [[Numeral system]]
* [[Unum (number format)|Unum]]
* [[Posit (number format)|Posit]]
==Notes and references==
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