Floating-point arithmetic: Difference between revisions

Revision as of 23:28, 28 January 2025

Revision as of 01:56, 29 January 2025

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{{Floating-point}}

In [[computing]], '''floating-point arithmetic''' ('''FP''') is [[arithmetic]] on subsets of [[real number]]s formed by a ''[[significand]]'' (a [[Sign (mathematics)|signed]] sequence of a fixed number of digits in some [[Radix|base]],) called a [[significand]], scaledmultiplied by an integer [[exponentinteger power]] of that base.

Numbers of this form are called '''floating-point numbers'''.<ref name="Muller_2010"/>{{rp|3}}<ref name="sterbenz1974fpcomp">{{cite book

|last1=Sterbenz

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In practice, most floating-point systems use [[Binary number|base two]], though base ten ([[decimal floating point]]) is also common.

Floating-point arithmetic operations, such as addition and division, approximate the corresponding real number arithmetic operations by [[rounding]] any result that is not a floating-point number itself to a nearby floating-point number.<ref name="Muller_2010"/>{{rp|22}}<ref name="sterbenz1974fpcomp"/>{{rp|10}}

For example, in a floating-point arithmetic with five base-ten digits, the sum 12.345 + 1.0001 = 13.3451 might be rounded to 13.345.

The term ''floating point'' refers to the fact that the number's [[radix point]] can "float" anywhere to the left, right, or between the [[significant digits]] of the number. This position is indicated by the exponent, so floating point can be considered a form of [[scientific notation]].

A floating-point system can be used to represent, with a fixed number of digits, numbers of very different [[Orders of magnitude (numbers)|orders of magnitude]] — such as the number of meters [[Orders of magnitude (length)#100_zettametres|between galaxies]] or [[Orders of magnitude (length)#10_femtometres|between protons in an atom]]. For this reason, floating-point arithmetic is often used to allow very small and very large real numbers that require fast processing times. The result of this [[dynamic range]] is that the numbers that can be represented are not uniformly spaced; the difference between two consecutive representable numbers varies with their exponent.<ref name="Smith_1997"/>