Editing Equality (mathematics) (section)

=== Approximate equality ===
{{Main|Approximation#Mathematics}}
[[File:Archimedes pi.svg|alt=diagram of a hexagon and pentagon circumscribed outside a circle|right|thumb|upright=1.35|The sequence given by the [[perimeter]]s of regular ''n''-sided [[polygon]]s that [[circumscribe]] the [[unit circle]] approximates <math>2\pi</math>]]
[[Numerical analysis]] is the study of [[Constructive proof|constructive]] methods and [[algorithms]] to find numerical [[approximation]]s (as opposed to [[Symbolic computation|symbolic manipulations]]) of solutions to problems in [[mathematical analysis]]. Especially those which cannot be [[Analytic solution|solved analytically]].<ref>{{Cite book |last=Kress |first=Rainer |date=1998 |title=Numerical Analysis |publisher=Springer |series=Graduate Texts in Mathematics |volume=181 |location=New York |pages=1–4 |doi=10.1007/978-1-4612-0599-9 |isbn=978-1-4612-6833-8 |issn=0072-5285}}</ref>

Calculations are likely to involve [[Round-off error|rounding errors]] and other [[approximation error]]s. [[Logarithm|Log tables]], slide rules, and calculators produce approximate answers to all but the simplest calculations. The results of computer calculations are normally an approximation, expressed in a limited number of significant digits, although they can be programmed to produce more precise results.<ref>{{Cite web |title=Numerical Computation Guide |url=http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html |url-status=dead |archive-url=https://web.archive.org/web/20160406101256/http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html |archive-date=2016-04-06 |access-date=2013-06-16}}</ref>

If viewed as a [[binary relation]], (denoted by the symbol <math>\approx</math>) between [[real number]]s or other things, if precisely defined, is not an equivalence relation since it's not transitive, even if modeled as a [[fuzzy relation]].<ref>{{Cite web |last1=Kerre |first1=Etienne E. |last2=De Cock |first2=Martine |date=2001 |title=Approximate Equality is no Fuzzy Equality |url=https://faculty.washington.edu/mdecock/papers/mdecock2001a.pdf}}</ref>

In [[computer science]], equality is given by some [[relational operator]]. Real numbers are often approximated by [[floating-point numbers]] (A sequence of some fixed number of digits of a given base, scaled by an integer [[exponent]] of that base),<ref>{{cite book |last1=Sterbenz |first1=Pat H. |url=https://archive.org/details/SterbenzFloatingPointComputation/mode/2up |title=Floating-Point Computation |date=1974 |publisher=Prentice-Hall |isbn=0-13-322495-3 |location=Englewood Cliffs, New Jersey}}</ref> thus it is common to store an [[Expression (computer science)|expression]] that denotes the real number as to not lose precision. However, the equality of two real numbers given by an expression is known to be [[undecidable problem|undecidable]] (specifically, real numbers defined by expressions involving the [[integer]]s, the basic [[arithmetic operation]]s, the [[logarithm]] and the [[exponential function]]). In other words, there cannot exist any [[algorithm]] for deciding such an equality (see [[Richardson's theorem]]).<ref>{{cite journal |last=Richardson |first=Daniel |year=1968 |title=Some Undecidable Problems Involving Elementary Functions of a Real Variable |journal=Journal of Symbolic Logic |volume=33 |issue=4 |pages=514–520 |jstor=2271358 |zbl=0175.27404}}</ref>