Editing Identity (mathematics)

{{Short description|Equation that is satisfied for all values of the variables}}
{{Distinguish|identity element|identity function|identity matrix}}
{{For|the basic notion of sameness in mathematics, sometimes called ''identity''|Equality (mathematics)}}
[[File:Trig functions on unit circle.svg|thumb|Visual proof of the [[Pythagorean identity]]: for any angle <math>\theta</math>, the point <math>(x, y) = (\cos\theta, \sin\theta)</math> lies on the [[unit circle]], which satisfies the equation <math>x^2 + y^2 =1</math>. Thus, <math>\cos^2\theta + \sin^2\theta =1</math>.]]

In [[mathematics]], an '''identity''' is an [[equality (mathematics)|equality]] relating one [[mathematical expression]] ''A''&nbsp;to another mathematical expression&nbsp;''B'', such that ''A'' and ''B'' (which might contain some [[variable (mathematics)|variables]]) produce the same value for all values of the variables within a certain [[domain of discourse]].<ref>Equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613</ref><ref>Pratt, Vaughan, "Algebra", The Stanford Encyclopedia of Philosophy (Winter 2022 Edition), Edward N. Zalta & Uri Nodelman (eds.), URL: https://plato.stanford.edu/entries/algebra/#Laws</ref> In other words, ''A''&nbsp;=&nbsp;''B'' is an identity if ''A'' and ''B'' define the same [[function (mathematics)|functions]], and an identity is an equality between functions that are differently defined. For example, <math>(a+b)^2 = a^2 + 2ab + b^2</math>  and <math>\cos^2\theta + \sin^2\theta =1</math> are identities.<ref>{{Cite web|url=https://www.mathwords.com/i/identity.htm|title=Mathwords: Identity|website=www.mathwords.com|access-date=2019-12-01}}</ref> Identities are sometimes indicated by the [[triple bar]] symbol {{math|≡}} instead of {{math|1==}}, the [[equals sign]].<ref>{{Cite web|url=https://www.mathopenref.com/identity.html|title=Identity – math word definition – Math Open Reference|website=www.mathopenref.com|access-date=2019-12-01}}</ref> Formally, an identity is a [[universally quantified]] equality.

== Common identities ==

=== Algebraic identities ===
{{See also|Factorization#Recognizable patterns}}
Certain identities, such as <math>a+0=a</math> and <math>a+(-a)=0</math>, form the basis of [[algebra]],<ref>{{Cite web|url=http://www.math.com/tables/algebra/basicidens.htm|title=Basic Identities|website=www.math.com|access-date=2019-12-01}}</ref> while other identities, such as <math>(a+b)^2 = a^2 + 2ab +b^2</math> and <math>a^2 - b^2 = (a+b)(a-b)</math>, can be useful in simplifying algebraic expressions and expanding them.<ref>{{Cite web|url=http://www.sosmath.com/tables/algiden/algiden.html|title=Algebraic Identities|website=www.sosmath.com|access-date=2019-12-01}}</ref>

=== Trigonometric identities ===
{{Main|List of trigonometric identities}}

Geometrically, [[trigonometric identity|trigonometric identities]] are identities involving certain functions of one or more [[angle]]s.<ref>{{Cite web|url=https://www.purplemath.com/modules/idents.htm|title=Trigonometric Identities|last=Stapel|first=Elizabeth|website=Purplemath|access-date=2019-12-01}}</ref> They are distinct from [[Trigonometry#Triangle identities|triangle identities]], which are identities involving both angles and side lengths of a [[triangle]]. Only the former are covered in this article.

These identities are useful whenever expressions involving trigonometric functions need to be simplified. Another important application is the [[integral|integration]] of non-trigonometric functions: a common technique which involves first using the [[Trigonometric substitution|substitution rule with a trigonometric function]], and then simplifying the resulting integral with a trigonometric identity.

One of the most prominent examples of trigonometric identities involves the equation <math>\sin^2 \theta + \cos^2 \theta = 1,</math> which is true for all [[real number|real]] values of <math>\theta</math>. On the other hand, the equation
:<math>\cos\theta = 1</math>

is only true for certain values of <math>\theta</math>, not all. For example, this equation is true when <math>\theta = 0,</math> but false when <math>\theta = 2</math>.

Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g. the double-angle identity <math>\sin(2\theta) = 2\sin\theta \cos\theta</math>, the addition formula for <math>\tan(x + y)</math>), which can be used to break down expressions of larger angles into those with smaller constituents.

=== Exponential identities ===
{{Main|Exponentiation}}

The following identities hold for all [[integer]] exponents, provided that the base is non-zero:
:<math>\begin{align}
      b^{m + n} &= b^m \cdot b^n \\
        (b^m)^n &= b^{m\cdot n} \\
  (b \cdot c)^n &= b^n \cdot c^n
\end{align}</math>

Unlike addition and multiplication, exponentiation is not [[commutative]]. For example, {{nowrap|1=2 + 3 = 3 + 2 = 5}} and {{nowrap|1=2 · 3 = 3 · 2 = 6}}, but {{nowrap|1=2<sup>3</sup> = 8}} whereas {{nowrap|1=3<sup>2</sup> = 9}}.

Also unlike addition and multiplication, exponentiation is not [[associative]] either. For example, {{nowrap|1=(2 + 3) + 4 = 2 + (3 + 4) = 9}} and {{nowrap|1=(2 · 3) · 4 = 2 · (3 · 4) = 24}}, but 2<sup>3</sup> to the 4 is 8<sup>4</sup> (or 4,096) whereas 2 to the 3<sup>4</sup> is 2<sup>81</sup> (or 2,417,851,639,229,258,349,412,352). When no parentheses are written, by convention the order is top-down, not bottom-up:
:<math>b^{p^q} := b^{(p^q)} ,</math> &nbsp; whereas &nbsp; <math>(b^p)^q = b^{p \cdot q}.</math>

=== Logarithmic identities ===
{{Main|Logarithmic identities}}
Several important formulas, sometimes called ''logarithmic identities'' or ''log laws'', relate [[logarithm]]s to one another:{{efn|All statements in this section can be found in {{harvnb|Shirali|2002|p=|loc=Section 4}}, {{harvnb|Downing|2003|p=275}}, or {{harvnb|Kate|Bhapkar|2009|p=1-1}}, for example.}}

==== Product, quotient, power and root ====
The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the {{mvar|p}}th power of a number is {{mvar|p}} times the logarithm of the number itself; the logarithm of a {{mvar|p}}th root is the logarithm of the number divided by {{mvar|p}}. The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions  <math>x=b^{\log_b x},</math> and/or <math>y=b^{\log_b y},</math> in the left hand sides.

{| class="wikitable" style="margin:1em auto;"
!  !! Formula !! Example
|-
| product || <cite id=labegarithmProducts><math>\log_b(x y) = \log_b(x) + \log_b(y)</math></cite>|| <math>\log_3(243) = \log_3(9 \cdot 27) = \log_3(9) + \log_3(27) = 2 + 3 = 5</math>
|-
| quotient || <math>\log_b\! \left( \frac{x}{y} \right) = \log_b(x) - \log_b(y)</math>|| <math>\log_2(16) = \log_2\!\left( \frac{64}{4} \right) = \log_2(64) - \log_2(4) = 6 - 2 = 4</math>
|-
| power || <cite id=labelLogarithmPowers><math>\log_b(x^p) = p \log_b(x)</math></cite>|| <math>\log_2(64) = \log_2(2^6) = 6 \log_2 (2) = 6</math>
|-
| root || <math>\log_b\! \sqrt[p]{x} = \frac{\log_b(x)} p</math>|| <math>\log_{10}\! \sqrt{1000} = \frac{1}{2}\log_{10} 1000 = \frac{3}{2} = 1.5</math>
|}

==== Change of base ====
The logarithm log<sub>''b''</sub>(''x'') can be computed from the logarithms of ''x'' and ''b'' with respect to an arbitrary base ''k'' using the following formula:
: <cite id=labelLogarithmBaseChange><math> \log_b(x) = \frac{\log_k(x)}{\log_k(b)}.</math></cite>
Typical [[scientific calculator]]s calculate the logarithms to bases 10 and [[e (mathematical constant)|''e'']].<ref>{{Citation | last1=Bernstein | first1=Stephen | last2=Bernstein | first2=Ruth | title=Schaum's outline of theory and problems of elements of statistics. I, Descriptive statistics and probability | publisher=[[McGraw-Hill]] | location=New York | series=Schaum's outline series | isbn=978-0-07-005023-5 | year=1999 | url-access=registration | url=https://archive.org/details/schaumsoutlineof00bern }}, p. 21</ref> Logarithms with respect to any base ''b'' can be determined using either of these two logarithms by the previous formula:
:<math> \log_b (x) = \frac{\log_{10} (x)}{\log_{10} (b)} = \frac{\log_{e} (x)}{\log_{e} (b)}. </math>
Given a number ''x'' and its logarithm log<sub>''b''</sub>(''x'') to an unknown base ''b'', the base is given by:
: <math> b = x^\frac{1}{\log_b(x)}.</math>

=== Hyperbolic function identities ===
{{Main|Hyperbolic function}}
The hyperbolic functions satisfy many identities, all of them similar in form to the [[trigonometric identity|trigonometric identities]]. In fact, '''Osborn's rule'''<ref>{{cite journal|jstor=3602492|title=109. Mnemonic for Hyperbolic Formulae|journal=The Mathematical Gazette|first=G.|last=Osborn|date=1 January 1902|volume=2|issue=34|pages=189|doi=10.2307/3602492|url=https://zenodo.org/record/1449741}}</ref> states that one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integer powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term which contains a product of an [[parity (mathematics)|even]] number of hyperbolic sines.<ref>{{cite book
 |title=Technical mathematics with calculus
 |edition=3rd
 |first1=John Charles
 |last1=Peterson
 |publisher=Cengage Learning
 |year=2003
 |isbn=0-7668-6189-9
 |page=1155
 |url=https://books.google.com/books?id=PGuSDjHvircC}}, [https://books.google.com/books?id=PGuSDjHvircC&pg=PA1155 Chapter 26, page 1155]</ref>

The [[Gudermannian function]] gives a direct relationship between the trigonometric functions and the hyperbolic ones that does not involve [[complex number]]s.

== Logic and universal algebra ==
Formally, an identity is a true [[universal quantifier|universally quantified]] [[Well-formed formula#Predicate logic|formula]] of the form <math>\forall x_1,\ldots,x_n: s=t,</math> where {{math|''s''}} and {{math|''t''}} are [[term (logic)|terms]] with no other [[free variable]]s than <math>x_1,\ldots,x_n.</math> The quantifier prefix <math>\forall x_1,\ldots,x_n</math> is often left implicit, when it is stated that the formula is an identity. For example, the [[axiom]]s of a [[monoid]] are often given as the formulas
:<math>\forall x,y,z: x*(y*z)=(x*y)*z,\quad \forall x: x*1=x, \quad \forall x: 1*x=x,</math>
or, shortly,
:<math>x*(y*z)=(x*y)*z,\qquad x*1=x, \qquad 1*x=x.</math>

So, these formulas are identities in every monoid. As for any equality, the formulas without quantifier are often called [[equation]]s. In other words, an identity is an equation that is true for all values of the variables.<ref>{{cite book | editor=Jan van Leeuwen | editor-link=Jan van Leeuwen | title=Formal Models and Semantics | publisher=Elsevier | series=Handbook of Theoretical Computer Science | volume=B | year=1990 | author1=Nachum Dershowitz | author2= Jean-Pierre Jouannaud | author1-link=Nachum Dershowitz | author2-link=Jean-Pierre Jouannaud | contribution=Rewrite Systems | pages=243–320 }}</ref><ref>{{cite book | isbn=3-540-54280-9 | author=Wolfgang Wechsler | title=Universal Algebra for Computer Scientists | location=Berlin | publisher=Springer | editor1=Wilfried Brauer | editor2= Grzegorz Rozenberg |editor3= Arto Salomaa | editor1-link=Wilfried Brauer | editor2-link= Grzegorz Rozenberg |editor3-link= Arto Salomaa| series=[[EATCS]] Monographs on Theoretical Computer Science | volume=25 | year=1992 }} Here: Def.1 of Sect.3.2.1, p.160.</ref>

== See also ==
* [[Accounting identity]]
* [[List of mathematical identities]]
* [[Law (mathematics)]]

== References ==
=== Notes ===
{{notelist}}

=== Citations ===
{{reflist}}

=== Sources ===
{{refbegin}}
*{{cite book|last=Downing|first=Douglas|title=Algebra the Easy Way|url=https://books.google.com/books?id=RiX-TJLiQv0C|year=2003|publisher=Barrons Educational Series|isbn=978-0-7641-1972-9}}
*{{cite book|last1=Kate|first1=S.K.|last2=Bhapkar|first2=H.R.|title=Basics Of Mathematics|year=2009|publisher=Technical Publications|isbn=978-81-8431-755-8}}
*{{cite book|last=Shirali|first=S.|title=Adventures in Problem Solving|url=https://books.google.com/books?id=TPE0fXGnYtMC&pg=PP1|date=2002|publisher=Universities Press|isbn=978-81-7371-413-9}}
* {{Cite book |last=Efthimiou |first=Costas |url=https://books.google.com/books?id=-DN-AwAAQBAJ |title=Introduction to Functional Equations |publisher=[[American Mathematical Society]] |year=2011 |isbn=978-0-8218-5314-6 |archive-url=https://web.archive.org/web/20230603025341/http://www.msri.org/people/staff/levy/files/MCL/Efthimiou/100914book.pdf |archive-date=June 3, 2023}}
* {{cite book |author=Christopher G. Small |url=https://books.google.com/books?id=2D2RYbb22nMC |title=Functional Equations and How to Solve Them |date=3 April 2007 |publisher=Springer Science & Business Media |isbn=978-0-387-48901-8}}
* {{Cite journal |last1=Adkins |first1=William A. |last2=Davidson |first2=Mark G. |date=2012 |title=Ordinary Differential Equations |url=https://link.springer.com/book/10.1007/978-1-4614-3618-8 |journal=Undergraduate Texts in Mathematics |location=New York, NY |language=en |doi=10.1007/978-1-4614-3618-8 |isbn=978-1-4614-3617-1 |issn=0172-6056}}
* {{Cite journal |last1=Brešar |first1=Matej |last2=Chebotar |first2=Mikhail A. |last3=Martindale |first3=Wallace S. |date=2007 |title=Functional Identities |url=https://link.springer.com/book/10.1007/978-3-7643-7796-0 |journal=Frontiers in Mathematics |location=Basel |language=en |doi=10.1007/978-3-7643-7796-0 |isbn=978-3-7643-7795-3 |issn=1660-8046}}
{{refend}}

== External links ==
*[https://web.archive.org/web/20190612171441/https://encyclopedia-of-equation.webnode.jp/ The Encyclopedia of Equation]   Online encyclopedia of mathematical identities (archived)
*[http://sites.google.com/site/tpiezas/Home/ A Collection of Algebraic Identities] {{Webarchive|url=https://web.archive.org/web/20111001021837/http://sites.google.com/site/tpiezas/Home |date=2011-10-01 }}

[[Category:Elementary algebra]]
[[Category:Mathematical identities| ]]
[[Category:Equivalence (mathematics)]]