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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an identity is an
equality Equality may refer to: Society * Political equality, in which all members of a society are of equal standing ** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elite ...
relating one
mathematical expression In mathematics, an expression or mathematical expression is a finite combination of Glossary of mathematical symbols, symbols that is well-formed formula, well-formed according to rules that depend on the context. Mathematical symbols can desi ...
''A'' to another mathematical expression ''B'', such that ''A'' and ''B'' (which might contain some variables) produce the same value for all values of the variables within a certain range of validity. In other words, ''A'' = ''B'' is an identity if ''A'' and ''B'' define the same functions, and an identity is an equality between functions that are differently defined. For example, (a+b)^2 = a^2 + 2ab + b^2 and \cos^2\theta + \sin^2\theta =1 are identities. Identities are sometimes indicated by the
triple bar The triple bar, or tribar ≡, is a symbol with multiple, context-dependent meanings. It has the appearance of an equals sign  sign with a third line. The triple bar character in Unicode is code point .. The closely related code point ...
symbol instead of , the
equals sign The equals sign (British English, Unicode) or equal sign (American English), also known as the equality sign, is the mathematical symbol , which is used to indicate equality in some well-defined sense. In an equation, it is placed between two ...
.


Common identities


Algebraic identities

Certain identities, such as a+0=a and a+(-a)=0, form the basis of
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
, while other identities, such as (a+b)^2 = a^2 + 2ab +b^2 and a^2 - b^2 = (a+b)(a-b), can be useful in simplifying algebraic expressions and expanding them.


Trigonometric identities

Geometrically,
trigonometric identities In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
are identities involving certain functions of one or more
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
s. They are distinct from triangle identities, which are identities involving both angles and side lengths of a
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
. 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
integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
of non-trigonometric functions: a common technique which involves first using the 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 \sin^2 \theta + \cos^2 \theta = 1, which is true for all
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
values of \theta. On the other hand, the equation :\cos\theta = 1 is only true for certain values of \theta, not all. For example, this equation is true when \theta = 0, but false when \theta = 2. Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g. the double-angle identity \sin(2\theta) = 2\sin\theta \cos\theta, the addition formula for \tan(x + y)), which can be used to break down expressions of larger angles into those with smaller constituents.


Exponential identities

The following identities hold for all
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
exponents, provided that the base is non-zero: :\begin b^ &= b^m \cdot b^n \\ (b^m)^n &= b^ \\ (b \cdot c)^n &= b^n \cdot c^n \end Unlike addition and multiplication, exponentiation is not
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
. For example, and , but whereas . Also unlike addition and multiplication, exponentiation is not
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
either. For example, and , but 23 to the 4 is 84 (or 4,096) whereas 2 to the 34 is 281 (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: :b^ := b^ ,   whereas   (b^p)^q = b^.


Logarithmic identities

Several important formulas, sometimes called ''logarithmic identities'' or ''log laws'', relate
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
s to one another:


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 th power of a number is times the logarithm of the number itself; the logarithm of a th root is the logarithm of the number divided by . The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions x=b^, and/or y=b^, in the left hand sides.


Change of base

The logarithm log''b''(''x'') can be computed from the logarithms of ''x'' and ''b'' with respect to an arbitrary base ''k'' using the following formula: : \log_b(x) = \frac. Typical
scientific calculator A scientific calculator is an electronic calculator, either desktop or handheld, designed to perform mathematical operations. They have completely replaced slide rules and are used in both educational and professional settings. In some areas ...
s calculate the logarithms to bases 10 and ''e''. Logarithms with respect to any base ''b'' can be determined using either of these two logarithms by the previous formula: : \log_b (x) = \frac = \frac. Given a number ''x'' and its logarithm log''b''(''x'') to an unknown base ''b'', the base is given by: : b = x^\frac.


Hyperbolic function identities

The hyperbolic functions satisfy many identities, all of them similar in form to the
trigonometric identities In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
. In fact, Osborn's rule 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
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire game w ...
number of hyperbolic sines. The
Gudermannian function In mathematics, the Gudermannian function relates a hyperbolic angle measure \psi to a circular angle measure \phi called the ''gudermannian'' of \psi and denoted \operatorname\psi. The Gudermannian function reveals a close relationship betwee ...
gives a direct relationship between the trigonometric functions and the hyperbolic ones that does not involve
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s.


Logic and universal algebra

Formally, an identity is a true
universally quantified In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other w ...
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
of the form \forall x_1,\ldots,x_n: s=t, where and are terms with no other
free variable In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
s than x_1,\ldots,x_n. The quantifier prefix \forall x_1,\ldots,x_n is often left implicit, when it is stated that the formula is an identity. For example, the
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s of a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...
are often given as the formulas :\forall x,y,z: x*(y*z)=(x*y)*z,\quad \forall x: x*1=x, \quad \forall x: 1*x=x, or, shortly, :x*(y*z)=(x*y)*z,\qquad x*1=x, \qquad 1*x=x. So, these formulas are identities in every monoid. As for any equality, the formulas without quantifier are often called
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
s. In other words, an identity is an equation that is true for all values of the variables. Here: Def.1 of Sect.3.2.1, p.160.


See also

*
Accounting identity In accounting, finance and economics, an accounting identity is an equality that must be true regardless of the value of its variables, or a statement that by definition (or construction) must be true. Where an accounting identity applies, any devia ...
*
List of mathematical identities This article lists mathematical identities, that is, ''identically true relations'' holding in mathematics. * Bézout's identity (despite its usual name, it is not, properly speaking, an identity) * Binomial inverse theorem * Binomial identity ...


References


Notes


Citations


Sources

* * *


External links

{{Commons category
The Encyclopedia of Equation
Online encyclopedia of mathematical identities (archived)
A Collection of Algebraic Identities
Elementary algebra Equivalence (mathematics)