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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, an identity is an equality relating one mathematical expression ''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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
, 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 featuring the equal sign The equals sign (British English, Unicode Consortium) or equal sign (American English), ...
symbol instead of , the
equals sign The equals sign (, ) or equal sign (), formerly known as the equality sign, is the , which is used to indicate in some sense. In an , it is placed between two that have the same value, or for which one studies the conditions under which the ...

equals sign
.


Common identities


Algebraic identities

Certain identities, such as a+0=a and a+(-a)=0, form the basis of algebra, 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 are identities involving certain functions of one or more
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

angle
s. They are distinct from , which are identities involving both angles and side lengths of a
triangle A triangle is a polygon In geometry, a polygon () is a plane (mathematics), plane Shape, figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The ...

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 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
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

complex
values of \theta (since the complex numbers \mathbb form the domain of sine and cosine). On the other hand, the equation :\cos \theta = 1 is only true for certain values of \theta, not all (nor for all values in a
neighborhood A neighbourhood (British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial codification of grammar ...
). 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 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
. For example, and , but , whereas . And unlike addition and multiplication, exponentiation is not
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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). Without parentheses to modify the order of calculation, by convention the order is top-down, not bottom-up: :b^ = b^ \ne (b^p)^q = b^ = b^ .


Logarithmic identities

Several important formulas, sometimes called ''logarithmic identities'' or ''log laws'', relate logarithms 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 calculators 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 mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. In fact, Osborn's rule states that one can convert any trigonometric identity into a hyperbolic identity by expanding it completely in terms of integral 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 number of hyperbolic sines. The
Gudermannian function of the Gudermannian function The Gudermannian function, named after Christoph Gudermann (1798–1852), relates the circular functions and hyperbolic function Hyperbolic is an adjective describing something that resembles or pertains to a hyperbo ...
gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers.


Logic and universal algebra

Formally, an identity is a true
universally quantified Universality most commonly refers to: * Universality (philosophy) * Universality (dynamical systems) Universality principle may refer to: * In statistics, Random matrix#Bulk statistics, universality principle, a property of systems that can be ...
formula In , a formula is a concise way of expressing information symbolically, as in a mathematical formula or a . The informal use of the term ''formula'' in science refers to the . The plural of ''formula'' can be either ''formulas'' (from the mos ...
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 Mathematical notation, notation (symbol) that specifies places in an expression (mathematics), expressio ...
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 truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

axiom
s of a
monoid In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

equation
s. In other word, an identity is an equation that is true for every values of the variables. Here: Def.1 of Sect.3.2.1, p.160.


See also

* Accounting identity * List of mathematical identities


References


Notes


Citations


Sources

* * *


External links

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