In
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 ''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,
and
are identities.
Identities are sometimes indicated by the
triple bar symbol instead of , the
equals sign.
Common identities
Algebraic identities
Certain identities, such as
and
, 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
and
, 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, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s. They are distinct from
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
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
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
. On the other hand, the equation
:
is only true for certain values of
, not all. For example, this equation is true when
but false when
.
Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g. the double-angle identity
, the addition formula for
),
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:
:
Unlike addition and multiplication, exponentiation is not
commutative. 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 ...
either. For example, and , but 2
3 to the 4 is 8
4 (or 4,096) whereas 2 to the 3
4 is 2
81 (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:
:
whereas
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
and/or
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:
:
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:
:
Given a number ''x'' and its logarithm log
''b''(''x'') to an unknown base ''b'', the base is given by:
:
Hyperbolic function identities
The hyperbolic functions satisfy many identities, all of them similar in form to the
trigonometric identities. 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 wh ...
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 betwe ...
gives a direct relationship between the trigonometric functions and the hyperbolic ones that does not involve
complex numbers.
Logic and universal algebra
Formally, an identity is a true
universally quantified
In mathematical logic, a universal quantification is a type of Quantification (logic), quantifier, a logical constant which is interpretation (logic), interpreted as "given any" or "for all". It expresses that a predicate (mathematical logic), pr ...
formula of the form
where and are
terms with no other
free variables than
The quantifier prefix
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 a ...
are often given as the formulas
:
or, shortly,
:
So, these formulas are identities in every monoid. As for any equality, the formulas without quantifier are often called
equations. 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 dev ...
*
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)