Identity (mathematics)
In mathematics, 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 domain of discourse.
[1][2] 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.
[3] Identities are sometimes indicated by the triple bar symbol ≡ instead of =, the equals sign.
[4] Formally, an identity is a universally quantified equality.
, form the basis of algebra,[5] while other identities, such as
, can be useful in simplifying algebraic expressions and expanding them.
[7] They are distinct from triangle identities, which are identities involving both angles and side lengths of a triangle.
These identities are useful whenever expressions involving trigonometric functions need to be simplified.
Another important application is the integration 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 values of
On the other hand, the equation is only true for certain values of
Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g. the double-angle identity
), which can be used to break down expressions of larger angles into those with smaller constituents.
The following identities hold for all integer exponents, provided that the base is non-zero: Unlike addition and multiplication, exponentiation is not commutative.
Also unlike addition and multiplication, exponentiation is not associative either.
When no parentheses are written, by convention the order is top-down, not bottom-up: Several important formulas, sometimes called logarithmic identities or log laws, relate logarithms to one another:[a] 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 pth power of a number is p times the logarithm of the number itself; the logarithm of a pth root is the logarithm of the number divided by p. The following table lists these identities with examples.
Each of the identities can be derived after substitution of the logarithm definitions
in the left hand sides.
The logarithm logb(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.[8] 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 logb(x) to an unknown base b, the base is given by: The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities.
In fact, Osborn's rule[9] 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 number of hyperbolic sines.
[10] The Gudermannian function gives a direct relationship between the trigonometric functions and the hyperbolic ones that does not involve complex numbers.
Formally, an identity is a true universally quantified formula of the form
is often left implicit, when it is stated that the formula is an identity.
For example, the axioms of a monoid 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.
