Movatterモバイル変換

[0]ホーム

Jump to content

Identity (mathematics)

Edit links

From Wikipedia, the free encyclopedia

Equation that is satisfied for all values of the variables

Not to be confused withidentity element,identity function, oridentity matrix.

For the basic notion of sameness in mathematics, sometimes calledidentity, seeEquality (mathematics).

Visual proof of thePythagorean identity: for any angle $\theta$ , the point $(x,y)=(\cos \theta ,\sin \theta )$ lies on theunit circle, which satisfies the equation $x^{2}+y^{2}=1$ . Thus, $\cos ^{2}\theta +\sin ^{2}\theta =1$ .

{\displaystyle \theta } — Visual proof of thePythagorean identity: for any angle $\theta$ , the point $(x,y)=(\cos \theta ,\sin \theta )$ lies on theunit circle, which satisfies the equation $x^{2}+y^{2}=1$ . Thus, $\cos ^{2}\theta +\sin ^{2}\theta =1$ .

Inmathematics, anidentity is anequality relating onemathematical expressionA to another mathematical expression B, such thatA andB (which might contain somevariables) produce the same value for all values of the variables within a certaindomain of discourse.^[1]^[2] In other words,A = B is an identity ifA andB define the samefunctions, 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.^[3] Identities are sometimes indicated by thetriple bar symbol≡ instead of=, theequals sign.^[4] Formally, an identity is auniversally quantified equality.

Common identities

[edit]

Algebraic identities

[edit]

Certain identities, such as $a+0=a$ and $a+(-a)=0$ , form the basis ofalgebra,^[5] 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.^[6]

Trigonometric identities

[edit]

Main article:List of trigonometric identities

Geometrically,trigonometric identities are identities involving certain functions of one or moreangles.^[7] They are distinct fromtriangle identities, which are identities involving both angles and side lengths of atriangle. 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 theintegration of non-trigonometric functions: a common technique which involves first using thesubstitution 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 allreal 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

[edit]

Main article:Exponentiation

The following identities hold for allinteger exponents, provided that the base is non-zero:

{\begin{aligned}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{aligned}}

Unlike addition and multiplication, exponentiation is notcommutative. For example,2 + 3 = 3 + 2 = 5 and2 · 3 = 3 · 2 = 6, but2³ = 8 whereas3² = 9.

Also unlike addition and multiplication, exponentiation is notassociative either. For example,(2 + 3) + 4 = 2 + (3 + 4) = 9 and(2 · 3) · 4 = 2 · (3 · 4) = 24, 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^{p^{q}}:=b^{(p^{q})},

whereas

(b^{p})^{q}=b^{p\cdot q}.

Logarithmic identities

[edit]

Main article:Logarithmic identities

Several important formulas, sometimes calledlogarithmic identities orlog laws, relatelogarithms to one another:^[a]

Product, quotient, power and root

[edit]

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 thepth power of a number isp times the logarithm of the number itself; the logarithm of apth root is the logarithm of the number divided byp. The following table lists these identities with examples. Each of the identities can be derived after substitution of the logarithm definitions $x=b^{\log _{b}x},$ and/or $y=b^{\log _{b}y},$ in the left hand sides.

	Formula	Example
product	$\log _{b}(xy)=\log _{b}(x)+\log _{b}(y)$	$\log _{3}(243)=\log _{3}(9\cdot 27)=\log _{3}(9)+\log _{3}(27)=2+3=5$
quotient	$\log _{b}\!\left({\frac {x}{y}}\right)=\log _{b}(x)-\log _{b}(y)$	$\log _{2}(16)=\log _{2}\!\left({\frac {64}{4}}\right)=\log _{2}(64)-\log _{2}(4)=6-2=4$
power	$\log _{b}(x^{p})=p\log _{b}(x)$	$\log _{2}(64)=\log _{2}(2^{6})=6\log _{2}(2)=6$
root	$\log _{b}\!{\sqrt[{p}]{x}}={\frac {\log _{b}(x)}{p}}$	$\log _{10}\!{\sqrt {1000}}={\frac {1}{2}}\log _{10}1000={\frac {3}{2}}=1.5$

Change of base

[edit]

The logarithm log_b(x) can be computed from the logarithms ofx andb with respect to an arbitrary basek using the following formula:

\log _{b}(x)={\frac {\log _{k}(x)}{\log _{k}(b)}}.

Typicalscientific calculators calculate the logarithms to bases 10 ande.^[8] Logarithms with respect to any baseb can be determined using either of these two logarithms by the previous formula:

\log _{b}(x)={\frac {\log _{10}(x)}{\log _{10}(b)}}={\frac {\log _{e}(x)}{\log _{e}(b)}}.

Given a numberx and its logarithm log_b(x) to an unknown baseb, the base is given by:

b=x^{\frac {1}{\log _{b}(x)}}.

Hyperbolic function identities

[edit]

Main article:Hyperbolic function

The hyperbolic functions satisfy many identities, all of them similar in form to thetrigonometric 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 aneven number of hyperbolic sines.^[10]

TheGudermannian function gives a direct relationship between the trigonometric functions and the hyperbolic ones that does not involvecomplex numbers.

Logic and universal algebra

[edit]

Formally, an identity is a trueuniversally quantified formula of the form $\forall x_{1},\ldots ,x_{n}:s=t,$ wheres andt areterms with no otherfree variables 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, theaxioms of amonoid 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 calledequations. In other words, an identity is an equation that is true for all values of the variables.^[11]^[12]