In mathematics , a multiplicative inverse or reciprocal for a number X , denoted by 1/ X or X^{-1} , is a number which when multiplied by X yields the multiplier identity , 1. The inverse of a multiplier of the numerator a / b is b / a . For the multiplicative inverse of a real number, divide 1 by the number. For example, the inverse of 5 is one fifth (1/5 or 0.2), and the inverse of 0.25 is 1 divided by 0.25, or 4. The reciprocal function , the function f (x ) that maps x to 1 / x , is the simplest example of a function which has its own inverse (an intricacy ).

*Multiplying a number is the same as dividing its inverse and vice versa. For example, multiplying by 4/5 (or 0.8) will give the same result as dividing by 5/4 (or 1.25). Therefore, after multiplying by a number, its reciprocal multiplication gives the original number (since their product is 1).*

The term reciprocal was in common use as of the third edition of the Encyclopdia Britannica (1797),to describe two numbers whose product is 1; Geometric quantities in inverse ratios aredescribed as reciprocals in a 1570 translation of Euclid ‘s Elements .

In the phrase multiplicative inverse , the qualifier multiplier is often omitted and then understood silently (as opposed to additive inverse ). Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it is possible that now ba ; Then “inverse” usually denotes that an element has both a left and a right inverse .

The notation f^{−1} is also sometimes used to refer to the inverse function of the function f , which is not normally equal to the multiplicative inverse. For example, the multiplicative inverse 1/(sin x ) = (sin x )^{−1} is the cosecant of x , not the inverse sine of x as denoted by sin^{−1} x or arcsin x . Only for linear maps are they strongly related (see below). reciprocal vs inverseThe terminology difference is not enough to make this distinction, as many authors prefer the opposite naming convention, probably for historical reasons (e.g. in French , the inverse function is preferably called a bijection reciprocity).

**Examples and counterexamples**

In real numbers, zero has no reciprocal because no real number when multiplied by 0 produces 1 (the product of any number containing zero is zero). With the exception of zero, the inverses of every real number are real, the inverses of every rational number are rational , and the inverses of every complex number are complex . The property in which every element other than zero has a multiplicative inverse is part of the definition of a field , of which these are all examples. On the other hand, any integer other than 1 and -1 is an integer reciprocal, and therefore integers are not a field.

In modular arithmetic , the modular multiplier of an inverse is also defined: it is a number x such that ax 1 (mod n ) . This multiplicative inverse exists if and only if a and n are coprime . For example, the inverse of 3 modulo 11 is 4 because 4 3 1 (mod 11) . The extended Euclidean algorithm can be used to calculate this. Sedenions is an algebra in which every nonzero element is a multiplicative inverse, has nonzero elements but still has divisors of zero, that is, x , y such that xy = 0.

A square matrix has an inverse if and only if the coefficients of its determinant are inverse to the ring . A linear map that contains a matrix A -1 with respect to some base is the reciprocal function of a map containing A as a matrix in the same base . Thus, while the two different notions of the inverse of a function are strongly related in this case, they should be carefully distinguished in the general case (as explained above).

Trigonometric functions are related to the mutual identity: the cotangent is the reciprocal of the tangent; secant is the inverse of cosine; Cosecant is the inverse of sine. A ring in which every non-zero element has a multiplicative inverse is a division ring ; Similarly an algebra in which it holds is a division algebra.

**Complex data**

As explained above, every non-zero complex number *z* = *a* + *bi* has an inverse complex. This can be found by multiplying both *the top and bottom of 1/z* by its complex conjugate and using the property that , the absolute value of *z squares, which is the real number a *^{2} + *b *^{2} :

{\bar {z}} = a-biz {\bar {z}} = \ | z \ | ^ {2}

{\frac {1}{z}}={\frac {\bar {z}}{z{\bar {z}}}}={\frac {\bar {z}}{\|z\|^{2}}}={\frac {a-bi}{a^{2}+b^{2}}}={\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}i.

the intuition is that

{\displaystyle {\frac {\bar {z}} {\| z \|}}}

gives us the complex conjugate of magnitude with a reduced magnitude to the value of , so dividing again ensures that the magnitude is now equal to the inverse of the original magnitude, so: *1 | z |*

{\displaystyle {\frac {1} {z}} = {\frac {\bar {z}} {\| z \| ^ {2}}}}

In particular, if || *z* ||=1 ( the unit of *z* is magnitude), then . Consequently, the imaginary units, ± *i* , have the additive inverse equal to the multiplicative inverse, and are the only complex numbers with this property. For example, the additive and multiplicative inverses of *i* are – ( *i* ) = – *i* and 1 / *i* = – *i* , respectively.

\frac{1}{z} = {\bar {z}}

For a complex number in polar form *z* = *r* (cos + *i* sin ) , the inverse takes only the inverse of the magnitude and the negative of the angle:

{\frac {1}{z}}={\frac {1}{r}}\left(\cos(-\varphi )+i\sin(-\varphi )\right).

**Calculation**

In real calculus, the derivative of 1 / *x* = *x *^{-1} is given by the power rule with power -1:

{\frac {d}{dx}}x^{-1}=(-1)x^{(-1)-1}=-x^{-2}=-{\frac {1}{x^{2}}}.

The power rule for integrals (Cavalieri’s quadrature formula) cannot be used to calculate the integral of 1/ *x* , because doing so would divide by 0:

{\displaystyle \int {\frac {dx}{x}}={\frac {x^{0}}{0}}+C}

Instead the integral is given by:

{\displaystyle \int _{1}^{a}{\frac {dx}{x}}=\ln a,}

{\displaystyle \int {\frac {dx}{x}}=\ln x+C.}

where ln is the natural logarithm. To show this, note that , then if and , we have:

{\textstyle {\frac {d}{dx}}e^{x}=e^{x}}y=e^{x}x=\ln y

{\displaystyle {\frac {dy}{dx}}=y\quad \Rightarrow \quad {\frac {dy}{y}}=dx\quad \Rightarrow \quad \int {\frac {dy}{y}}=\int dx\quad \Rightarrow \quad \int {\frac {dy}{y}}=x+C=\ln y+C.}

**Algorithm**

The inverse can be calculated by hand using long division.

Reciprocal calculation is important in many division algorithms, because the quotient *a* / *b can be* calculated by first computing 1/ *b* and then multiplying it *by a. *Noted that a is zero at *x* = 1 / *b* , Newton’s method can find that zero, starting with an approximation and iterating using the rule:

f(x)=1/x-bx_{0}

x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}=x_{n}-{\frac {1/x_{n}-b}{-1/x_{n}^{2}}}=2x_{n}-bx_{n}^{2}=x_{n}(2-bx_{n}).

This continues until the desired accuracy is reached. For example, suppose we want to calculate 1/17 of 0.0588 with 3 digits of precision. Taking *x *0 = 0.1 , the following sequence is produced:

*x*_{1}= 0.1(2 − 17 × 0.1) = 0.03*x*_{2}= 0.03(2 − 17 × 0.03) = 0.0447*x*_{3}= 0.0447(2 − 17 × 0.0447) ≈ 0.0554*x*_{4}= 0.0554(2 − 17 × 0.0554) ≈ 0.0586*x*_{5}= 0.0586(2 − 17 × 0.0586) ≈ 0.0588

A typical initial guess can be found by rounding *b to a nearby power of 2, then using a bit shift to calculate its reciprocal.*

In constructive mathematics, for a real number *x* to have a reciprocal, it is not sufficient that *x* a 0. There should instead be given *rational* number *r* such that 0 < *r* <| *X* |. In the context of the approximation algorithm described above, it is necessary to prove that the change in *y* will eventually become arbitrarily small.

This iteration can also be generalized to a wide variety of inversions; For example, matrix inverse.

**Inverses of Irrational Numbers**

Every real or complex number except zero has an inverse, and the inverse of some irrational numbers can have important special properties. Examples include the reciprocal of *e* ( 0.3 0.367879) and the reciprocal of the golden ratio (≈ 0.618034). The first inverse is special because no other positive number can produce a lesser number when put into its own power; is the global minimum of . The second number is the only positive number that is equal to its reciprocal plus one: . Its additive inverse is the only negative number that is equal to its reciprocal minus one:

f(1/e)f(x)=x^{x}{\displaystyle \varphi =1/\varphi +1}{\displaystyle -\varphi =-1/\varphi -1}

The program returns an infinite number of irrational numbers that are differentiable with their reciprocal by an integer. For example, irrational . Its reciprocal , less than exactly . Such irrational numbers share an obvious property: they have the same fractional part as their reciprocal, because these numbers are different from an integer.

{\textstyle f(n)=n+{\sqrt {(n^{2}+1)}},n\in \mathbb {N} ,n>0}f(2)2+{\sqrt {5}}1/(2+{\sqrt {5}})-2+{\sqrt {5}}4

**Further comments**

If multiplication is associative, then the element with the multiplication inverse *x* cannot have a zero divisor ( *x* is a zero divisor if there is some nonzero *y* , *xy* = 0 ). To see this, it is enough to multiply the equation *xy* = 0 by the inverse of *x* (on the left), and then simplify using associativity. In the absence of affiliation, Sedanians provide a counterexample.

The inverse does not occur: An element that is not a zero divisor is not guaranteed to have a multiplicative inverse. **All integers within Z** except -1, 0, 1 provide examples; They are not zero divisors nor do they have an inverse in **Z. **If the ring or algebra is finite, however, all elements of *a* that are not zero divisors have a (left and right) inverse. For , first see that the map *f* ( *x* ) = *ax* must be injected: *f* ( *x* ) = *f* ( *y* ) implies *x* = *y* :

{\begin{aligned}ax&=ay&\quad \Rightarrow &\quad ax-ay=0\\&&\quad \Rightarrow &\quad a(x-y)=0\\&&\quad \Rightarrow &\quad x-y=0\\&&\quad \Rightarrow &\quad x=y.\end{aligned}}

Different elements map to different elements, so the image has the same number of elements, and the map is necessarily surjective. In particular, (ie multiplied by *a* ) must map some element *x* 1, for *ax* = 1 , so that *x* contains an inverse for *a* .

**Application**

*The inverse 1/ expansion of q* in any base can also serve as a source of pseudorandom numbers ^{[3]} , if *q* is a “suitable” safe prime, then 2 is a prime of the form *p* + 1 where *p* There is also one prime. *A sequence of pseudo-random numbers of length q* -1 will be produced by expansion .