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Multiplicative inverse

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Number which when multiplied by x equals 1

"Reciprocal (mathematics)" redirects here; not to be confused withReciprocation (geometry).

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Graph showing the diagrammatic representation of limits approaching infinity — The reciprocal function,y = 1/x. For every non-zerox coordinate, the correspondingy coordinate on the graph represents its multiplicative inverse. The graph forms arectangular hyperbola.

Inmathematics, amultiplicative inverse orreciprocal for anumberx, denoted by 1/x orx⁻¹, is a number which whenmultiplied byx yields themultiplicative identity, 1. The multiplicative inverse of afractiona/b isb/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. Thereciprocal function, thefunctionf(x) that mapsx to 1/x, is one of the simplest examples of a function which is its own inverse (aninvolution).

Multiplying by a number is the same asdividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication by its reciprocal yields the original number (since the product of the number and its reciprocal is 1).

The termreciprocal was in common use at least as far back as the third edition ofEncyclopædia Britannica (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described asreciprocall in a 1570 translation ofEuclid'sElements.^[1]

In the phrasemultiplicative inverse, the qualifiermultiplicative is often omitted and then tacitly understood (in contrast to theadditive inverse). Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen thatab ≠ba; then "inverse" typically implies that an element is both a left and rightinverse.

The notationf⁻¹ is sometimes also used for theinverse function of the functionf, which is for most functions not equal to the multiplicative inverse. For example, the multiplicative inverse1/(sinx) = (sinx)⁻¹ is thecosecant of x, and not theinverse sine ofx denoted bysin⁻¹x orarcsinx. The terminology differencereciprocal versusinverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example inFrench, the inverse function is preferably called thebijection réciproque).

Examples and counterexamples

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In the real numbers,zero does not have a reciprocal (division by zero isundefined) because no real number multiplied by 0 produces 1 (the product of any number with zero is zero). With the exception of zero, reciprocals of everyreal number are real, reciprocals of everyrational number are rational, and reciprocals of everycomplex number are complex. The property that every element other than zero has a multiplicative inverse is part of the definition of afield, of which these are all examples. On the other hand, nointeger other than 1 and −1 has an integer reciprocal, and so the integers are not a field.

Inmodular arithmetic, themodular multiplicative inverse ofa is also defined: it is the numberx such thatax ≡ 1 (modn). This multiplicative inverse existsif and only ifa andn arecoprime. For example, the inverse of 3 modulo 11 is 4 because4 ⋅ 3 ≡ 1 (mod 11). Theextended Euclidean algorithm may be used to compute it.

Thesedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, that is, nonzero elementsx,y such thatxy = 0.

Asquare matrix has an inverseif and only if itsdeterminant has an inverse in the coefficientring. The linear map that has the matrixA⁻¹ with respect to some base is then the inverse function of the map havingA as matrix in the same base. Thus, the two distinct notions of the inverse of a function are strongly related in this case, but they still do not coincide, since the multiplicative inverse ofAx would be (Ax)⁻¹, notA⁻¹x.

These two notions of an inverse function do sometimes coincide, for example for the function $f(x)=x^{i}=e^{i\ln(x)}$ where $\ln$ is theprincipal branch of the complex logarithm and $e^{-\pi }<|x|<e^{\pi }$ :

((1/f)\circ f)(x)=(1/f)(f(x))=1/(f(f(x)))=1/e^{i\ln(e^{i\ln(x)})}=1/e^{ii\ln(x)}=1/e^{-\ln(x)}=x

Thetrigonometric functions are related by the reciprocal identity: the cotangent is the reciprocal of the tangent; the secant is the reciprocal of the cosine; the cosecant is the reciprocal of the sine.

A ring in which every nonzero element has a multiplicative inverse is adivision ring; likewise analgebra in which this holds is adivision algebra.

Complex numbers

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As mentioned above, the reciprocal of every nonzero complex number $z=a+bi$ is complex. It can be found by multiplying both top and bottom of 1/z by itscomplex conjugate ${\bar {z}}=a-bi$ and using the property that $z{\bar {z}}=\|z\|^{2}$ , theabsolute value ofz squared, which is the real numbera² +b²:

{\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

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

gives us thecomplex conjugate with amagnitude reduced to a value of $1 {\displaystyle 1}$ , so dividing again by $\|z\|$ ensures that the magnitude is now equal to the reciprocal of the original magnitude as well, hence:

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

In particular, if ||z||=1 (z has unit magnitude), then $1/z={\bar {z}}$ . Consequently, theimaginary units,±i, haveadditive inverse equal to multiplicative inverse, and are the only complex numbers with this property. For example, additive and multiplicative inverses ofi are−(i) = −i and1/i = −i, respectively.

For a complex number in polar formz =r(cos φ +i sin φ), the reciprocal simply takes the reciprocal of the magnitude and the negative of the angle:

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

Geometric intuition for the integral of 1/x. The three integrals from 1 to 2, from 2 to 4, and from 4 to 8 are all equal. Each region is the previous region halved vertically and doubled horizontally. Extending this, the integral from 1 to 2^k isk times the integral from 1 to 2, just as ln 2^k =k ln 2.

Calculus

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In realcalculus, thederivative of1/x =x⁻¹ is given by thepower rule with the 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 compute the integral of 1/x, because doing so would result in division by 0: $\int {\frac {dx}{x}}={\frac {x^{0}}{0}}+C$ Instead the integral is given by: $\int _{1}^{a}{\frac {dx}{x}}=\ln a,$ $\int {\frac {dx}{x}}=\ln x+C.$ where ln is thenatural logarithm. To show this, note that ${\textstyle {\frac {d}{dy}}e^{y}=e^{y}}$ , so if $x=e^{y}$ and $y=\ln x$ , we have:^[2] ${\begin{aligned}&{\frac {dx}{dy}}=x\quad \Rightarrow \quad {\frac {dx}{x}}=dy\\[10mu]&\quad \Rightarrow \quad \int {\frac {dx}{x}}=\int dy=y+C=\ln x+C.\end{aligned}}$

Algorithms

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The reciprocal may be computed by hand with the use oflong division.

Computing the reciprocal is important in manydivision algorithms, since the quotienta/b can be computed by first computing 1/b and then multiplying it bya. Noting that $f(x)=1/x-b$ has azero atx = 1/b,Newton's method can find that zero, starting with a guess $x_{0}$ and iterating using the rule:

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 precision is reached. For example, suppose we wish to compute 1/17 ≈ 0.0588 with 3 digits of precision. Takingx₀ = 0.1, the following sequence is produced:

x₁ = 0.1(2 − 17 × 0.1) = 0.03

x₂ = 0.03(2 − 17 × 0.03) = 0.0447

x₃ = 0.0447(2 − 17 × 0.0447) ≈ 0.0554

x₄ = 0.0554(2 − 17 × 0.0554) ≈ 0.0586

x₅ = 0.0586(2 − 17 × 0.0586) ≈ 0.0588

A typical initial guess can be found by roundingb to a nearby power of 2, then usingbit shifts to compute its reciprocal.

Inconstructive mathematics, for a real numberx to have a reciprocal, it is not sufficient thatx ≠ 0. There must instead be given arational numberr such that 0 < r < |x|. In terms of the approximationalgorithm described above, this is needed to prove that the change iny will eventually become arbitrarily small.

Graph of f(x) =x^x showing the minimum at (1/e,e^−1/e).

This iteration can also be generalized to a wider sort of inverses; for example,matrix inverses.

Reciprocals of irrational numbers

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Every real or complex number excluding zero has a reciprocal, and reciprocals of certainirrational numbers can have important special properties. Examples include the reciprocal ofe (≈ 0.367879) and thegolden ratio's reciprocal (≈ 0.618034). The first reciprocal is special because no other positive number can produce a lower number when put to the power of itself; $f(1/e)$ is theglobal minimum of $f(x)=x^{x}$ . The second number is the only positive number that is equal to its reciprocal plus one: $\varphi =1/\varphi +1$ . Itsadditive inverse is the only negative number that is equal to its reciprocal minus one: $-\varphi =-1/\varphi -1$ .

The function ${\textstyle f(n)={\tfrac {1}{2}}n+{\sqrt {{\bigl (}{\tfrac {1}{2}}n{\bigr )}{\vphantom {)}}^{2}+1}}}$ can be used to find the irrational number which differs from its reciprocal by an integer⁠ $n {\displaystyle n}$ ⁠, because in general⁠ $\textstyle f(n)-f(n)^{-1}=n$ ⁠. For example, $f(4)$ is the irrational number $2+{\sqrt {5}}$ . Its reciprocal is $1/{\bigl (}2+{\sqrt {5}}~\!{\bigr )}=-2+{\sqrt {5}}$ , exactly $4 {\displaystyle 4}$ less. Such irrational numbers share an evident property: they have the samefractional part as their reciprocal, since these numbers differ by an integer.

The reciprocal function plays an important role insimple continued fractions, which have a number of remarkable properties relating to the representation of (both rational and) irrational numbers.

Further remarks

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If the multiplication is associative, an elementx with a multiplicative inverse cannot be azero divisor (x is a zero divisor if some nonzeroy,xy = 0). To see this, it is sufficient to multiply the equationxy = 0 by the inverse ofx (on the left), and then simplify using associativity. In the absence of associativity, thesedenions provide a counterexample.

The converse does not hold: an element which is not azero divisor is not guaranteed to have a multiplicative inverse.WithinZ, all integers except −1, 0, 1 provide examples; they are not zero divisors nor do they have inverses inZ.If the ring or algebra isfinite, however, then all elementsa which are not zero divisors do have a (left and right) inverse. For, first observe that the mapf(x) =ax must beinjective:f(x) =f(y) impliesx =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}}

Distinct elements map to distinct elements, so the image consists of the same finite number of elements, and the map is necessarilysurjective. Specifically, ƒ (namely multiplication bya) must map some elementx to 1,ax = 1, so thatx is an inverse fora.

Applications

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The expansion of the reciprocal 1/q in any base can also act^[3] as a source ofpseudo-random numbers, ifq is a "suitable"safe prime, a prime of the form 2p + 1 wherep is also a prime. A sequence of pseudo-random numbers of lengthq − 1 will be produced by the expansion.

Notes

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^"In equall Parallelipipedons the bases are reciprokall to their altitudes".OED "Reciprocal" §3a. SirHenry Billingsley translation of Elements XI, 34.
^Anthony, Dr."Proof that INT(1/x)dx = lnx".Ask Dr. Math. Drexel University. Retrieved22 March 2013.
^Mitchell, Douglas W., "A nonlinear random number generator with known, long cycle length",Cryptologia 17, January 1993, 55–62.