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In , a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when by ''x'' yields the , 1. The multiplicative inverse of a ''a''/''b'' is ''b''/''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. The reciprocal function, the function ''f''(''x'') that maps ''x'' to 1/''x'', is one of the simplest examples of a function which is its own inverse (an ). Multiplying by a number is the same as 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 term ''reciprocal'' was in common use at least as far back as the third edition of ' (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as in a 1570 translation of 's '. In the phrase ''multiplicative inverse'', the qualifier ''multiplicative'' is often omitted and then tacitly understood (in contrast to the ). Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ; then "inverse" typically implies that an element is both a left and right . The notation ''f'' −1 is sometimes also used for the of the function ''f'', which is not in general equal to the multiplicative inverse. For example, the multiplicative inverse is the of x, and not the denoted by or . Only for s are they strongly related (see below). The terminology difference ''reciprocal'' versus ''inverse'' is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example in , the inverse function is preferably called the ).

Examples and counterexamples

In the real numbers, 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 every are real, reciprocals of every are rational, and reciprocals of every are complex. The property that every element other than zero has a multiplicative inverse is part of the definition of a , of which these are all examples. On the other hand, no other than 1 and −1 has an integer reciprocal, and so the integers are not a field. In , the of ''a'' is also defined: it is the number ''x'' such that . This multiplicative inverse exists ''a'' and ''n'' are . For example, the inverse of 3 modulo 11 is 4 because . The may be used to compute it. The s are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, that is, nonzero elements ''x'', ''y'' such that ''xy'' = 0. A has an inverse its has an inverse in the coefficient . The linear map that has the matrix ''A''−1 with respect to some base is then the reciprocal function of the map having ''A'' as matrix in the same base. Thus, the two distinct notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case (as noted above). The 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 a ; likewise an in which this holds is a .

Complex numbers

As mentioned above, the reciprocal of every nonzero complex number is complex. It can be found by multiplying both top and bottom of 1/''z'' by its $\bar z = a - bi$ and using the property that $z\bar z = \, z\, ^2$, the of ''z'' squared, which is the real number : :$\frac = \frac = \frac = \frac = \frac - \fraci.$ The intuition is that :$\frac$ gives us the with a reduced to a value of $1$, so dividing again by $\, z\,$ ensures that the magnitude is now equal to the reciprocal of the original magnitude as well, hence: :$\frac = \frac$ In particular, if , , ''z'', , =1 (''z'' has unit magnitude), then $1/z = \bar z$. Consequently, the s, , have equal to multiplicative inverse, and are the only complex numbers with this property. For example, additive and multiplicative inverses of are and , respectively. For a complex number in polar form , the reciprocal simply takes the reciprocal of the magnitude and the negative of the angle: :$\frac = \frac\left\left(\cos\left(-\varphi\right) + i \sin\left(-\varphi\right)\right\right).$ Calculus

In real , the of is given by the with the power −1: :$\frac x^ = \left(-1\right)x^ = -x^ = -\frac.$ The power rule for integrals () cannot be used to compute the integral of 1/''x'', because doing so would result in division by 0: $\int \frac = \frac + C$ Instead the integral is given by: $\int_1^a \frac = \ln a,$ $\int \frac = \ln x + C.$ where ln is the . To show this, note that $\frac e^x = e^x$, so if $y = e^x$ and $x = \ln y$, we have: $\frac = y\quad \Rightarrow \quad \frac = dx \quad\Rightarrow\quad \int \frac = \int dx \quad\Rightarrow\quad \int \frac = x + C = \ln y + C.$

Algorithms

The reciprocal may be computed by hand with the use of . Computing the reciprocal is important in many s, since the quotient ''a''/''b'' can be computed by first computing 1/''b'' and then multiplying it by ''a''. Noting that $f\left(x\right) = 1/x - b$ has a at ''x'' = 1/''b'', can find that zero, starting with a guess $x_0$ and iterating using the rule: :$x_ = x_n - \frac = x_n - \frac = 2x_n - bx_n^2 = x_n\left(2 - bx_n\right).$ This continues until the desired precision is reached. For example, suppose we wish to compute 1/17 ≈ 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 s to compute its reciprocal. In , for a real number ''x'' to have a reciprocal, it is not sufficient that ''x'' ≠ 0. There must instead be given a ''rational'' number ''r'' such that 0 < ''r'' < , ''x'', . In terms of the approximation described above, this is needed to prove that the change in ''y'' will eventually become arbitrarily small. This iteration can also be generalized to a wider sort of inverses; for example, .

Reciprocals of irrational numbers

Every real or complex number excluding zero has a reciprocal, and reciprocals of certain s can have important special properties. Examples include the reciprocal of ' (≈ 0.367879) and the (≈ 0.618034). The first reciprocal is special because no other positive number can produce a lower number when put to the power of itself; $f\left(1/e\right)$ is the of $f\left(x\right)=x^x$. The second number is the only positive number that is equal to its reciprocal plus one:$\varphi = 1/\varphi + 1$. Its is the only negative number that is equal to its reciprocal minus one:$-\varphi = -1/\varphi - 1$. The function $f(n) = n + \sqrt, n \in \N, n>0$ gives an infinite number of irrational numbers that differ with their reciprocal by an integer. For example, $f\left(2\right)$ is the irrational $2+\sqrt 5$. Its reciprocal $1 / \left(2 + \sqrt 5\right)$ is $-2 + \sqrt 5$, exactly $4$ less. Such irrational numbers share an evident property: they have the same as their reciprocal, since these numbers differ by an integer.

Further remarks

If the multiplication is associative, an element ''x'' with a multiplicative inverse cannot be a (''x'' is a zero divisor if some nonzero ''y'', ). To see this, it is sufficient to multiply the equation by the inverse of ''x'' (on the left), and then simplify using associativity. In the absence of associativity, the s provide a counterexample. The converse does not hold: an element which is not a is not guaranteed to have a multiplicative inverse. Within Z, all integers except −1, 0, 1 provide examples; they are not zero divisors nor do they have inverses in Z. If the ring or algebra is , however, then all elements ''a'' which are not zero divisors do have a (left and right) inverse. For, first observe that the map must be : implies : :$\begin ax &= ay &\quad \rArr & \quad ax-ay = 0 \\ & &\quad \rArr &\quad a\left(x-y\right) = 0 \\ & &\quad \rArr &\quad x-y = 0 \\ & &\quad \rArr &\quad x = y. \end$ Distinct elements map to distinct elements, so the image consists of the same finite number of elements, and the map is necessarily . Specifically, ƒ (namely multiplication by ''a'') must map some element ''x'' to 1, {{nowrap, 1=''ax'' = 1, so that ''x'' is an inverse for ''a''.

Applications

The expansion of the reciprocal 1/''q'' in any base can also act Mitchell, Douglas W., "A nonlinear random number generator with known, long cycle length," ' 17, January 1993, 55–62. as a source of , if ''q'' is a "suitable" , a prime of the form 2''p'' + 1 where ''p'' is also a prime. A sequence of pseudo-random numbers of length ''q'' − 1 will be produced by the expansion.