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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 multiplicative identity, 1. The multiplicative inverse of a fraction *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 involution).

**Mathematics** includes the study of such topics as quantity, structure, space, and change.

**Multiplication** is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.

In mathematics, a **rational number** is any number that can be expressed as the quotient or fraction *p*/*q* of two integers, a numerator *p* and a non-zero denominator *q*. Since *q* may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "**the rationals**", the **field of rationals** or the **field of rational numbers** is usually denoted by a boldface **Q** ; it was thus denoted in 1895 by Giuseppe Peano after *quoziente*, Italian for "quotient".

- Examples and counterexamples
- Complex numbers
- Calculus
- Algorithms
- Reciprocals of irrational numbers
- Further remarks
- Applications
- See also
- Notes
- References

The term *reciprocal* was in common use at least as far back as the third edition of * Encyclopædia Britannica * (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as *reciprocall* in a 1570 translation of Euclid's * Elements *.^{ [1] }

The ** Encyclopædia Britannica**, formerly published by Encyclopædia Britannica, Inc., is a general knowledge English-language encyclopaedia. It was written by about 100 full-time editors and more than 4,000 contributors. The 2010 version of the 15th edition, which spans 32 volumes and 32,640 pages, was the last printed edition.

**Euclid**, sometimes given the name **Euclid of Alexandria** to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "founder of geometry" or the "father of geometry". He was active in Alexandria during the reign of Ptolemy I. His *Elements* is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics from the time of its publication until the late 19th or early 20th century. In the *Elements*, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory, and rigor.

The * Elements* is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC. It is a collection of definitions, postulates, propositions, and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines.

In the phrase *multiplicative inverse*, the qualifier *multiplicative* is often omitted and then tacitly understood (in contrast to the additive inverse). Multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that *ab* ≠ *ba*; then "inverse" typically implies that an element is both a left and right inverse.

In mathematics, the **additive inverse** of a number a is the number that, when added to a, yields zero. This number is also known as the **opposite** (number), **sign change**, and **negation**. For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself.

In abstract algebra, the idea of an **inverse element** generalises concepts of a negation in relation to addition, and a reciprocal in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element. While the precise definition of an inverse element varies depending on the algebraic structure involved, these definitions coincide in a group.

The notation *f*^{−1} is sometimes also used for the inverse function of the function *f*, which is not in general equal to the multiplicative inverse. For example, the multiplicative inverse 1/(sin *x*) = (sin *x*)^{−1} is the cosecant of x, and not the inverse sine of *x* denoted by sin^{−1}*x* or arcsin *x*. Only for linear maps 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 French, the inverse function is preferably called bijection réciproque).

In mathematics, an **inverse function** is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa, i.e., *f*(*x*) = *y* if and only if *g*(*y*) = *x*.

In mathematics, the **inverse trigonometric functions** are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

**French** is a Romance language of the Indo-European family. It descended from the Vulgar Latin of the Roman Empire, as did all Romance languages. French evolved from Gallo-Romance, the spoken Latin in Gaul, and more specifically in Northern Gaul. Its closest relatives are the other langues d'oïl—languages historically spoken in northern France and in southern Belgium, which French (Francien) has largely supplanted. French was also influenced by native Celtic languages of Northern Roman Gaul like Gallia Belgica and by the (Germanic) Frankish language of the post-Roman Frankish invaders. Today, owing to France's past overseas expansion, there are numerous French-based creole languages, most notably Haitian Creole. A French-speaking person or nation may be referred to as **Francophone** in both English and French.

In the real numbers, zero does not have a reciprocal 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 real number are real, reciprocals of every rational number are rational, and reciprocals of every complex number are complex. The property that 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, no integer other than 1 and −1 has an integer reciprocal, and so the integers are not a field.

In mathematics, **division by zero** is division where the divisor (denominator) is zero. Such a division can be formally expressed as *a*/0 where *a* is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is no number which, when multiplied by 0, gives *a*, and so division by zero is undefined. Since any number multiplied by zero is zero, the expression 0/0 is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to *a*/0 is contained in George Berkeley's criticism of infinitesimal calculus in 1734 in *The Analyst*.

In mathematics, a **real number** is a value of a continuous quantity that can represent a distance along a line. The adjective *real* in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.

A **complex number** is a number that can be expressed in the form *a* + *bi*, where *a* and *b* are real numbers, and *i* is a solution of the equation *x*^{2} = −1. Because no real number satisfies this equation, *i* is called an imaginary number. For the complex number *a* + *bi*, *a* is called the **real part**, and *b* is called the **imaginary part**. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.

In modular arithmetic, the modular multiplicative inverse of *a* is also defined: it is the 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 may be used to compute it.

In mathematics, **modular arithmetic** is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the **modulus**. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book *Disquisitiones Arithmeticae*, published in 1801.

In mathematics, in particular the area of number theory, a **modular multiplicative inverse** of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In the standard notation of modular arithmetic this congruence is written as

In logic and related fields such as mathematics and philosophy, **if and only if** is a biconditional logical connective between statements.

The sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, i.e. nonzero elements *x*, *y* such that *xy* = 0.

A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring. 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 trigonometric 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 a division ring; likewise an algebra in which this holds is a division algebra.

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 its complex conjugate and using the property that , the absolute value of *z* squared, which is the real number *a*^{2} + *b*^{2}:

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

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

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

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:

Instead the integral is given by:

where ln is the natural logarithm. To show this, note that , so if and , we have:^{ [2] }

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

Computing the reciprocal is important in many division algorithms, since the quotient *a*/*b* can be computed by first computing 1/*b* and then multiplying it by *a*. Noting that has a zero at *x* = 1/*b*, Newton's method can find that zero, starting with a guess and iterating using the rule:

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 bit shifts to compute its reciprocal.

In constructive mathematics, 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 algorithm described above, this is needed to prove that the change in *y* will eventually become arbitrarily small.

This iteration can also be generalised to a wider sort of inverses, e.g. matrix inverses.

Every number excluding zero has a reciprocal, and reciprocals of certain irrational numbers can have important special properties. Examples include the reciprocal of e (≈ 0.367879) and the golden 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; 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:.

The function gives an infinite number of irrational numbers that differ with their reciprocal by an integer. For example, is the irrational . Its reciprocal is , exactly less. Such irrational numbers share a curious property: they have the same fractional part as their reciprocal.

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

The converse does not hold: an element which is not a zero divisor 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 finite, however, then all elements *a* which are not zero divisors do have a (left and right) inverse. For, first observe that the map *f*(*x*) = *ax* must be injective: *f*(*x*) = *f*(*y*) implies *x* = *y*:

Distinct elements map to distinct elements, so the image consists of the same finite number of elements, and the map is necessarily surjective. Specifically, ƒ (namely multiplication by *a*) must map some element *x* to 1, *ax* = 1, so that *x* is an inverse for *a*.

The expansion of the reciprocal 1/*q* in any base can also act ^{ [3] } as a source of pseudo-random numbers, if *q* is a "suitable" safe prime, 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.

- ↑ " In equall Parallelipipedons the bases are reciprokall to their altitudes".
*OED*"Reciprocal" §3a. Sir Henry Billingsley translation of Elements XI, 34. - ↑ Anthony, Dr. "Proof that INT(1/x)dx = lnx".
*Ask Dr. Math*. Drexel University. Retrieved 22 March 2013. - ↑ Mitchell, Douglas W., "A nonlinear random number generator with known, long cycle length,"
*Cryptologia*17, January 1993, 55–62.

**Euler's formula**, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:

In mathematics, a **finite field** or **Galois field** is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod *p* when *p* is a prime number.

In mathematics, the **logarithm** is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the *base* b, must be raised, to produce that number x. In the simplest case, the logarithm counts repeated multiplication of the same factor; e.g., since 1000 = 10 × 10 × 10 = 10^{3}, the "logarithm to base 10" of 1000 is 3. The logarithm of x to *base*b is denoted as log_{b} (*x*) (or, without parentheses, as log_{b} *x*, or even without explicit base as log *x*, when no confusion is possible). More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm for any two positive real numbers b and x where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is:

In probability theory, the **normal****distribution** is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be **normally distributed** and is called a **normal deviate**.

The **natural logarithm** of a number is its logarithm to the base of the mathematical constant *e*, where *e* is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of *x* is generally written as ln *x*, log_{e}*x*, or sometimes, if the base *e* is implicit, simply log *x*. Parentheses are sometimes added for clarity, giving ln(*x*), log_{e}(*x*) or log(*x*). This is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity.

In mathematics, the **polar coordinate system** is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

In mathematics, a **square root** of a number *a* is a number *y* such that *y*^{2} = *a*; in other words, a number *y* whose *square* (the result of multiplying the number by itself, or *y* ⋅ *y*) is *a*. For example, 4 and −4 are square roots of 16 because 4^{2} = (−4)^{2} = 16. Every nonnegative real number *a* has a unique nonnegative square root, called the *principal square root*, which is denoted by √*a*, where √ is called the *radical sign* or *radix*. For example, the principal square root of 9 is 3, which is denoted by √9 = 3, because 3^{2} = 3 · 3 = 9 and 3 is nonnegative. The term (or number) whose square root is being considered is known as the *radicand*. The radicand is the number or expression underneath the radical sign, in this example 9.

In number theory, **Euler's totient function** counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as *φ*(*n*) or *ϕ*(*n*), and may also be called **Euler's phi function**. It can be defined more formally as the number of integers k in the range 1 ≤ *k* ≤ *n* for which the greatest common divisor gcd(*n*, *k*) is equal to 1. The integers k of this form are sometimes referred to as totatives of n.

In algebra, the **discriminant** of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them. For example, the discriminant of the quadratic polynomial is which is zero if and only if the polynomial has a double root, and is positive if and only if the polynomial has two real roots.

**Exponentiation** is a mathematical operation, written as *b*^{n}, involving two numbers, the *base*b and the *exponent*n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, *b*^{n} is the product of multiplying n bases:

In calculus, and more generally in mathematical analysis, **integration by parts** or **partial integration** is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be readily derived by integrating the product rule of differentiation.

In mathematics, the **complex conjugate** of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. For example, the complex conjugate of 3 + 4*i* is 3 − 4*i*.

In mathematics, a **homogeneous function** is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

In mathematics, an **asymptotic expansion**, **asymptotic series** or **Poincaré expansion** is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by Dingle (1973) revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function.

A **cardioid** is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion.

The **multiple integral** is a definite integral of a function of more than one real variable, for example, *f*(*x*, *y*) or *f*(*x*, *y*, *z*). Integrals of a function of two variables over a region in **R**^{2} are called double integrals, and integrals of a function of three variables over a region of **R**^{3} are called triple integrals.

This is a summary of **differentiation rules**, that is, rules for computing the derivative of a function in calculus.

In the mathematical analysis, and especially in real and harmonic analysis, a **Birnbaum–Orlicz space** is a type of function space which generalizes the L^{p} spaces. Like the L^{p} spaces, they are Banach spaces. The spaces are named for Władysław Orlicz and Zygmunt William Birnbaum, who first defined them in 1931.

- Maximally Periodic Reciprocals, Matthews R.A.J.
*Bulletin of the Institute of Mathematics and its Applications*vol 28 pp 147–148 1992

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