In mathematics, a multiplicative inverse or reciprocal for a number or , is a number which when multiplied by yields the multiplicative identity, 1. The multiplicative inverse of a fraction Dividing 1 by a real number yields its multiplicative inverse. 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 that maps is one of the simplest examples of a function which is its own inverse (an involution).

Multiplying by a number is the same as dividing 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 Encyclopædia Britannica (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as in a 1570 translation of Euclid's Elements.

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 ; then "inverse" typically implies that an element is both a left and right inverse.

The notation is sometimes also used for the inverse function of the function is the cosecant of , and not the inverse sine of x denoted by or . 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 the ).

Examples and counterexamples

In the real numbers, zero does not have a reciprocal (division by zero is undefined) 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 modular arithmetic, the modular multiplicative inverse of is also defined: it is the number such that . This multiplicative inverse exists if and only if and are coprime. For example, the inverse of is four because . The extended Euclidean algorithm may be used to compute it.

The sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, that is, nonzero elements , such that .

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 with respect to some base is then the inverse function of the map having 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 of would be , not .

where is the principal branch of the complex logarithm

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.

Complex numbers

As mentioned above, the reciprocal of every nonzero complex number by its complex conjugate, the absolute value of squared, which is the real number :

gives us the complex conjugate with a magnitude reduced to a value of 1, so dividing again by In particular, if ( Consequently, the imaginary units, , have additive inverse 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:

Geometrically in the complex plane the inverse of a complex number can be found by performing an inversion in the unit circle followed by a reflection over the real axis (see drawing).

Calculus

In real calculus, the derivative is given by the power rule

The power rule for integrals (Cavalieri's quadrature formula because doing so would result in division by zero where is the natural logarithm, we have:

Algorithms

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

Computing the reciprocal is important in many division algorithms and then multiplying it by has a zero 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 with three digits of precision. Taking , the following sequence is produced:

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

In constructive mathematics, for a real number to have a reciprocal, it is not sufficient that . There must instead be given a rational number such that . In terms of the approximation algorithm described above, this is needed to prove that the change in will eventually become arbitrarily small.

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

Reciprocals of irrational numbers

Every real or complex number excluding zero has a reciprocal, and reciprocals of certain irrational numbers can have important special properties. Examples include the reciprocal of (≈0.367879) and the golden ratio's reciprocal (≈ is the global minimum of Its additive inverse

The function

can be used to find the irrational number which differs from its reciprocal by an integer , because in general . For example:

Such irrational numbers share an evident property: they have the same fractional part as their reciprocal, since these numbers differ by an integer.

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

Further remarks

If the multiplication is associative, an element with a multiplicative inverse cannot be a zero divisor ( is a zero divisor if some nonzero , ). To see this, it is sufficient to multiply the equation by the inverse of (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 all integers except provide examples; they are not zero divisors nor do they have inverses in If the ring or algebra is finite, however, then all elements which are not zero divisors do have a (left and right) inverse. For, first observe that the map must be injective: implies Distinct elements map to distinct elements, so the image consists of the same finite number of elements, and the map is necessarily surjective. Specifically, (namely multiplication by ) must map some element to , , so that is an inverse for .

Applications

in any base can also act as a source of pseudo-random numbers, if is a "suitable" safe prime, a prime of the form where is also a prime. A sequence of pseudo-random numbers of length will be produced by the expansion.

See also

• Division (mathematics)

• Exponential decay

• Fraction

• Group (mathematics)

• Hyperbola

• Inverse distribution

• Inverse element

• List of sums of reciprocals

• Repeating decimal

• 6-sphere coordinates

• Unit fractions – reciprocals of integers

• Zeros and poles

Notes

References

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