Multiplicative inverse

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).

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]

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

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).

Hyperbola one over x
The reciprocal function: y = 1/x. For every x except 0, y represents its multiplicative inverse. The graph forms a rectangular hyperbola.

Examples and counterexamples

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 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.

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.

Complex numbers

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 a2 + b2:

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:

Reciprocal integral
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 scaled vertically down by 50%, then horizontally by 200%. Extending this, the integral from 1 to 2k is k times the integral from 1 to 2, just as ln 2k = k ln 2.

Calculus

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]

Algorithms

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 x0 = 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 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.

X to x power showing minimum
Graph of f(x) = xx showing the minimum at (1/e, e−1/e).

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

Reciprocals of irrational numbers

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.

Further remarks

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.

Applications

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 2p + 1 where p is also a prime. A sequence of pseudo-random numbers of length q − 1 will be produced by the expansion.

See also

Notes

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

References

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

Backhouse's constant is a mathematical constant named after Nigel Backhouse. Its value is approximately 1.456 074 948.

It is defined by using the power series such that the coefficients of successive terms are the prime numbers,

and its multiplicative inverse as a formal power series,

Then:

(sequence A072508 in the OEIS).

This limit was conjectured to exist by Backhouse (1995), and the conjecture was later proven by Philippe Flajolet (1995).

Cos-1

Cos-1, COS-1, cos-1, or cos−1 may refer to:

Cos-1, one of two commonly used COS cell lines

cos x−1 = cos(x)−1 = −(1−cos(x)) = −ver(x) or negative versine of x, the additive inverse (or negation) of an old trigonometric function

cos−1y = cos−1(y), sometimes interpreted as arccos(y) or arccosine of y, the compositional inverse of the trigonometric function cosine (see below for ambiguity)

cos−1x = cos−1(x), sometimes interpreted as (cos(x))−1 = 1/cos(x) = sec(x) or secant of x, the multiplicative inverse (or reciprocal) of the trigonometric function cosine (see above for ambiguity)

cos x−1, sometimes interpreted as cos(x−1) = cos(1/x), the cosine of the multiplicative inverse (or reciprocal) of x (see below for ambiguity)

cos x−1, sometimes interpreted as (cos(x))−1 = 1/cos(x) = sec(x) or secant of x, the multiplicative inverse (or reciprocal) of the trigonometric function cosine (see above for ambiguity)

Cot-1

Cot-1, COT-1, cot-1, or cot−1 may refer to:

Cot-1 DNA, used in comparative genomic hybridization

cot−1y = cot−1(y), sometimes interpreted as arccot(y) or arccotangent of y, the compositional inverse of the trigonometric function cotangent (see below for ambiguity)

cot−1x = cot−1(x), sometimes interpreted as (cot(x))−1 = 1/cot(x) = tan(x) or tangent of x, the multiplicative inverse (or reciprocal) of the trigonometric function cotangent (see above for ambiguity)

cot x−1, sometimes interpreted as cot(x−1) = cot(1/x), the cotangent of the multiplicative inverse (or reciprocal) of x (see below for ambiguity)

cot x−1, sometimes interpreted as (cot(x))−1 = 1/cot(x) = tan(x) or tangent of x, the multiplicative inverse (or reciprocal) of the trigonometric function cotangent (see above for ambiguity)

Csc-1

Csc-1, CSC-1, csc-1, or csc−1 may refer to:

csc x−1 = csc(x)−1 = excsc(x) or excosecant of x, an old trigonometric function

csc−1y = csc−1(y), sometimes interpreted as arccsc(y) or arccosecant of y, the compositional inverse of the trigonometric function cosecant (see below for ambiguity)

csc−1x = csc−1(x), sometimes interpreted as (csc(x))−1 = 1/csc(x) = sin(x) or sine of x, the multiplicative inverse (or reciprocal) of the trigonometric function cosecant (see above for ambiguity)

csc x−1, sometimes interpreted as csc(x−1) = csc(1/x), the cosecant of the multiplicative inverse (or reciprocal) of x (see below for ambiguity)

csc x−1, sometimes interpreted as (csc(x))−1 = 1/csc(x) = sin(x) or sine of x, the multiplicative inverse (or reciprocal) of the trigonometric function cosecant (see above for ambiguity)

Cycles per instruction

In computer architecture, cycles per instruction (aka clock cycles per instruction, clocks per instruction, or CPI) is one aspect of a processor's performance: the average number of clock cycles per instruction for a program or program fragment. It is the multiplicative inverse of instructions per cycle.

Division ring

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring in which every nonzero element a has a multiplicative inverse, i.e., an element x with a·x = x·a = 1. Stated differently, a ring is a division ring if and only if the group of units equals the set of all nonzero elements. A division ring is a type of noncommutative ring under the looser definition where noncommutative ring refers to rings which are not necessarily commutative.

Division rings differ from fields only in that their multiplication is not required to be commutative. However, by Wedderburn's little theorem all finite division rings are commutative and therefore finite fields. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields".All division rings are simple, i.e. have no two-sided ideal besides the zero ideal and itself.

Extended Euclidean algorithm

In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor of integers a and b, also the coefficients of Bézout's identity, which are integers x and y such that

This is a certifying algorithm, because the gcd is the only number that can simultaneously satisfy this equation and divide the inputs. It allows one to compute also, with almost no extra cost, the quotients of a and b by their greatest common divisor.

Extended Euclidean algorithm also refers to a very similar algorithm for computing the polynomial greatest common divisor and the coefficients of Bézout's identity of two univariate polynomials.

The extended Euclidean algorithm is particularly useful when a and b are coprime. With that provision, x is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. Similarly, the polynomial extended Euclidean algorithm allows one to compute the multiplicative inverse in algebraic field extensions and, in particular in finite fields of non prime order. It follows that both extended Euclidean algorithms are widely used in cryptography. In particular, the computation of the modular multiplicative inverse is an essential step in RSA public-key encryption method.

Gross Rent Multiplier

Gross Rent Multiplier (GRM) is the ratio of the price of a real estate investment to its annual rental income before accounting for expenses such as property taxes, insurance, and utilities; GRM is the number of years the property would take to pay for itself in gross received rent. For a prospective real estate investor, a lower GRM represents a better opportunity.

The GRM is useful for comparing and selecting investment properties where depreciation effects, periodic costs (such as property taxes and insurance) and costs to the investor incurred by a potential renter (such as utilities and repairs) can be expected to be uniform across the properties (either as uniform values or uniform fractions of the gross rental income) or insignificant in comparison to gross rental income. As these costs are also often more difficult to predict than market rental return, the GRM serves as an alternative to a measure of net investment return where such a measure would be difficult to determine.

Example; $200,000 Sale Price / (750 per month rent * 12 months) = 22.22

Today, it is quite common for GRM to be quoted by real estate professionals using annual rents rather than monthly rents. A 100 GRM (monthly rents) = 8.33 GRM (annual rents). An 8.33 GRM calculated on annual rents suggests the gross rent will pay for the property in 8.33 years.

The common measure of rental real estate value based on net return rather than gross rental income is the Capitalization Rate or Cap Rate. In contrast to the GRM, the Cap Rate is not a multiplier but a rate of annual return. A similar multiplier to the GRM derived from net return would be the multiplicative inverse of the Cap Rate.

Identity element

In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic structures such as groups and rings. The term identity element is often shortened to identity (as will be done in this article) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.

Let (S, ∗) be a set S with a binary operation ∗ on it. Then an element e of S is called a left identity if e ∗ a = a for all a in S, and a right identity if a ∗ e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.

An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). These need not be ordinary addition and multiplication, but rather arbitrary operations. The distinction is used most often for sets that support both binary operations, such as rings and fields. The multiplicative identity is often called unity in the latter context (a ring with unity). This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. Unity itself is necessarily a unit.

Inequality (mathematics)

In mathematics, an inequality is a relation that holds between two values when they are different (see also: equality).

The notation a ≠ b means that a is not equal to b.It does not say that one is greater than the other, or even that they can be compared in size.If the values in question are elements of an ordered set, such as the integers or the real numbers, they can be compared in size.

The notation a < b means that a is less than b.

The notation a > b means that a is greater than b.In either case, a is not equal to b. These relations are known as strict inequalities. The notation a < b may also be read as "a is strictly less than b".In contrast to strict inequalities, there are two types of inequality relations that are not strict:

The notation a ≤ b or a ⩽ b means that a is less than or equal to b (or, equivalently, not greater than b, or at most b); "not greater than" can also be represented by a ≯ b, the symbol for "greater than" bisected by a vertical line, "not". (The Unicode characters are: U+2264 ≤ LESS-THAN OR EQUAL TO, U+2A7D ⩽ LESS-THAN OR SLANTED EQUAL TO, and U+226F ≯ NOT GREATER-THAN.)

The notation a ≥ b or a ⩾ b means that a is greater than or equal to b (or, equivalently, not less than b, or at least b); "not less than" can also be represented by a ≮ b, the symbol for "less than" bisected by a vertical line, "not". (The Unicode characters are: U+2265 ≥ GREATER-THAN OR EQUAL TO, U+2A7E ⩾ GREATER-THAN OR SLANTED EQUAL TO, and U+226E ≮ NOT LESS-THAN.)In engineering sciences, a less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude. This implies that the lesser value can be neglected with little effect on the accuracy of an approximation. (For example, the ultrarelativistic limit in physics.)

The notation a ≪ b means that a is much less than b. (In measure theory, however, this notation is used for absolute continuity, an unrelated concept.)

The notation a ≫ b means that a is much greater than b.

Instructions per cycle

In computer architecture, instructions per cycle (IPC) is one aspect of a processor's performance: the average number of instructions executed for each clock cycle. It is the multiplicative inverse of cycles per instruction.

Inverse second

The inverse second, reciprocal second, or per second (s−1) is a unit of frequency, defined as the multiplicative inverse of the second (a unit of time). It is dimensionally equivalent to:

the unit hertz – the SI unit for cycles per second

the unit becquerel – the SI unit for aperiodic or stochastic radionuclide events per second

the unit baud – the unit for symbol rate over a communication link

strain rate – the velocity gradient (comprising the shear rate and directional expansion rate) of a fluid or solidIt also provides the denominator for temporal rates, such as that of angular frequency in radians per second.

Mean reciprocal rank

The mean reciprocal rank is a statistic measure for evaluating any process that produces a list of possible responses to a sample of queries, ordered by probability of correctness. The reciprocal rank of a query response is the multiplicative inverse of the rank of the first correct answer: 1 for first place, ​12 for second place, ​13 for third place and so on. The mean reciprocal rank is the average of the reciprocal ranks of results for a sample of queries Q:

where refers to the rank position of the first relevant document for the i-th query.

The reciprocal value of the mean reciprocal rank corresponds to the harmonic mean of the ranks.

Modular multiplicative inverse

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

which is the shorthand way of writing the statement that m divides (evenly) the quantity ax − 1, or, put another way, the remainder after dividing ax by the integer m is 1. If a does have an inverse modulo m there are an infinite number of solutions of this congruence which form a congruence class with respect to this modulus. Furthermore, any integer that is congruent to a (i.e., in a's congruence class) will have any element of x's congruence class as a modular multiplicative inverse. Using the notation of to indicate the congruence class containing w, this can be expressed by saying that the modulo multiplicative inverse of the congruence class is the congruence class such that:

where the symbol denotes the multiplication of equivalence classes modulo m. Written in this way the analogy with the usual concept of a multiplicative inverse in the set of rational or real numbers is clearly represented, replacing the numbers by congruence classes and altering the binary operation appropriately.

As with the analogous operation on the real numbers, a fundamental use of this operation is in solving, when possible, linear congruences of the form,

Finding modular multiplicative inverses also has practical applications in the field of cryptography, i.e. public-key cryptography and the RSA Algorithm. A benefit for the computer implementation of these applications is that there exists a very fast algorithm (the extended Euclidean algorithm) that can be used for the calculation of modular multiplicative inverses.

Operation (mathematics)

In mathematics, an operation is a calculation from zero or more input values (called operands) to an output value. The number of operands is the arity of the operation. The most commonly studied operations are binary operations, (that is, operations of arity 2) such as addition and multiplication, and unary operations (operations of arity 1), such as additive inverse and multiplicative inverse. An operation of arity zero, or nullary operation, is a constant. The mixed product is an example of an operation of arity 3, also called ternary operation. Generally, the arity is supposed to be finite. However, infinitary operations are sometimes considered, in which context the "usual" operations of finite arity are called finitary operations.

Sec-1

Sec-1, SEC-1, sec-1, or sec−1 may refer to:

sec x−1 = sec(x)−1 = exsec(x) or exsecant of x, an old trigonometric function

sec−1y = sec−1(y), sometimes interpreted as arcsec(y) or arcsecant of y, the compositional inverse of the trigonometric function secant (see below for ambiguity)

sec−1x = sec−1(x), sometimes interpreted as (sec(x))−1 = 1/sec(x) = cos(x) or cosine of x, the multiplicative inverse (or reciprocal) of the trigonometric function secant (see above for ambiguity)

sec x−1, sometimes interpreted as sec(x−1) = sec(1/x), the secant of the multiplicative inverse (or reciprocal) of x (see below for ambiguity)

sec x−1, sometimes interpreted as (sec(x))−1 = 1/sec(x) = cos(x) or cosine of x, the multiplicative inverse (or reciprocal) of the trigonometric function secant (see above for ambiguity)

Sin-1

Sin-1, SIN-1, sin-1, or sin−1 may refer to:

SIN-1, Linsidomine (3-morpholinosydnonimine), a drug acting as a vasodilator

sin x−1 = sin(x)−1 = −(1−sin(x)) = −cvs(x) or negative coversine of x, the additive inverse (or negation) of an old trigonometric function

sin−1y = sin−1(y), sometimes interpreted as arcsin(y) or arcsine of y, the compositional inverse of the trigonometric function sine (see below for ambiguity)

sin−1x = sin−1(x), sometimes interpreted as (sin(x))−1 = 1/sin(x) = csc(x) or cosecant of x, the multiplicative inverse (or reciprocal) of the trigonometric function sine (see above for ambiguity)

sin x−1, sometimes interpreted as sin(x−1) = sin(1/x), the sine of the multiplicative inverse (or reciprocal) of x (see below for ambiguity)

sin x−1, sometimes interpreted as (sin(x))−1 = 1/sin(x) = csc(x) or cosecant of x, the multiplicative inverse (or reciprocal) of the trigonometric function sine (see above for ambiguity)

Tan-1

Tan-1, TAN-1, tan-1, or tan−1 may refer to:

tan−1y = tan−1(y), sometimes interpreted as arctan(y) or arctangent of y, the compositional inverse of the trigonometric function tangent (see below for ambiguity)

tan−1x = tan−1(x), never interpreted as (tan(x))−1 = 1/tan(x) = cot(x) or cotangent of x, the multiplicative inverse (or reciprocal) of the trigonometric function tangent (see above for ambiguity)

tan x−1, sometimes interpreted as tan(x−1) = tan(1/x), the tangent of the multiplicative inverse (or reciprocal) of x (see below for ambiguity)

tan x−1, sometimes interpreted as (tan(x))−1 = 1/tan(x) = cot(x) or cotangent of x, the multiplicative inverse (or reciprocal) of the trigonometric function tangent (see above for ambiguity)

Zodiac (cipher)

In cryptography, Zodiac is a block cipher designed in 2000 by Chang-Hyi Lee for the Korean firm SoftForum.

Zodiac uses a 16-round Feistel network structure with key whitening. The round function uses only XORs and S-box lookups. There are two 8×8-bit S-boxes: one based on the discrete exponentiation 45x as in SAFER, the other using the multiplicative inverse in the finite field GF(28), as introduced by SHARK.

Zodiac is theoretically vulnerable to impossible differential cryptanalysis, which can recover a 128-bit key in 2119 encryptions.

This page is based on a Wikipedia article written by authors (here).
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.