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Encyclopedia > Absolute value

In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3. In computer programming, the mathematical function used to perform this calculation is usually given the name abs(). Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... A negative number is a number that is less than zero, such as −3. ... “Programming” redirects here. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...


Generalizations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... Definitions In abstract algebra, an ordered ring is a commutative ring with a a total order such that if and , then if and , then . ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...


The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts. The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

The graph of the absolute value function for real numbers.
The graph of the absolute value function for real numbers.

Contents

Wikipedia does not have an article with this exact name. ...

Real numbers

For any real number a the absolute value or modulus of a is denoted[2] by | a | and is defined as In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

|a| = begin{cases} a, & mbox{if } a ge 0 -a, & mbox{if } a < 0. end{cases}

As can be seen from the above definition, the absolute value of a is always either positive or zero, but never negative. A negative number is a number that is less than zero, such as &#8722;3. ... For other senses of this word, see zero or 0. ... A negative number is a number that is less than zero, such as −3. ...


From a geometric point of view, the absolute value of a real number is the distance along the real number line of that number from zero, and more generally the absolute value of the difference of two real numbers is the distance between them. Indeed the notion of an abstract distance function in mathematics can be seen to be a generalization of the absolute value of the difference (see "Distance" below). Distance is a numerical description of how far apart objects are at any given moment in time. ... In mathematics, the real line is simply the set of real numbers. ... For distance between people, see proxemics. ...


The following proposition, gives an identity which is sometimes used as an alternative (and equivalent) definition of the absolute value: In mathematics, the term identity has several important uses: An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an equality which is true under more particular conditions. ...


PROPOSITION 1:

|a| = sqrt{a^2}

The absolute value has the following four fundamental properties:


PROPOSITION 2:

|a| ge 0 Non-negativity
|a| = 0 iff a = 0 Positive-definiteness
|ab| = |a||b|, Multiplicativeness
|a+b| le |a| + |b| Subadditivity

Other important properties of the absolute value include: In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then f(ab) = f(a) f(b). ... A function f(x) is subadditive if for all x and y in the domain of f. ...


PROPOSITION 3:

|-a| = |a|, Symmetry
|a - b| = 0 iff a = b Identity of indiscernibles (equivalent to positive-definiteness)
|a - b| le |a - c| +|c - b| Triangle inequality (equivalent to subadditivity)
|a/b| = |a| / |b| mbox{ (if } b ne 0) , Preservation of division (equivalent to multiplicativeness)
|a-b| ge |a| - |b| (equivalent to subadditivity)

Two other useful inequalities are: Sphere symmetry group o. ... The identity of indiscernibles is an ontological principle; i. ... In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ...

|a| le b iff -b le a le b
|a| ge b iff a le -b mbox{ or } b le a

The above are often used in solving inequalities; for example:

|x-3| le 9 iff -9 le x-3 le 9
iff -6 le x le 12

Complex numbers

image:complex.png -from the creater, wshun 02:20, 24 Dec 2004 (UTC) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

Since the complex numbers are not ordered, the definition given above for the real absolute value cannot be directly generalized for a complex number. However the identity given in Proposition 1: In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... ...

|a| = sqrt{a^2}

can be seen as motivating the following definition.


For any complex number

z = x + iy,

where x and y are real numbers, the absolute value or modulus of z is denoted | z | , and is defined as

|z| = sqrt{x^2 + y^2}.

It follows that the absolute value of a real number x is equal to its absolute value considered as a complex number since:

|x + i0| = sqrt{x^2 + 0^2} = sqrt{x^2} = |x|.

Similar to the geometric interpretation of the absolute value for real numbers, it follows from the Pythagorean theorem that the absolute value of a complex number is the distance in the complex plane of that complex number from the origin, and more generally, that the absolute value of the difference of two complex numbers is equal to the distance between those two complex numbers. In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ...


The complex absolute value shares all the properties of the real absolute value given in Propositions 2 and 3 above. In addition, If

z = x + i y = r (cos phi + i sin phi ) ,

and

bar{z} = x - iy

is the complex conjugate of z, then it is easily seen that In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

|z| = r,
|z|=|bar{z}|
|z| = sqrt{zbar{z}}.

The latter formula is the complex analogue of proposition 1 mentioned above in the real case...


Since the positive reals form a subgroup of the complex numbers under multiplication, we may think of absolute value as an endomorphism of the multiplicative group of the complex numbers. In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... In mathematics, multiplicative group in group theory may mean any group G written in multiplicative notation (rather than additive notation for an abelian group) for its binary operation or in particular the multiplicative group of a field F, namely F{0} under multiplication, written F* or Fx. ...


Absolute value functions

The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0. It is monotonically decreasing on the interval (-∞, 0] and monotonically increasing on the interval [0, ∞). Since a real number and its negative have the same absolute value, it is an even function, and is hence not invertible. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... For a non-technical overview of the subject, see Calculus. ... A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right). ... A monotonically increasing function (it is strictly increasing on the left and just non-decreasing on the right). ... In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking negatives. ... In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ...


The complex absolute value function is continuous everywhere but (complex) differentiable nowhere (One way to see this is to show that it does not obey the Cauchy-Riemann equations). In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary but not sufficient condition for a function to be holomorphic. ...


Both the real and complex functions are idempotent. In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...


It is a nonlinear function. To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ...


Ordered rings

The definition of absolute value given for real numbers above can easily be extended to any ordered ring. That is, if a is an element of an ordered ring R, then the absolute value of a, denoted by | a | , is defined to be: Definitions In abstract algebra, an ordered ring is a commutative ring with a a total order such that if and , then if and , then . ...

|a| = begin{cases} a, & mbox{if } a ge 0 -a, & mbox{if } a < 0, end{cases}

where − a is the additive inverse of a, and 0 is the additive identity element. The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ... 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. ...


Distance

The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them. Distance is a numerical description of how far apart objects are at any given moment in time. ...


The standard Euclidean distance between two points In mathematics, the Euclidean distance or Euclidean metric is the ordinary distance between two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ...

a = (a_1, a_2, cdots , a_n)

and

b = (b_1, b_2, cdots , b_n)

in Euclidean n-space is defined as: Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

sqrt{(a_1-b_1)^2 + (a_2-b_2)^2 + cdots + (a_n-b_n)^2}.

This can be seen to be a generalization of | a − b | , since if a, b are real, then by Proposition 1,

|a - b| = sqrt{(a - b)^2}

while if

a = a_1 + i a_2 ,

and

b = b_1 + i b_2 ,

are complex numbers, then

|a - b| , = |(a_1 + i a_2) - (b_1 + i b_2)|,
= |(a_1 - b_1) + i(a_2 - b_2)|,
= sqrt{(a_1 - b_1)^2 + (a_2 - b_2)^2}

The above shows that the "absolute value" distance for the real numbers or the complex numbers, agrees with the standard Euclidean distance they inherit as a result of considering them as the one and two-dimensional Euclidean spaces respectively.


The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given in Propositions 2 and 3 above, can be seen to motivate the more general notion of a distance function as follows: For distance between people, see proxemics. ...


A real valued function d on a set X times X is called a distance function (or a metric) for X, if it satisfies the following four axioms:

d(a, b) ge 0 Non-negativity
d(a, b) = 0 iff a = b Identity of indiscernibles
d(a, b) = d(b, a) , Symmetry
d(a, b) le d(a, c) + d(c, b) Triangle inequality

Derivatives

The derivative of the real absolute value function is the signum function, sgn(x), which is defined as For a non-technical overview of the subject, see Calculus. ... Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ...

sgn (x) = frac{x}{|x|}

for x ≠ 0. The absolute value function is not differentiable at x = 0. Where the absolute value function of a real number returns a value without respect to its sign, the signum function returns a number's sign without respect to its value. Therefore x = sgn(x)abs(x). The signum function is a form of the Heaviside step function used in signal processing, defined as: The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of...

u(x) = begin{cases} 0, & x < 0 frac{1}{2}, & x = 0 1, & x > 0 end{cases}

Where the value of the Heaviside function at zero is conventional. So we have at all nonzero points on the real number line, In mathematics, the real line is simply the set of real numbers. ...

u(x) = frac{sgn(x) +1}{2}.,

The absolute value function has no concavity at any point, the sign function is constant at all points. Therefore the second derivative of |x| with respect to x is zero everywhere except zero, where it is undefined.


The absolute value function is also integrable. Its antiderivative is In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ...

int|x|dx=frac{x|x|}{2}+C.

Fields

The fundamental properties of the absolute value for real numbers given in Proposition 2 above, can be used to generalize the notion of absolute value to an arbitrary field, as follows.


A real-valued function v on a field F is called an absolute value (also a modulus, magnitude, value, or valuation) if it satisfies the following four axioms: In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, an absolute value is a function which measures the size of elements in a field or integral domain. ...

v(a) ge 0 Non-negativity
v(a) = 0 iff a = mathbf{0} Positive-definiteness
v(ab) = v(a) v(b) , Multiplicativeness
v(a+b) le v(a) + v(b) Subadditivity or the triangle inequality

Where mathbf{0} denotes the additive identity element of F. It follows from positive-definiteness and multiplicativeness that v(mathbf{1}) = 1, where mathbf{1} denotes the multiplicative identity element of F. The real and complex absolute values defined above are examples of absolute values for an arbitrary field. 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. ...


If v is an absolute value on F, then the function d on F times F, defined by d(a,b) = v(a − b), is a metric and the following are equivalent:

  • vBig(sum_{k=1}^n mathbf{1}Big) le 1 for every n in mathbb{N}.
  • v(a) le 1 Rightarrow v(1+a) le 1 for all a in F.
  • v(a + b) le mathrm{max}{v(a), v(b)} for all a, b in F.

An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwise it is said to be Archimedean.[3] In mathematics, an ultrametric space is a special kind of metric space. ... In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. ... An archimedean field is an ordered field with the archimedean property. ...


Vector spaces

Main article: Norm (mathematics)

Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalize the notion to an arbitrary vector space. In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...


A real valued function ||·|| on a vector space V over a field F, is called an absolute value (or more usually a norm) if it satisfies the following axioms: In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...


For all a in F, and mathbf{v}, mathbf{u} in V,

|mathbf{v}| ge 0 Non-negativity
|mathbf{v}| = 0 iff mathbf{v} = 0 Positive-definiteness
|a mathbf{v}| = |a| |mathbf{v}| Positive homogeneity or positive scalability
|mathbf{v} + mathbf{u}| le |mathbf{v}| + |mathbf{u}| Subadditivity or triangle inequality

The norm of a vector is also called its length or magnitude.


In the case of Euclidean space Rn, the function defined by Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

|(x_1, x_2, cdots , x_n) | = sqrt{sum_{i=1}^{n}(x_i)^2}

is a norm called the Euclidean norm. When the real numbers R are considered as the one-dimensional vector space R1, the absolute value is a norm, and is the p-norm for any p. In fact the absolute value is the "only" norm on R1, in the sense that, for every norm ||·|| on R1, ||x||=||1||·|x|. The complex absolute value is a special case of the norm in an inner product space. It is identical to the Euclidean norm, if the complex plane is identified with the Euclidean plane R2. In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...


Algorithms

In the C programming language, the abs(), labs(), llabs() (in C99), fabs(), fabsf(), and fabsl() functions compute the absolute value of an operand. Coding the integer version of the function is trivial, ignoring the boundary case where the largest negative integer is input: C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ... 

 int abs (int i) { if (i < 0) return -i; else return i; }

The floating-point versions are trickier, as they have to contend with special codes for infinity and not-a-numbers. This article or section is in need of attention from an expert on the subject. ... The infinity symbol ∞ in several typefaces. ... In computing, NaN (Not a Number) is a value or symbol that is usually produced as the result of an operation on invalid input operands, especially in floating-point calculations. ...


The function for absolute value in Fortran, Matlab, and GNU_Octave is abs. It handles integer, real as well as complex numbers. Fortran (previously FORTRAN[1]) is a general-purpose[2], procedural,[3] imperative programming language that is especially suited to numeric computation and scientific computing. ... MATLAB is a numerical computing environment and programming language. ... Octave is a free computer program for performing numerical computations which is mostly compatible with MATLAB. It is part of the GNU project. ...


Using assembly language, it is possible to take the absolute value of a register in just three instructions (example shown for a 32-bit register on an x86 architecture, Intel syntax): See the terminology section, below, regarding inconsistent use of the terms assembly and assembler. ... In computer architecture, a processor register is a small amount of very fast computer memory used to speed the execution of computer programs by providing quick access to frequently used values—typically, these values are involved in multiple expression evaluations occurring within a small region on the program. ... It has been suggested that x86 assembly language be merged into this article or section. ... Intel Corporation (NASDAQ: INTC, SEHK: 4335), founded in 1968 as Integrated Electronics Corporation, is an American multinational corporation that is best known for designing and manufacturing microprocessors and specialized integrated circuits. ... 

 cdq xor eax, edx sub eax, edx

cdq extends the sign bit of eax into edx. If eax is nonnegative, then edx becomes zero, and the latter two instructions have no effect, leaving eax unchanged. If eax is negative, then edx becomes 0xFFFFFFFF, or -1. The next two instructions then become a two's complement inversion, giving the absolute value of the negative value in eax. The twos complement of a binary number is the value obtained by subtracting the number from a large power of two (specifically, from 2N for an N-bit twos complement). ...


Assuming 32-bit register and x86 platform (where sign bit is left-most bit) this can be even shortened to: It has been suggested that x86 assembly language be merged into this article or section. ... 

 and eax, 0x7fffffff

which simply clears the sign bit. Or without using constant value written using two instructions:

 shl eax, 1 shr eax, 1

shl multiplies value of eax by two, and sar divides it by two, but setting sign bit to zero.


This can be also written in C, however it must be remembered that if function takes as an argument signed variable it must be first casted to unsigned:

 int fast_abs(int i) { return ((unsigned)a << 1) >> 1 }

Notes

  1. ^ Jean-Robert Argand, is credited with introducing the term "modulus" in 1806, see: Nahin, O'Connor and Robertson, and functions.Wolfram.com.
  2. ^ functions.Wolfram.com credits Karl Weierstrass with introducing the notation | x | , in 1841.
  3. ^ Schechter, p 260-261.

References

Jean-Robert Argand was an accountant and bookkeeper in Paris who was only an amateur mathematician. ... 1806 was a common year starting on Wednesday (see link for calendar). ... Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... 1841 is a common year starting on Friday (link will take you to calendar). ... Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

See also


  Results from FactBites:
 
Positive and negative numbers (1681 words)
The absolute value of a number is always a positive number (or zero).
The x value or x-coordinate tells how many steps left or right the point is from the point (0,0), just like on the number line (negative is left of the origin, positive is right of the origin).
The y value or y-coordinate tells how many steps up or down the point is from the point (0,0), (negative is down from the origin, positive is up from the origin).
Absolute value Summary (1786 words)
Absolute value is an operation in mathematics, written as bars on either side of the expression.
The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.
From a geometric point of view, the absolute value of a real number is the distance along the real number line of that number from zero, and more generally the absolute value of the difference of two real numbers is the distance between them.
  More results at FactBites »


 

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