» 
Arabic Bulgarian Chinese Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hindi Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Malagasy Norwegian Persian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Thai Turkish Vietnamese
Arabic Bulgarian Chinese Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hindi Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Malagasy Norwegian Persian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Thai Turkish Vietnamese

definitions - absolute value

absolute value (n.)

1.a real number regardless of its sign

   Advertizing ▼

definition (more)

definition of Wikipedia

synonyms - absolute value

absolute value (n.)

intrinsic value, numerical value

   Advertizing ▼

analogical dictionary

Wikipedia

Absolute value

                   
  The graph of the absolute value function for real numbers

In mathematics, the absolute value (or modulus) | a | of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of –3 is also 3. The absolute value of a number may be thought of as its distance from zero.

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. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

Contents

  Terminology and notation

Jean-Robert Argand introduced the term "module" 'unit of measure' in French in 1806 specifically for the complex absolute value[1][2] and it was borrowed into English in 1866 as the Latin equivalent "modulus".[1] The term "absolute value" has been used in this sense since at least 1806 in French[3] and 1857 in English.[4] The notation | a | was introduced by Karl Weierstrass in 1841.[5] Other names for absolute value include "the numerical value"[1] and "the magnitude".[1]

The same notation is used with sets to denote cardinality; the meaning depends on context.

  Definition and properties

  Real numbers

For any real number a the absolute value or modulus of a is denoted by | a | (a vertical bar on each side of the quantity) and is defined as[6]

|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.

From an analytic geometry point of view, the absolute value of a real number is that number's distance from zero along the real number line, 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).

Since the square-root notation without sign represents the positive square root, it follows that

|a| = \sqrt{a^2} (1)

which is sometimes used as a definition of absolute value.[7]

The absolute value has the following four fundamental properties:

|a| \ge 0 (2) Non-negativity
|a| = 0 \iff a = 0 (3) Positive-definiteness
|ab| = |a||b|\, (4) Multiplicativeness
|a+b|  \le |a| + |b|  (5) Subadditivity

Other important properties of the absolute value include:

||a|| = |a|\, (6) Self-composition (the absolute value of the absolute value is the absolute value)
|-a| = |a|\, (7) Symmetry
|a - b| = 0 \iff a = b (8) Identity of indiscernibles (equivalent to positive-definiteness)
|a - b|  \le |a - c| +|c - b|  (9) Triangle inequality (equivalent to subadditivity)
|a/b| = |a| / |b| \mbox{ (if } b \ne 0) \, (10) Preservation of division (equivalent to multiplicativeness)
|a-b| \ge ||a| - |b|| (11) (equivalent to subadditivity)

If b > 0, two other useful properties concerning inequalities are:

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

These relations may be used to solve inequalities involving absolute values. For example:

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

Absolute value is used to define the absolute difference, the standard metric on the real numbers.

  Complex numbers

  The absolute value of a complex number z is the distance r from z to the origin. It is also seen in the picture that z and its complex conjugate z have the same absolute value.

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 geometric interpretation of the absolute value of a real number as its distance from 0 can be generalized. The absolute value of a complex number is defined as its distance in the complex plane from the origin using the Pythagorean theorem. More generally the absolute value of the difference of two complex numbers is equal to the distance between those two complex numbers.

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 given by

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

When the complex part y is zero this is the same as the absolute value of the real number x.

When a complex number z is expressed in polar form as

z = r e^{i \theta}

with r ≥ 0 and θ real, its absolute value is

|z| = r.

The absolute value of a complex number can be written in the complex analogue of equation (1) above as:

|z| = \sqrt{z \cdot \overline{z}}

where \overline z is the complex conjugate of z.

The complex absolute value shares all the properties of the real absolute value given in equations (2)–(11) above.

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.[citation needed]

  Absolute value function

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.

Both the real and complex functions are idempotent.

It is a nonlinear convex function.

  Derivative

The derivative of the real absolute value function is given by the step function[8][9]

\frac{d|x|}{dx} = \frac{x}{|x|} = \begin{cases} -1 & x<0 \\ \text{undefined} & x = 0 \\ 1 & x>0 \end{cases}

The subdifferential of |x| at x = 0 is the interval [−1,1].[10]

The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann equations.[8]

The second derivative of | x | with respect to x is zero everywhere except zero, where it is undefined. As a generalized function, the second derivative may be taken to equal two times the Dirac delta function.

  Antiderivative

The antiderivative (indefinite integral) of the absolute value function is

\int|x|dx=\frac{x|x|}{2}+C,

where C is an arbitrary constant of integration.

  Relationship to other functions

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. The following equations show the relationship between these two functions:

|a| = a \sgn(a),
\sgn(a) = \frac{|a|}{a}.

The real absolute value function is also related to a form of the Heaviside step function used in signal processing, defined as:

 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 for all nonzero points on the real number line,

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

  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.

The standard Euclidean distance between two points

a = (a_1, a_2, \dots , a_n)

and

b = (b_1, b_2, \dots , b_n)

in Euclidean n-space is defined as:

\sqrt{\sum_{i=1}^n(a_i-b_i)^2}.

This can be seen to be a generalization of | ab |, since if a and b are real, then by equation (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 above, can be seen to motivate the more general notion of a distance function as follows:

A real valued function d on a set X × X is called a distance function (or a metric) on X, if it satisfies the following four axioms:[11]

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

  Generalizations

  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:[12]

|a| = \begin{cases} a, & \mbox{if }  a \ge 0  \\ -a,  & \mbox{if } a \le 0 \end{cases} \;

where −a is the additive inverse of a, and 0 is the additive identity element.

  Fields

The fundamental properties of the absolute value for real numbers given in (2)–(5) 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)[13] if it satisfies the following four axioms:

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 0 denotes the additive identity element of F. It follows from positive-definiteness and multiplicativeness that v(1) = 1, where 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.

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

  • d satisfies the ultrametric inequality d(x, y) \leq \max(d(x,z),d(y,z)) for all x, y, z in F.
  •  v\Big({\textstyle \sum_{k=1}^n } \mathbf{1}\Big) \le 1 \text{ for every } n \in \mathbb{N}.
  •  v(a) \le 1 \Rightarrow v(1+a) \le 1 \text{ for all } a \in F.
  •  v(a + b) \le \mathrm{max}\{v(a), v(b)\} \text{ 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.[14]

  Vector spaces

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.

A real-valued function on a vector space V over a field F, represented as ||V||, is called an absolute value (or more usually a norm) if it satisfies the following axioms:

For all a in F, and v, 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 the 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

\|(x_1, x_2, \dots , 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.

  See also

  Notes

  1. ^ a b c d Oxford English Dictionary, Draft Revision, June 2008
  2. ^ Nahin, O'Connor and Robertson, and functions.Wolfram.com.; for the French sense, see Littré, 1877
  3. ^ Lazare Nicolas M. Carnot, Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques pris dans l'espace, p. 105 at Google Books
  4. ^ James Mill Peirce, A Text-book of Analytic Geometry at Google Books. The oldest citation in the 2nd edition of the Oxford English Dictionary is from 1907. The term "absolute value" is also used in contrast to "relative value".
  5. ^ Nicholas J. Higham, Handbook of writing for the mathematical sciences, SIAM. ISBN 0-89871-420-6, p. 25
  6. ^ Mendelson, p. 2.
  7. ^ Stewart, James B. (2001). Calculus: concepts and contexts. Australia: Brooks/Cole. ISBN 0-534-37718-1. , p. A5
  8. ^ a b Weisstein, Eric W. Absolute Value. From MathWorld--A Wolfram Web Resource.
  9. ^ Bartel and Sherbert, p. 163
  10. ^ Peter Wriggers, Panagiotis Panatiotopoulos, eds., New Developments in Contact Problems, 1999, ISBN 3-211-83154-1, p. 31–32
  11. ^ These axioms are not minimal; for instance, non-negativity can be derived from the other three: 0 = d(a, a) ≤ d(a, b) + d(b, a) = 2d(a, b).
  12. ^ Mac Lane, p.264.
  13. ^ Shechter, p. 260. This meaning of valuation is rare. Usually, a valuation is the logarithm of the inverse of an absolute value
  14. ^ Shechter, pp. 260–261.

  References

  External links

   
               

 

All translations of absolute value


sensagent's content

  • definitions
  • synonyms
  • antonyms
  • encyclopedia

Dictionary and translator for handheld

⇨ New : sensagent is now available on your handheld

   Advertising ▼

sensagent's office

Shortkey or widget. Free.

Windows Shortkey: sensagent. Free.

Vista Widget : sensagent. Free.

Webmaster Solution

Alexandria

A windows (pop-into) of information (full-content of Sensagent) triggered by double-clicking any word on your webpage. Give contextual explanation and translation from your sites !

Try here  or   get the code

SensagentBox

With a SensagentBox, visitors to your site can access reliable information on over 5 million pages provided by Sensagent.com. Choose the design that fits your site.

Business solution

Improve your site content

Add new content to your site from Sensagent by XML.

Crawl products or adds

Get XML access to reach the best products.

Index images and define metadata

Get XML access to fix the meaning of your metadata.


Please, email us to describe your idea.

WordGame

The English word games are:
○   Anagrams
○   Wildcard, crossword
○   Lettris
○   Boggle.

Lettris

Lettris is a curious tetris-clone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.

boggle

Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !

English dictionary
Main references

Most English definitions are provided by WordNet .
English thesaurus is mainly derived from The Integral Dictionary (TID).
English Encyclopedia is licensed by Wikipedia (GNU).

Copyrights

The wordgames anagrams, crossword, Lettris and Boggle are provided by Memodata.
The web service Alexandria is granted from Memodata for the Ebay search.
The SensagentBox are offered by sensAgent.

Translation

Change the target language to find translations.
Tips: browse the semantic fields (see From ideas to words) in two languages to learn more.

last searches on the dictionary :

3844 online visitors

computed in 0.094s

   Advertising ▼

I would like to report:
section :
a spelling or a grammatical mistake
an offensive content(racist, pornographic, injurious, etc.)
a copyright violation
an error
a missing statement
other
please precise:

Advertize

Partnership

Company informations

My account

login

registration

   Advertising ▼

2011 Panini Absolute Memorabilia Football Value Pack Lot (24 Packs!) (37.95 USD)

Commercial use of this term

2011 Panini Absolute Memorabilia Football Value Pack (5.99 USD)

Commercial use of this term

AKROBATIK "ABSOLUTE VALUE" CD NM (16.0 AUD)

Commercial use of this term

The Absolute Value of -1 2010 by Steve Brezenoff 0761354174 (4.48 USD)

Commercial use of this term

Justin Verlander – 2005 Absolute Memorabilia – Rookie Card – Book Value = $10.00 (2.5 USD)

Commercial use of this term

1910 OLD MAGAZINE PRINT AD, PETREL TOURING CAR, THE MOTOR CAR OF ABSOLUTE VALUE! (12.99 USD)

Commercial use of this term

2013 Panini Absolute Football NFL 2 PCS Jumbo Value/Rack Pack! (29.95 USD)

Commercial use of this term

2012/13 Panini Absolute Basketball Value Rack Pack (Lot of 12) (30.95 USD)

Commercial use of this term

The Absolute Value of Mike, Erskine, Kathryn, Good Book (1.0 USD)

Commercial use of this term

MATH TEACHER WITH ABSOLUTE VALUES Warning Sign school gift funny mathematician (7.15 USD)

Commercial use of this term

2012 Panini Absolute Football Value Rack Pack (Lot of 12) (30.95 USD)

Commercial use of this term

Absolute Honesty : Building a Corporate Culture That Values Straight Talk and... (2.99 USD)

Commercial use of this term

The Absolute Value of -1 (Carolrhoda Ya), Steve Brezenoff, Good Condition, Book (5.74 USD)

Commercial use of this term

11-221: Absolute Activist Value Master V. Ficeto (9.94 USD)

Commercial use of this term

The Absolute Value of -1 (5.51 USD)

Commercial use of this term

Absolute Value : What Makes Customers Buy in the Age of Complete Access and... (16.99 USD)

Commercial use of this term

Absolute Beginners : Guitar Value Pack (2004, Book / CD / DVD)NEW FACTORY SEALED (9.99 USD)

Commercial use of this term

Absolute Value: What Really Influences Customers in the Age of (Nearly) Perfect (27.99 USD)

Commercial use of this term

Absolute Honesty: Building a Corporate Culture That Values Straight T 0814407811 (4.48 USD)

Commercial use of this term