The graph of a function | graph of the absolute value function for real numbers

In mathematics, the absolute value ( or modulus ) of a real number is the numerical value of 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.

Jean-Robert Argand introduced the term " module " ' unit of measure ' in French in 1806 specifically for the complex absolute value and it was borrowed into English in 1866 as the Latin equivalent " modulus ".

The term " absolute value " has been used in this sense since at least 1806 in French and 1857 in English.

Other names for absolute value include " the numerical value " and " the magnitude ".

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

or, more commonly, using the absolute value, with

This close approximation to the modern value of − 273. 15 ° C for the zero of the air-thermometer was further improved upon in 1779 by Johann Heinrich Lambert, who observed that − 270 ° C might be regarded as absolute cold.

In astronomy, to derive absolute magnitude from the observed apparent magnitude of a celestial object its value is corrected from distance to its observer.

The most basic overdrive effect involves clipping the signal when its absolute value exceeds a certain threshold.

Small could mean less than 3 dB below ( i. e., power output < 1 / 2 or voltage output < 0. 707 of ) the maximum value, or more rarely 10 dB below, or it could mean below a certain absolute value.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

* The set of real ( or complex ) numbers is a Banach algebra with norm given by the absolute value.

* The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.

This, however, cannot be done by a community whose very experience of truth is confused and incoherent: it has no absolute standard, and consequently cannot distinguish the absolute from the contingent.

Truly, that Liberals should choose Louis 14, as a bogey-symbol of conservatism is grotesquely ironic, considering the Louis 14, character of their Grand Monarque, FDR: not only in his accretion of absolute power and personal deification, ( le roi gouverne par lui meme ), but in the disastrous effects of his spending and war policies.

What Sam Rayburn's life proves to us all is the magnificent lesson in political science that one can devotedly and with absolute dedication represent the seemingly provincial interests of one's own community, one's own district, one's own State, and by that help himself represent even better the sweep and scope of the problems of this the greatest nation of all time.

Whenever the Secretary of the Treasury, or the Comptroller General of the United States, as the case may be, shall find that any person is entitled to any such payment, after such payment shall have been received by such person, it shall be an absolute bar to recovery by any other person against the United States, its officers, agents, or employees with respect to such payment.

It is relatively easy to go a step further and reason that an attacker, in possession of such absolute power, would simultaneously destroy his opponent's cities and people.

The result of the observations is Af where the phase angle, Q, is measured in degrees from new moon and the probable errors include absolute as well as relative errors.

Often work of this sort is presented as calligraphy -- the pure utterance of the brush stroke seeking only absolute painteresque values.

It is a program of prepayment of health costs with absolute freedom of choice guaranteed.

It is worth mentioning that the Nepōhualtzintzin amounted to the rank from 10 to the 18 in floating point, which calculated stellar as well as infinitesimal amounts with absolute precision, meant that no round off was allowed, when translated into modern computer arithmetic.

In Australia, the right to make a plea in mitigation is absolute.

This does not mean that it is claimed that they are true in some absolute sense.

The president is elected for a five year term by the people ( absolute majority with 2nd round if necessary ).

It is not an absolute perfection but a perfection in love.

Its chemical and physical properties are so similar to that of natural amethyst that it can not be differentiated with absolute certainty without advanced gemnological testing ( which is often cost-prohibitive ).

Thucydides and perhaps Machiavelli's principe are most closely related to me owing to the absolute determination which they show of refusing to deceive themselves and of seeing reason in reality -- not in " rationality ," and still less in " morality.

These measures are closely related to the concepts of absolute and relative risk aversion developed by Kenneth Arrow and John W. Pratt.

A closely related but different concept is that of an absolute presentation of a group.

It is used to describe the concept of absolute freedom to live in a community or be part of an organization whose values or culture are closely related to one's preferences ; or, on a more basic level, to associate with any individual one chooses.

As a large-scale landowner tied closely to the doomed ancien regime, led by people from the aristocracy, and philosophically opposed to many of the fundamental principles of the revolution, the Church, like the absolute monarchy and the feudal nobility, was a target of the revolution even in the early phases, when leading revolutionaries such as Lafayette were still well-disposed toward King Louis XVI as an individual.

" This freedom from absolute, arbitrary power, is so necessary to, and closely joined with a man's preservation, that he cannot part with it, but by what forfeits his preservation and life together: for a man, not having the power of his own life, cannot, by compact, or his own consent, enslave himself to any one, nor put himself under the absolute, arbitrary power of another, to take away his life, when he pleases.

The concept of relative rest is closely linked to that of inertial observers and the statement that nothing is at absolute rest is loosely equivalent to stating that there are no frames of reference which are truly inertial.

In Brazil and Turkey, Unilever's Omo is the absolute leading brand of laundry detergent, with " Omo " closely becoming a genericized trademark in these countries.

This addressing mode is closely related to the indexed absolute addressing mode.

Aside from the lack of absolute certainty as to identification, Air Vice Marshal Bill Bostock, Air Officer Commanding RAAF Command, contended that naming him would change the impact of the news upon Newton's fellow No. 22 Squadron members " from the impersonal to the closely personal " and hence " seriously affect morale ".

Rather than designing Domo's arms for absolute precision, Edsinger and Weber designed the arms to work more closely to that of a human.

ITS-90 is designed to represent the thermodynamic ( absolute ) temperature scale ( referencing absolute zero ) as closely as possible throughout its range.

If the distances are assumed to be known very well, then the absolute magnitude and the line width are closely related.