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.

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.

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.

In addition, virtually an unlimited number of slots can be provided with 20, including the SBC slot, as a practical though not an absolute limit.

where is the difference in density of particles separated by a height difference of, is Boltzmann's constant ( namely, the ratio of the universal gas constant,, to Avogadro's number, ), and is the absolute temperature.

It is usual for a decimal number whose absolute value is less than one to have a leading zero.

However, in enlightened absolutism ( also known as benevolent despotism ), which came to prominence in 18th century Europe, absolute monarchs used their authority to institute a number of reforms in the political systems and societies of their countries.

An empirical formula makes no reference to isomerism, structure, or absolute number of atoms.

* Logarithmic number systems represent a real number by the logarithm of its absolute value and a sign bit.

Using their data on the time occupied by travel to work, Table 2 below shows the absolute number of people who responded with travel times " at least 15 but less than 20 minutes " is higher than the numbers for the categories above and below it.

That is, ƒ ( z ) is the absolute value ( or modulus ) of the complex number z.

Shannon's entropy represents an absolute limit on the best possible lossless compression of any communication, under certain constraints: treating messages to be encoded as a sequence of independent and identically distributed random variables, Shannon's source coding theorem shows that, in the limit, the average length of the shortest possible representation to encode the messages in a given alphabet is their entropy divided by the logarithm of the number of symbols in the target alphabet.

* The absolute value of a real or complex number ( | a | )

The race Directors retain the absolute discretion to exclude any horse from the race, or exempt any horse from the ballot on the race, but in order to reduce the field to the safety limit of 24, horses are balloted out based on a number of factors which include:

A non-zero element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1 / n, for n a standard natural number.

Road safety in Russia is poor with road accident deaths per million population higher than all countries in the G8 and the other BRIC countries, although the absolute number is actually less than China, India and USA.

However, if no candidate receives the required number of votes ( usually an absolute majority or 40-45 % with a winning margin of 5-15 %), then those candidates having less than a certain proportion of the votes, or all but the two candidates receiving the most votes, are eliminated, and a second round of voting occurs.

If no candidate receives an absolute majority in the first round then the candidate ( s ) with the fewest number of votes is eliminated and excluded from further ballots.

Since the constitution of 1978 came into effect, there have not been any coalition governments, even if the party with the largest number of seats has failed to obtained absolute majority, though in such cases the party in government has had to rely on the support of minority parties to gain confidence and to approve the State's budgets.

The number of sunspots correlates with the intensity of solar radiation over the period since 1979, when satellite measurements of absolute radiative flux became available.

It gives the co-ordinates in 1900 epoch, the secular variation, the proper motion, the weighted average absolute parallax and its standard error, the number of parallax observations, quality of interagreement of the different values, the visual magnitude and various cross-identifications with other catalogues.

Alternate definitions are, " the entropy of all systems and of all states of a system is smallest at absolute zero ," or equivalently " it is impossible to reach the absolute zero of temperature by any finite number of processes ".

The absolute number of tuberculosis cases has been decreasing since 2006, and new cases have decreased since 2002.

Such an interferometer provides a means of absolute temperature measurement, and may be used as an alternative to the gas thermometer.

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.

Some jurisdictions hold this as an absolute right, and in its absence, a sentence may potentially be overturned, with the result that a new sentencing hearing must be held.

Those who ignore the effects of long run trade deficits may be confusing David Ricardo's principle of comparative advantage with Adam Smith's principle of absolute advantage, specifically ignoring the latter.

These sizes are not absolute, as luthiers may choose variations around these nominal scale-lengths ;

It may be rejected out right by an absolute majority of MEPs.

" Freedom of contract is a qualified and not an absolute right ...," Hughes declared in upholding an Iowa law that voided contracts limiting the legal rights of railroad workers: The state may " interfere where the parties do not stand upon an equality ...." Using similar reasoning, the associate justice upheld a California law that mandated a forty-eight-hour work-week for women in various industries and allowed a federal statute to override a contract between an interstate railroad and its employees.

Even if one region has an absolute advantage as to the ratio of its outputs to inputs in every type of output, it may still specialize in the output in which it has a comparative advantage and thereby gain from trading with a region that lacks any absolute advantage but has a comparative advantage in producing something else.

::: The President may in absolute discretion refuse to dissolve Dáil Éireann on the advice of a Taoiseach who has ceased to retain the support of a majority in Dáil Éireann.

I John Audubon, having this day mutual consent with Ferdinand Rozier, dissolved and forever closed the partnership and firm of Audubon and Rozier, and having Received from said Ferdinand Rozier payment and notes to the full amount of my part of the goods and debts of the late firm of Audubon and Rozier, I the said John Audubon one of the firm aforesaid do hereby release and forever quit claim to all and any interest which I have or may have in the stock on hand and debts due to the late firm of Audubon and Rozier assign, transfer and set over to said Ferdinand Rozier, all my rights, titles, claims and interest in the goods, merchandise and debts due to the late firm of Audubon and Rozier, and do hereby authorize and empower him for my part, to collect the same in any manner what ever either privately or by suit or suits in law or equity hereby declaring him sole and absolute proprietor and rightful owner of all goods, merchandise and debts of this firm aforesaid, as completely as they were the goods and property of the late firm Audubon and Rozier.

His aim, of course, is to emphasize the absolute significance of Christ, so that all that ever existed of virtue and truth may be referred to him.

Relocating machine code may involve re-targeting of absolute jumps, loads and stores.

Monarchs may be autocrats ( absolute monarchy ) or ceremonial heads of state who exercise little or no power or only reserve power, with actual authority vested in a parliament or other body ( constitutional monarchy ).

While having relinquished some of his formerly absolute power, the Prince of Monaco remains head of state and retains most of the country's governing power ; however, the principality's judicial and legislative bodies may operate independent of his control.

Moderate vacuum pressures are often ambiguous, as they may represent absolute pressure or gauge pressure without a negative sign.

In gauges intended to sense small pressures or pressure differences, or require that an absolute pressure be measured, the gear train and needle may be driven by an enclosed and sealed bellows chamber, called an aneroid, which means " without liquid ".

Byzantine music uses the eight natural, non-tempered scales called Ēkhoi, " sounds ", exclusively, and therefore the absolute pitch of each note may slightly vary each time, depending on the particular Ēkhos used.

It is important to note that the actual nut hand may not be the same as the absolute nut hand ; for example, if the board is < font color = red > 7 ♥</ font > 2 ♣ K ♠ < font color = red > K ♥ 3 ♦</ font > a player with K ♣ < font color = red > K ♦</ font > has the absolute nut hand.

Such a conception of reality may be thoroughly predestinarian, if God is the personal cause of continued existence and the orchestrator or determiner of the relationship between each present event and each subsequent present event ; but, it is only predestination if in this conception God acts with absolute freedom and entire knowledge of Himself.

Man's will may be called free and responsible, but not in an absolute sense ; the choice of good or of evil must be uncoerced to be free, but it is never uncreated or uncaused.

That theory was brought to a focus on the question of whether man exists for the sake of government, or government for the sake of man – a distinction which may derive from, or lead to, the question of natural and absolute rights, and whether property is one of them.

It generalizes Galileo's principle of relativity — that all uniform motion is relative, and that there is no absolute and well-defined state of rest ( no privileged reference frames )— from mechanics to all the laws of physics, including both the laws of mechanics and of electrodynamics, whatever they may be.