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.

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.

However, his real position was that motion is absolute.

In Science and Health with Key to the Scriptures, Eddy argues that given the absolute goodness and perfection of God, sin, disease, and death were not created by Him, and therefore cannot be truly real.

The space R of real numbers and the space C of complex numbers ( with the metric given by the absolute value ) are complete, and so is Euclidean space R < sup > n </ sup >, with the usual distance metric.

Cantor's construction of the real numbers is similar to the above construction ; the real numbers are the completion of the rational numbers using the ordinary absolute value to measure distances.

A geometric interpretation can be given to the value of the determinant of a square matrix with real entries: the absolute value of the determinant gives the scale factor by which area or volume is multiplied under the associated linear transformation, while its sign indicates whether the transformation preserves orientation.

** " Has the same absolute value " on the set of real numbers

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

The idea that nothing is real except in the mind of the individual finds its roots in the Greek Sophists, who argued that since nothing can be perceived except through the senses — and since all individuals sense, and therefore perceive, things differently — there is no absolute truth, only relative truth.

Another version of Hahn – Banach theorem states that if V is a vector space over the scalar field K ( either the real numbers R or the complex numbers C ), if is a seminorm, and is a K-linear functional on a K-linear subspace U of V which is dominated by on U in absolute value,

" This absolute protection or full value view treats intellectual property as another type of ' real ' property, typically adopting its law and rhetoric.

The absolute value function of a real or complex argument, and the floor function of a real argument are idempotent.

The absolute definition of morality in wuxia is understandably a reaction to the real world where it is not quite so clear what or who is purely good or otherwise, consider the context and the historicity of Hong Kong at the time of Louis Cha's works.

In Ricardo and other classical economists, this definition serves as a measure of " real cost ", " absolute value ", or a " measure of value " invariable under changes in distribution and technology.

The only other spheres that admit the structure of a Lie group are the 0-sphere S < sup > 0 </ sup > ( real numbers with absolute value 1 ) and the circle S < sup > 1 </ sup > ( complex numbers with absolute value 1 ).

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

* The circle group S < sup > 1 </ sup > consisting of angles mod 2π under addition or, alternatively, the complex numbers with absolute value 1 under multiplication.

This theory is far more complex than it seems, in that the truth of a claim depends on the context in which the claim is made, unlike basic ideas in daily life where truth is absolute.

* the special orthogonal group SO ( 2 ) consisting of all rotations about a fixed point ; it is also called the circle group S < sup > 1 </ sup >, the multiplicative group of complex numbers of absolute value 1.

It can also be used to define an angle in complex inner product spaces, by taking the absolute value of the right hand side, as is done when extracting a metric from quantum fidelity.

* The absolute value function when defined on the set of real numbers or complex numbers is not everywhere analytic because it is not differentiable at 0.

In the general context of a G-valued series, a distinction is made between absolute and unconditional convergence, and the assertion that a real or complex series which is not absolutely convergent is necessarily conditionally convergent ( meaning not unconditionally convergent ) is then a theorem, not a definition.

The hue represents the Arg ( mathematics ) | complex argument, the luminance represents the absolute value.

As the coefficients in the linear combination are now complex, the probability must now come from the absolute value of the multiplication of the wave function by its complex conjugate.

Where the wave equation is represented as a complex vector, the probability will be extracted from the absolute value of an inner-product of the coefficient matrix and its complex conjugate.

In this picture, the hue of a point z is determined by the argument ( complex analysis ) | argument of W ( z ) and the brightness by the absolute value of W ( z ).

Non-harming manifests perspectives both absolute and relative, particularly in the ever-increasingly complex ethics of global culture.

The matrices R < sub > 1 </ sub >,..., R < sub > k </ sub > give conjugate pairs of eigenvalues lying on the unit circle in the complex plane ; so this decomposition confirms that all eigenvalues have absolute value 1.