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Absolute value - Wikipedia

https://en.wikipedia.org/wiki/Absolute_value

In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a positive x, |x| = −x for a negative x (in which case −x is positive), and |0| = 0. 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. Generalisations 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

The graph of the absolute value function for real numbers

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

Terminology and notation Definition and properties Real numbers

The absolute value of a number may be thought of as its distance from zero.

Complex numbers Proof of the complex triangle inequality Absolute value function Relationship to the sign function Derivative Antiderivative Distance Generalizations Ordered rings Fields Vector spaces Composition algebras Notes References External links

In 1806, Jean-Robert Argand introduced the term module, meaning unit of measure in French, 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 from at least 1806 in French[3] and 1857 in English.[4] The notation |x|, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841.[5] Other names for absolute value

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Absolute value - Wikipedia

https://en.wikipedia.org/wiki/Absolute_value

include numerical value[1] and magnitude.[1] In programming languages and computational software packages, the absolute value of x is generally represented by abs(x), or a similar expression. The vertical bar notation also appears in a number of other mathematical contexts: for example, when applied to a set, it denotes its cardinality; when applied to a matrix, it denotes its determinant. Vertical bars denote the absolute value only for algebraic objects for which the notion of an absolute value is defined, notably an element of a normed division algebra like a real number, complex number, quaternion. A closely related but distinct notation is the use of vertical bars for either the euclidean norm[6] or sup norm[7] of a vector in subscripts (

and

, although double vertical bars with

, respectively) are a more common and less ambiguous notation.

Real numbers For any real number x, the absolute value or modulus of x is denoted by |x| (a vertical bar on each side of the quantity) and is defined as[8]

The absolute value of x is thus always either positive or zero, but never negative: when x itself is negative (x < 0), then its absolute value is necessarily positive (|x| = −x > 0). 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 generalisation of the absolute value of the difference (see "Distance" below). Since the square root symbol represents the unique positive square root (when applied to a positive number), it follows that

is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.[9] The absolute value has the following four fundamental properties (a, b are real numbers), that are used for generalization of this notion to other domains:

Non-negativity Positive-definiteness Multiplicativity Subadditivity, specifically the triangle inequality Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that one of the two alternatives of taking s as either –1 or +1 guarantees that Now, since has

for all real

. Consequently,

and

, it follows that, whichever is the value of s, one , as desired. (For a

generalization of this argument to complex numbers, see "Proof of the triangle inequality for complex numbers"

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below.) Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above.

Idempotence (the absolute value of the absolute value is the absolute value) Evenness (reflection symmetry of the graph) Identity of indiscernibles (equivalent to positivedefiniteness) Triangle inequality (equivalent to subadditivity) (if

)

Preservation of division (equivalent to multiplicativity) Reverse triangle inequality (equivalent to subadditivity)

Two other useful properties concerning inequalities are:

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

The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard metric on the real numbers.

Since the complex numbers are not

Absolute value - Wikipedia

https://en.wikipedia.org/wiki/Absolute_value

(and θ ∈ arg(z) is the argument

with (or phase) of z), its absolute value is

. Since the product of any complex number z and its complex conjugate with the same absolute value, is always the non-negative real number

, the absolute value of a complex number can be

conveniently expressed as

resembling the alternative definition for reals: The complex absolute value shares the four fundamental properties given above for the real absolute value. In the language of group theory, the multiplicative property may be rephrased as follows: the absolute value is a group homomorphism from the multiplicative group of the complex numbers onto the group under multiplication of positive real numbers.[11]

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

Importantly, the property of subadditivity ("triangle inequality") extends to any finite collection of n complex numbers

as

This inequality also applies to infinite families, provided that the infinite series

is absolutely convergent. If

Lebesgue integration is viewed as the continuous analog of summation, then this inequality is analogously obeyed by complex-valued, measurable functions

when integrated over a measurable subset

(This includes Riemann-integrable functions over a bounded interval

:

as a special case.)

Proof of the complex triangle inequality The triangle inequality, as given by

, can be demonstrated by applying three easily verified properties of the

complex numbers: Namely, for every complex number

(i): there exists (ii): .

such that

Also, for a family of complex numbers

(iii): if Proof of

,

and

,

. In particular,

, then : Choose

such that

;

. and

(summed over

). The

following computation then affords the desired inequality:

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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 opposite have the same absolute value, it is an even function, and is hence not invertible. The real absolute value function is a piecewise linear, convex function. Both the real and complex functions are idempotent. The graph of the absolute value function for real numbers

The absolute value function of a real number returns its value irrespective of its sign, whereas the sign (or signum) function returns a number's sign irrespective of its value. The following equations show the relationship between these two functions:

or

and for x ≠ 0,

The real absolute value function has a derivative for every x ≠ 0, but is not differentiable at x = 0. Its derivative for x ≠ 0 is given by the step function:[13][14]

Absolute value - Wikipedia

https://en.wikipedia.org/wiki/Absolute_value

The subdifferential of |x| at x = 0 is the interval [−1,1].[15] The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann equations.[13] The second derivative of |x| with respect to x is zero everywhere except zero, where it does not exist. As a generalised function, the second derivative may be taken as two times the Dirac delta function.

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

where C is an arbitrary constant of integration. This is not a complex antiderivative because complex antiderivatives can only exist for complex-differentiable (holomorphic) functions, which the complex absolute value function is not.

f n

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