How Do You Find The Absolute Value

Distance from zero to a number

The graph of the absolute value part for existent numbers

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

In mathematics, the accented value or modulus of a existent number ${\displaystyle x}$ , denoted ${\displaystyle |10|}$ , is the non-negative value of ${\displaystyle ten}$ without regard to its sign. Namely, ${\displaystyle |x|=x}$ if x is a positive number, and ${\displaystyle |x|=-ten}$ if ${\displaystyle x}$ is negative (in which case negating ${\displaystyle x}$ makes ${\displaystyle -x}$ positive), and ${\displaystyle |0|=0}$ . For instance, the accented value of iii is three, and the accented value of −3 is also 3. The accented value of a number may be idea of as its distance from zero.

Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For instance, an absolute value is also defined for the circuitous numbers, the quaternions, ordered rings, fields and vector spaces. The accented value is closely related to the notions of magnitude, altitude, and norm in diverse mathematical and physical contexts.

Terminology and notation [edit]

In 1806, Jean-Robert Argand introduced the term module, meaning unit of measurement of measure in French, specifically for the circuitous accented value,^{[1] [2]} and it was borrowed into English in 1866 equally the Latin equivalent modulus.^[i] The term absolute value has been used in this sense from at to the lowest degree 1806 in French^[3] and 1857 in English.^[4] The notation | ten |, with a vertical bar on each side, was introduced by Karl Weierstrass in 1841.^[5] Other names for absolute value include numerical value ^[1] and magnitude.^[1] In programming languages and computational software packages, the absolute value of x is by and large represented past abs(10), or a similar expression.

The vertical bar notation also appears in a number of other mathematical contexts: for instance, when practical to a gear up, 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, for example a real number, a complex number, or a quaternion. A closely related but singled-out note is the use of vertical bars for either the Euclidean norm^[6] or sup norm^[7] of a vector in ${\displaystyle \mathbb {R} ^{n}}$ , although double vertical bars with subscripts ( ${\displaystyle \|\cdot \|_{2}}$ and ${\displaystyle \|\cdot \|_{\infty }}$ , respectively) are a more mutual and less ambiguous notation.

Definition and properties [edit]

Existent numbers [edit]

For any real number ${\displaystyle x}$ , the accented value or modulus of ${\displaystyle x}$ is denoted by ${\displaystyle |x|}$ , with a vertical bar on each side of the quantity, and is divers as^[8]

$${\displaystyle |x|={\brainstorm{cases}x,&{\text{if }}x\geq 0\\-x,&{\text{if }}x<0.\terminate{cases}}}$$

The absolute value of ${\displaystyle x}$ is thus ever either a positive number or zero, but never negative. When ${\displaystyle x}$ itself is negative ( ${\displaystyle ten<0}$ ), then its absolute value is necessarily positive ( ${\displaystyle |10|=-ten>0}$ ).

From an analytic geometry point of view, the absolute value of a real number is that number's distance from zippo along the real number line, and more generally the absolute value of the deviation of ii existent numbers is the distance between them. The notion of an abstruse distance function in mathematics tin exist seen to be a generalisation of the absolute value of the difference (come across "Distance" below).

Since the foursquare root symbol represents the unique positive square root, when applied to a positive number, it follows that

$${\displaystyle |x|={\sqrt {x^{2}}}.}$$

This is equivalent to the definition above, and may exist used as an alternative definition of the absolute value of real numbers.^[nine]

The absolute value has the following iv fundamental backdrop (a, b are real numbers), that are used for generalization of this notion to other domains:

Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To encounter that subadditivity holds, first note that ${\displaystyle |a+b|=southward(a+b)}$ where ${\displaystyle southward=\pm 1}$ , with its sign called to brand the consequence positive. At present, since ${\displaystyle -1\cdot x\leq |10|}$ and ${\displaystyle +ane\cdot 10\leq |x|}$ , it follows that, whichever of ${\displaystyle \pm 1}$ is the value of ${\displaystyle due south}$ , one has ${\displaystyle south\cdot x\leq |ten|}$ for all real ${\displaystyle x}$ . Consequently, ${\displaystyle |a+b|=s\cdot (a+b)=south\cdot a+south\cdot b\leq |a|+|b|}$ , every bit desired.

Some boosted useful properties are given below. These are either firsthand consequences of the definition or implied by the four key properties above.

${\displaystyle {\bigl |}\left|a\right|{\bigr |}=|a|}$	Idempotence (the absolute value of the accented value is the absolute value)
${\displaystyle \left|-a\right|=|a|}$	Evenness (reflection symmetry of the graph)
${\displaystyle |a-b|=0\iff a=b}$	Identity of indiscernibles (equivalent to positive-correctness)
${\displaystyle |a-b|\leq |a-c|+|c-b|}$	Triangle inequality (equivalent to subadditivity)
${\displaystyle \left|{\frac {a}{b}}\right|={\frac {|a|}{|b|}}\ }$ (if ${\displaystyle b\neq 0}$ )	Preservation of division (equivalent to multiplicativity)
${\displaystyle |a-b|\geq {\bigl |}\left|a\right|-\left|b\right|{\bigr |}}$	Reverse triangle inequality (equivalent to subadditivity)

Two other useful properties concerning inequalities are:

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

The absolute value, equally "altitude from zero", is used to define the absolute difference betwixt arbitrary real numbers, the standard metric on the real numbers.

Complex numbers [edit]

Since the complex numbers are not ordered, the definition given at the top for the real absolute value cannot be direct practical to circuitous numbers. However, the geometric interpretation of the absolute value of a real number equally its distance from 0 can be generalised. The absolute value of a complex number is divers by the Euclidean distance of its corresponding point in the complex plane from the origin. This tin be computed using the Pythagorean theorem: for any complex number

$${\displaystyle z=x+iy,}$$

where ${\displaystyle x}$ and ${\displaystyle y}$ are real numbers, the accented value or modulus of ${\displaystyle z}$ is denoted ${\displaystyle |z|}$ and is defined past^[10]

$${\displaystyle |z|={\sqrt {\operatorname {Re} (z)^{two}+\operatorname {Im} (z)^{2}}}={\sqrt {x^{two}+y^{2}}},}$$

the Pythagorean improver of ${\displaystyle x}$ and ${\displaystyle y}$ , where ${\displaystyle \operatorname {Re} (z)=x}$ and ${\displaystyle \operatorname {Im} (z)=y}$ denote the real and imaginary parts of ${\displaystyle z}$ , respectively. When the imaginary part ${\displaystyle y}$ is zero, this coincides with the definition of the accented value of the existent number ${\displaystyle 10}$ .

When a complex number ${\displaystyle z}$ is expressed in its polar form every bit ${\displaystyle z=re^{i\theta },}$ its absolute value is ${\displaystyle |z|=r.}$

Since the product of any complex number ${\displaystyle z}$ and its circuitous conjugate ${\displaystyle {\bar {z}}=x-iy}$ , with the same absolute value, is always the non-negative real number ${\displaystyle \left(x^{2}+y^{2}\right)}$ , the absolute value of a complex number ${\displaystyle z}$ is the foursquare root of ${\displaystyle z\cdot {\overline {z}},}$ which is therefore called the accented foursquare or squared modulus of ${\displaystyle z}$ :

$${\displaystyle |z|={\sqrt {z\cdot {\overline {z}}}}.}$$

This generalizes the alternative definition for reals: ${\textstyle |x|={\sqrt {ten\cdot x}}}$ .

The complex absolute value shares the iv fundamental properties given above for the existent accented value. The identity ${\displaystyle |z|^{2}=|z^{2}|}$ is a special instance of multiplicativity that is oft useful by itself.

Accented value function [edit]

The graph of the absolute value part for real numbers

The existent absolute value part is continuous everywhere. Information technology is differentiable everywhere except for 10 = 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 existent absolute value function is a piecewise linear, convex function.

For both real and circuitous numbers the absolute value function is idempotent (meaning that the absolute value of whatever absolute value is itself).

Relationship to the sign function [edit]

The absolute value function of a existent number returns its value irrespective of its sign, whereas the sign (or signum) role returns a number'due south sign irrespective of its value. The following equations evidence the relationship between these two functions:

${\displaystyle |x|=x\operatorname {sgn}(x),}$

or

${\displaystyle |x|\operatorname {sgn}(x)=x,}$

and for x ≠ 0,

${\displaystyle \operatorname {sgn}(x)={\frac {|x|}{x}}={\frac {x}{|x|}}.}$

Derivative [edit]

The real accented value function has a derivative for every x ≠ 0, just is not differentiable at x = 0. Its derivative for x ≠ 0 is given by the footstep function:^{[11] [12]}

${\displaystyle {\frac {d\left|10\right|}{dx}}={\frac {x}{|10|}}={\brainstorm{cases}-i&10<0\\1&10>0.\stop{cases}}}$

The real absolute value function is an example of a continuous part that achieves a global minimum where the derivative does not be.

The subdifferential of| ten | at ten = 0 is the interval[−1, one].^[thirteen]

The complex absolute value role is continuous everywhere but circuitous differentiable nowhere because information technology violates the Cauchy–Riemann equations.^[eleven]

The second derivative of| x | with respect tox is null everywhere except zilch, where it does not exist. Every bit a generalised part, the second derivative may exist taken as 2 times the Dirac delta office.

Antiderivative [edit]

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

${\displaystyle \int \left|ten\right|dx={\frac {x\left|ten\right|}{2}}+C,}$

where C is an capricious abiding of integration. This is not a complex antiderivative considering circuitous antiderivatives can simply be for complex-differentiable (holomorphic) functions, which the complex absolute value part is not.

Distance [edit]

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 airplane, for complex numbers, and more generally, the accented value of the departure of two real or complex numbers is the distance between them.

The standard Euclidean distance between two points

${\displaystyle a=(a_{one},a_{2},\dots ,a_{n})}$

and

${\displaystyle b=(b_{1},b_{2},\dots ,b_{n})}$

in Euclidean n-infinite is defined as:

${\displaystyle {\sqrt {\textstyle \sum _{i=1}^{n}(a_{i}-b_{i})^{ii}}}.}$

This can be seen equally a generalisation, since for ${\displaystyle a_{1}}$ and ${\displaystyle b_{ane}}$ real, i.east. in a one-space, according to the alternative definition of the absolute value,

${\displaystyle |a_{1}-b_{1}|={\sqrt {(a_{1}-b_{one})^{2}}}={\sqrt {\textstyle \sum _{i=one}^{1}(a_{i}-b_{i})^{2}}},}$

and for ${\displaystyle a=a_{1}+ia_{2}}$ and ${\displaystyle b=b_{1}+ib_{2}}$ circuitous numbers, i.e. in a 2-space,

The above shows that the "absolute value"-distance, for real and complex numbers, agrees with the standard Euclidean distance, which they inherit every bit a result of considering them as one and ii-dimensional Euclidean spaces, respectively.

The properties of the absolute value of the difference of two real or circuitous numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, tin can be seen to motivate the more full general notion of a distance function as follows:

A real valued office d on a set 10 ×X is called a metric (or a altitude function) onX, if it satisfies the following iv axioms:^[14]

${\displaystyle d(a,b)\geq 0}$	Non-negativity
${\displaystyle d(a,b)=0\iff a=b}$	Identity of indiscernibles
${\displaystyle d(a,b)=d(b,a)}$	Symmetry
${\displaystyle d(a,b)\leq d(a,c)+d(c,b)}$	Triangle inequality

Generalizations [edit]

Ordered rings [edit]

The definition of absolute value given for existent numbers to a higher place tin can be extended to whatever ordered band. That is, ifa is an element of an ordered ringR, then the absolute value ofa, denoted by | a |, is defined to be:^[15]

${\displaystyle |a|=\left\{{\begin{array}{rl}a,&{\text{if }}a\geq 0\\-a,&{\text{if }}a<0.\end{array}}\correct.}$

where −a is the additive inverse ofa, 0 is the additive identity, and < and ≥ accept the usual meaning with respect to the ordering in the ring.

Fields [edit]

The four fundamental backdrop of the absolute value for real numbers tin exist used to generalise the notion of absolute value to an arbitrary field, as follows.

A real-valued functionv on a fieldF is chosen an accented value (also a modulus, magnitude, value, or valuation)^[16] if it satisfies the following four axioms:

Where 0 denotes the additive identity ofF. Information technology follows from positive-correctness and multiplicativity that five(1) = 1, where ane denotes the multiplicative identity ofF. The existent and complex absolute values defined above are examples of accented values for an arbitrary field.

If v is an absolute value onF, then the functiond on F ×F , defined by d(a,b) = v(a − b), is a metric and the post-obit are equivalent:

• d satisfies the ultrametric inequality ${\displaystyle d(x,y)\leq \max(d(10,z),d(y,z))}$ for all x, y, z inF.
• ${\textstyle \left\{5\left(\sum _{k=one}^{n}\mathbf {1} \right):due north\in \mathbb {N} \right\}}$ is bounded inR.
• ${\displaystyle v\left({\textstyle \sum _{m=1}^{northward}}\mathbf {1} \right)\leq 1\ }$ for every ${\displaystyle n\in \mathbb {N} }$ .
• ${\displaystyle 5(a)\leq i\Rightarrow five(1+a)\leq 1\ }$ for all ${\displaystyle a\in F}$ .
• ${\displaystyle five(a+b)\leq \max\{five(a),v(b)\}\ }$ for all ${\displaystyle a,b\in F}$ .

An absolute value which satisfies any (hence all) of the above conditions is said to be not-Archimedean, otherwise it is said to be Archimedean.^[17]

Vector spaces [edit]

Again the fundamental backdrop of the accented value for existent numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector infinite.

A real-valued function on a vector spaceV over a fieldF, represented every bit || · ||, is called an accented value, just more usually a norm, if it satisfies the following axioms:

For alla inF, and 5 , u inV,

The norm of a vector is likewise called its length or magnitude.

In the case of Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ , the function defined past

${\displaystyle \|(x_{i},x_{two},\dots ,x_{n})\|={\sqrt {\textstyle \sum _{i=i}^{n}x_{i}^{two}}}}$

is a norm called the Euclidean norm. When the existent numbers ${\displaystyle \mathbb {R} }$ are considered every bit the 1-dimensional vector space ${\displaystyle \mathbb {R} ^{1}}$ , the absolute value is a norm, and is the p-norm (see Fifty^p space) for anyp. In fact the accented value is the "just" norm on ${\displaystyle \mathbb {R} ^{ane}}$ , in the sense that, for every norm || · || on ${\displaystyle \mathbb {R} ^{one}}$ , || ten || = ||ane|| ⋅ | ten |.

The complex absolute value is a special case of the norm in an inner product infinite, which is identical to the Euclidean norm when the complex aeroplane is identified equally the Euclidean plane ${\displaystyle \mathbb {R} ^{ii}}$ .

Composition algebras [edit]

Every composition algebra A has an involution ten → x* called its conjugation. The product in A of an chemical element 10 and its cohabit x* is written N(10) = x x* and called the norm of 10.

The real numbers ${\displaystyle \mathbb {R} }$ , circuitous numbers ${\displaystyle \mathbb {C} }$ , and quaternions ${\displaystyle \mathbb {H} }$ are all composition algebras with norms given by definite quadratic forms. The absolute value in these division algebras is given by the square root of the composition algebra norm.

In general the norm of a composition algebra may be a quadratic class that is not definite and has null vectors. However, as in the case of division algebras, when an chemical element x has a non-nothing norm, and so x has a multiplicative inverse given past ten*/Northward(x).

Notes [edit]

1. ^ ^{a b c d} Oxford English Lexicon, Draft Revision, June 2008
2. ^ Nahin, O'Connor and Robertson, and functions.Wolfram.com.; for the French sense, run into Littré, 1877
3. ^ Lazare Nicolas Thou. Carnot, Mémoire sur la relation qui existe entre les distances respectives de cinq signal quelconques pris dans fifty'espace, p. 105 at Google Books
4. ^ James Mill Peirce, A Text-book of Analytic Geometry at Internet Annal. The oldest citation in the 2d edition of the Oxford English Lexicon is from 1907. The term accented 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. ^ Spivak, Michael (1965). Calculus on Manifolds. Boulder, CO: Westview. p. i. ISBN0805390219.
7. ^ Munkres, James (1991). Analysis on Manifolds. Boulder, CO: Westview. p. four. ISBN0201510359.
8. ^ Mendelson, p. 2.
9. ^ Stewart, James B. (2001). Calculus: concepts and contexts. Australia: Brooks/Cole. ISBN0-534-37718-one. , p. A5
10. ^ González, Mario O. (1992). Classical Circuitous Analysis. CRC Press. p. nineteen. ISBN9780824784157.
11. ^ ^{a b} "Weisstein, Eric W. Absolute Value. From MathWorld – A Wolfram Web Resource".
12. ^ Bartle and Sherbert, p. 163
13. ^ Peter Wriggers, Panagiotis Panatiotopoulos, eds., New Developments in Contact Problems, 1999, ISBN 3-211-83154-1, p. 31–32
14. ^ 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).
15. ^ Mac Lane, p. 264.
16. ^ Shechter, p. 260. This meaning of valuation is rare. Ordinarily, a valuation is the logarithm of the changed of an absolute value
17. ^ Shechter, pp. 260–261.

References [edit]

• Bartle; Sherbert; Introduction to existent analysis (4th ed.), John Wiley & Sons, 2011 ISBN 978-0-471-43331-vi.
• Nahin, Paul J.; An Imaginary Tale; Princeton University Press; (hardcover, 1998). ISBN 0-691-02795-i.
• Mac Lane, Saunders, Garrett Birkhoff, Algebra, American Mathematical Soc., 1999. ISBN 978-0-8218-1646-2.
• Mendelson, Elliott, Schaum's Outline of Beginning Calculus, McGraw-Hill Professional, 2008. ISBN 978-0-07-148754-two.
• O'Connor, J.J. and Robertson, Eastward.F.; "Jean Robert Argand".
• Schechter, Eric; Handbook of Analysis and Its Foundations, pp. 259–263, "Absolute Values", Bookish Press (1997) ISBN 0-12-622760-8.

External links [edit]

• "Absolute value". Encyclopedia of Mathematics. EMS Press. 2001 [1994].
• absolute value at PlanetMath.
• Weisstein, Eric W. "Absolute Value". MathWorld.

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

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