# Absolute Value

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• Chapter 1

Absolute value

For other uses, see Absolute value (disambiguation).In mathematics, the absolute value ormodulus |x| of a real number x is the non-negative value of x without regard

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

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. Forexample, the absolute value of 3 is 3, and the absolute value of 3 is also 3. The absolute value of a number may bethought 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 dened for the complex numbers, the quaternions, ordered rings, elds and vector spaces.The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical andphysical contexts.

1.1 Terminology and notationIn 1806, Jean-Robert Argand introduced the term module, meaning unit of measure in French, specically for thecomplex absolute value,[1][2] and it was borrowed into English in 1866 as the Latin equivalent modulus.[1] The termabsolute value has been used in this sense from at least 1806 in French[3] and 1857 in English.[4] The notation |x|, witha vertical bar on each side, was introduced by Karl Weierstrass in 1841.[5] Other names for absolute value includenumerical value[1] and magnitude.[1]

The same notation is used with sets to denote cardinality; the meaning depends on context.

1.2 Denition and properties

2

• 1.2. DEFINITION AND PROPERTIES 3

1.2.1 Real numbersFor any real number x the absolute value or modulus of x is denoted by |x| (a vertical bar on each side of thequantity) and is dened as[6]

jxj =(x; if x 0x; if x < 0

As can be seen from the above denition, the absolute value of x is always either positive or zero, but never negative.From an analytic geometry point of view, the absolute value of a real number is that numbers distance from zeroalong the real number line, and more generally the absolute value of the dierence of two real numbers is the distancebetween them. Indeed, the notion of an abstract distance function in mathematics can be seen to be a generalisationof the absolute value of the dierence (see Distance below).Since the square root notation without sign represents the positive square root, it follows that

which is sometimes used as a denition of absolute value of real numbers.[7]

The absolute value has the following four fundamental properties:

Other important properties of the absolute value include:

Two other useful properties concerning inequalities are:

jaj b () b a bjaj b () a b or b a

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

Absolute value is used to dene the absolute dierence, the standard metric on the real numbers.

1.2.2 Complex numbersSince the complex numbers are not ordered, the denition given above for the real absolute value cannot be directlygeneralised for a complex number. However the geometric interpretation of the absolute value of a real number as itsdistance from 0 can be generalised. The absolute value of a complex number is dened as its distance in the complexplane from the origin using the Pythagorean theorem. More generally the absolute value of the dierence of twocomplex numbers is equal to the distance between those two complex numbers.For any complex number

z = x+ iy;

where x and y are real numbers, the absolute value ormodulus of z is denoted |z| and is given by[8]

jzj =px2 + y2:

• 4 CHAPTER 1. ABSOLUTE VALUE

Im

Re

y

y

0 x

r

r

z=x+iy

z=xiyThe absolute value of a complex number z is the distance r from z to the origin. It is also seen in the picture that z and its complexconjugate z have the same absolute value.

When the imaginary part y is zero this is the same as the absolute value of the real number x.When a complex number z is expressed in polar form as

• 1.3. ABSOLUTE VALUE FUNCTION 5

z = rei

with r 0 and real, its absolute value is

jzj = rThe absolute value of a complex number can be written in the complex analogue of equation (1) above as:

jzj =pz z

where z is the complex conjugate of z. Notice that, contrary to equation (1):

jzj 6=pz2

The complex absolute value shares all the properties of the real absolute value given in equations (2)(11) above.Since the positive reals form a subgroup of the complex numbers under multiplication, we may think of absolutevalue as an endomorphism of the multiplicative group of the complex numbers.[9]

1.3 Absolute value function

1

2

3

4

3 2 1 1 2 30

y = |x|

The graph of the absolute value function for real numbers

The real absolute value function is continuous everywhere. It is dierentiable everywhere except for x = 0. It ismonotonically decreasing on the interval (,0] and monotonically increasing on the interval [0,+). Since a realnumber and its opposite have the same absolute value, it is an even function, and is hence not invertible.Both the real and complex functions are idempotent.It is a piecewise linear, convex function.

• 6 CHAPTER 1. ABSOLUTE VALUE

x

y

f (| x |)

| f (x )|

f (x )

Composition of absolute value with a cubic function in dierent orders

1.3.1 Relationship to the sign function

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

jxj = x sgn(x);

• 1.4. DISTANCE 7

or

jxj sgn(x) = x;

and for x 0,

sgn(x) = jxjx:

1.3.2 DerivativeThe real absolute value function has a derivative for every x 0, but is not dierentiable at x = 0. Its derivative for x 0 is given by the step function[10][11]

djxjdx

=x

jxj =(1 x < 01 x > 0:

The subdierential of |x| at x = 0 is the interval [1,1].[12]

The complex absolute value function is continuous everywhere but complex dierentiable nowhere because it violatesthe CauchyRiemann equations.[10]

The second derivative of |x| with respect to x is zero everywhere except zero, where it does not exist. As a generalisedfunction, the second derivative may be taken as two times the Dirac delta function.

1.3.3 AntiderivativeThe antiderivative (indenite integral) of the absolute value function is

Zjxjdx = xjxj

2+ C;

where C is an arbitrary constant of integration.

The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complexnumber is the distance from that number to the origin, along the real number line, for real numbers, or in the complexplane, for complex numbers, and more generally, the absolute value of the dierence of two real or complex numbersis the distance between them.The standard Euclidean distance between two points

a = (a1; a2; : : : ; an)

and

b = (b1; b2; : : : ; bn)

• 8 CHAPTER 1. ABSOLUTE VALUE

in Euclidean n-space is dened as:

vuut nXi=1

(ai bi)2:

This can be seen to be a generalisation of |a b|, since if a and b are real, then by equation (1),

ja bj =p(a b)2:

While if

a = a1 + ia2

and

b = b1 + ib2

are complex numbers, then

The above shows that the absolute value distance for the real numbers or the complex numbers, agrees with thestandard Euclidean distance they inherit as a result of considering them as the one and two-dimensional Euclideanspaces respectively.The properties of the absolute value of the dierence of two real or complex numbers: non-negativity, identity ofindiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion ofa distance function as follows:A real valued function d on a set X X is called a metric (or a distance function) on X, if it satises the followingfour axioms:[13]

1.5 Generalizations

1.5.1 Ordered ringsThe denition of absolute value given for real numbers above can be extended to any ordered ring. That is, if a is anelement of an ordered ring R, then the absolute value of a, denoted by |a|, is dened to be:[14]

jaj =(a; if a 0a; if a 0

where a is the additive inverse of a, and 0 is the additive identity element.

1.5.2 FieldsMain article: Absolute value (algebra)

• 1.5. GENERALIZATIONS 9

The fundamental properties of the absolute value for real numbers given in (2)(5) above, can be used to generalisethe notion of absolute value to an arbitrary eld, as follows.A real-valued function v on a eld F is called an absolute value (also a modulus, magnitude, value, or valuation)[15] ifit satises the following four axioms:

Where 0 denotes the additive identity element of F. It follows from positive-deniteness and multiplicativeness thatv(1) = 1, where 1 denotes the multiplicative identity element of F. The real and complex absolute values dened aboveare examples of absolute values for an arbitrary eld.If v is an absolute value on F, then the function d on F F, dened by d(a, b) = v(a b), is a metric and the followingare equivalent:

d satises the ultrametric inequality d(x; y) max(d(x; z); d(y; z)) for all x, y, z in F.

vPnk=11 : n 2 N is bounded in R. vPn

k=11 1 for every n 2 N:

v(a) 1) v(1 + a) 1 for all a 2 F:

v(a+ b) maxfv(a); v(b)g for all a; b 2 F:

An absolute value

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