# Absolute value

Last updated

In mathematics, the absolute value or modulus of a real number  x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, and |x| = −x if x is negative (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.

## Contents

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 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

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 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, for example a real number, a complex number, or a 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 ${\displaystyle \mathbb {R} ^{n}}$, although double vertical bars with subscripts (${\displaystyle \|\cdot \|_{2}}$ and ${\displaystyle \|\cdot \|_{\infty }}$, respectively) are a more common and less ambiguous notation.

## Definition and properties

### 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]

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

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. 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

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

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:

 ${\displaystyle |a|\geq 0}$ Non-negativity ${\displaystyle |a|=0\iff a=0}$ Positive-definiteness ${\displaystyle |ab|=\left|a\right|\left|b\right|}$ Multiplicativity ${\displaystyle |a+b|\leq |a|+|b|}$ 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 ${\displaystyle s\cdot (a+b)=|a+b|\geq 0.}$ Now, since ${\displaystyle -1\cdot x\leq |x|}$ and ${\displaystyle +1\cdot x\leq |x|}$, it follows that, whichever is the value of s, one has ${\displaystyle s\cdot x\leq |x|}$ for all real ${\displaystyle x}$. Consequently, ${\displaystyle |a+b|=s\cdot (a+b)=s\cdot a+s\cdot b\leq |a|+|b|}$, as desired. (For a generalization of this argument to complex numbers, see "Proof of the triangle inequality for complex numbers" below.)

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

 ${\displaystyle {\bigl |}\left|a\right|{\bigr |}=|a|}$ Idempotence (the absolute value of the absolute 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-definiteness) ${\displaystyle |a-b|\leq |a-c|+|c-b|}$ 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:

${\displaystyle |a|\leq b\iff -b\leq a\leq b}$
${\displaystyle |a|\geq b\iff a\leq -b\ }$ or ${\displaystyle a\geq b}$

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

 ${\displaystyle |x-3|\leq 9}$ ${\displaystyle \iff -9\leq x-3\leq 9}$ ${\displaystyle \iff -6\leq x\leq 12}$

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.

### Complex numbers

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

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

where x and y are real numbers, the absolute value or modulus of z is denoted |z| and is defined by [10]

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

where Re(z) = x and Im(z) = y denote the real and imaginary parts of z, respectively. When the imaginary part y is zero, this coincides with the definition of the absolute value of the real number x.

When a complex number z is expressed in its polar form as

${\displaystyle z=re^{i\theta },}$

with ${\textstyle r={\sqrt {[\operatorname {Re} (z)]^{2}+[\operatorname {Im} (z)]^{2}}}\geq 0}$ (and θ ∈ arg(z) is the argument (or phase) of z), its absolute value is

${\displaystyle |z|=r.}$

Since the product of any complex number z and its complex 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 z is the square root of ${\displaystyle z\cdot {\overline {z}},}$ which is therefore called the absolute square or squared modulus of z:

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

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

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]

Importantly, the property of subadditivity ("triangle inequality") extends to any finite collection of n complex numbers ${\textstyle (z_{k})_{k=1}^{n}}$ as

${\displaystyle \left|\sum _{k=1}^{n}z_{k}\right|\leq \sum _{k=1}^{n}\left|z_{k}\right|.}$

()

This inequality also applies to infinite families, provided that the infinite series ${\textstyle \sum _{k=1}^{\infty }z_{k}}$ 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 ${\displaystyle f:\mathbb {R} \to \mathbb {C} }$ when integrated over a measurable subset ${\displaystyle E}$:

${\displaystyle \left|\int _{E}f\,dx\right|\leq \int _{E}\left|f\right|dx.}$

(⁎⁎)

(This includes Riemann-integrable functions over a bounded interval ${\displaystyle [a,b]}$ 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 ${\displaystyle z\in \mathbb {C} }$,

1. there exists ${\displaystyle c\in \mathbb {C} }$ such that ${\displaystyle |c|=1}$ and ${\displaystyle |z|=c\cdot z}$;
2. ${\displaystyle \operatorname {Re} (z)\leq |z|}$.

Also, for a family of complex numbers ${\displaystyle (w_{k})_{k=1}^{n}}$,${\textstyle \sum _{k}w_{k}=\sum _{k}\operatorname {Re} (w_{k})+i\sum _{k}\operatorname {Im} (w_{k})}$. In particular,

1. if ${\textstyle \sum _{k}w_{k}\in \mathbb {R} }$, then ${\textstyle \sum _{k}w_{k}=\sum _{k}\operatorname {Re} (w_{k})}$.

Proof of ( ): Choose ${\displaystyle c\in \mathbb {C} }$ such that ${\displaystyle |c|=1}$ and ${\textstyle \left|\sum _{k}z_{k}\right|=c\left(\sum _{k}z_{k}\right)}$ (summed over ${\displaystyle k=1,\ldots ,n}$). The following computation then affords the desired inequality:

${\displaystyle \left|\sum _{k}z_{k}\right|\;{\overset {(1)}{=}}\;c\left(\sum _{k}z_{k}\right)=\sum _{k}cz_{k}\;{\overset {(3)}{=}}\;\sum _{k}\operatorname {Re} (cz_{k})\;{\overset {(2)}{\leq }}\;\sum _{k}|cz_{k}|=\sum _{k}\left|c\right|\left|z_{k}\right|=\sum _{k}\left|z_{k}\right|.}$

It is clear from this proof that equality holds in ( ) exactly if all the ${\displaystyle cz_{k}}$ are non-negative real numbers, which in turn occurs exactly if all nonzero ${\displaystyle z_{k}}$ have the same argument, i.e., ${\displaystyle z_{k}=a_{k}\zeta }$ for a complex constant ${\displaystyle \zeta }$ and real constants ${\displaystyle a_{k}\geq 0}$ for ${\displaystyle 1\leq k\leq n}$.

Since ${\displaystyle f}$ measurable implies that ${\displaystyle |f|}$ is also measurable, the proof of the inequality ( ⁎⁎ ) proceeds via the same technique, by replacing ${\textstyle \sum _{k}(\cdot )}$ with ${\textstyle \int _{E}(\cdot )\,dx}$ and ${\displaystyle z_{k}}$ with ${\displaystyle f(x)}$. [12]

## Absolute value function

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.

### 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 number's sign irrespective of its value. The following equations show 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

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]

${\displaystyle {\frac {d\left|x\right|}{dx}}={\frac {x}{|x|}}={\begin{cases}-1&x<0\\1&x>0.\end{cases}}}$

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

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

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

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.

## Distance

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 plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them.

The standard Euclidean distance between two points

${\displaystyle a=(a_{1},a_{2},\dots ,a_{n})}$

and

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

in Euclidean n-space is defined as:

${\displaystyle {\sqrt {\textstyle \sum _{i=1}^{n}(a_{i}-b_{i})^{2}}}.}$

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

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

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

 ${\displaystyle |a-b|}$ ${\displaystyle =|(a_{1}+ia_{2})-(b_{1}+ib_{2})|}$ ${\displaystyle =|(a_{1}-b_{1})+i(a_{2}-b_{2})|}$ ${\displaystyle ={\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}}}={\sqrt {\textstyle \sum _{i=1}^{2}(a_{i}-b_{i})^{2}}}.}$

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

The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given above, can be seen to motivate the more general notion of a 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 satisfies the following four axioms: [16]

 ${\displaystyle d(a,b)\geq 0}$ 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

### Ordered rings

The definition of absolute value given for real numbers above can be extended to any ordered ring. That is, if a is an element of an ordered ring R, then the absolute value of a, denoted by |a|, is defined to be: [17]

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

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

### Fields

The four fundamental properties of the absolute value for real numbers can be used to generalise the notion of absolute value to an arbitrary field, as follows.

A real-valued function v on a field  F is called an absolute value (also a modulus, magnitude, value, or valuation) [18] if it satisfies the following four axioms:

 ${\displaystyle v(a)\geq 0}$ Non-negativity ${\displaystyle v(a)=0\iff a=\mathbf {0} }$ Positive-definiteness ${\displaystyle v(ab)=v(a)v(b)}$ Multiplicativity ${\displaystyle v(a+b)\leq v(a)+v(b)}$ Subadditivity or the triangle inequality

Where 0 denotes the additive identity of F. It follows from positive-definiteness and multiplicativity that v(1) = 1, where 1 denotes the multiplicative identity of F. The real and complex absolute values defined above are examples of absolute values for an arbitrary field.

If v is an absolute value on F, then the function d on F × F, defined by d(a, b) = v(ab), is a metric and the following are equivalent:

• d satisfies the ultrametric inequality ${\displaystyle d(x,y)\leq \max(d(x,z),d(y,z))}$ for all x, y, z in F.
• ${\textstyle \left\{v\left(\sum _{k=1}^{n}\mathbf {1} \right):n\in \mathbb {N} \right\}}$ is bounded in R.
• ${\displaystyle v\left({\textstyle \sum _{k=1}^{n}}\mathbf {1} \right)\leq 1\ }$ for every ${\displaystyle n\in \mathbb {N} }$.
• ${\displaystyle v(a)\leq 1\Rightarrow v(1+a)\leq 1\ }$ for all ${\displaystyle a\in F.}$
• ${\displaystyle v(a+b)\leq \max\{v(a),v(b)\}\ }$ for all ${\displaystyle a,b\in F}$.

An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwise it is said to be Archimedean. [19]

### Vector spaces

Again the fundamental properties of the absolute value for real numbers can be used, with a slight modification, to generalise the notion to an arbitrary vector space.

A real-valued function on a vector space  V over a field F, represented as || · ||, is called an absolute value, but more usually a norm, if it satisfies the following axioms:

For all a in F, and v, u in V,

 ${\displaystyle \|\mathbf {v} \|\geq 0}$ Non-negativity ${\displaystyle \|\mathbf {v} \|=0\iff \mathbf {v} =0}$ Positive-definiteness ${\displaystyle \|a\mathbf {v} \|=|a|\|\mathbf {v} \|}$ Positive homogeneity or positive scalability ${\displaystyle \|\mathbf {v} +\mathbf {u} \|\leq \|\mathbf {v} \|+\|\mathbf {u} \|}$ Subadditivity or the triangle inequality

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

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

${\displaystyle \|(x_{1},x_{2},\dots ,x_{n})\|={\sqrt {\textstyle \sum _{i=1}^{n}x_{i}^{2}}}}$

is a norm called the Euclidean norm. When the real numbers ${\displaystyle \mathbb {R} }$ are considered as the one-dimensional vector space ${\displaystyle \mathbb {R} ^{1}}$, the absolute value is a norm, and is the p-norm (see Lp space) for any p. In fact the absolute value is the "only" norm on ${\displaystyle \mathbb {R} ^{1}}$, in the sense that, for every norm || · || on ${\displaystyle \mathbb {R} ^{1}}$, ||x|| = ||1|| ⋅ |x|.

The complex absolute value is a special case of the norm in an inner product space, which is identical to the Euclidean norm when the complex plane is identified as the Euclidean plane  ${\displaystyle \mathbb {R} ^{2}}$.

### Composition algebras

Every composition algebra A has an involution xx* called its conjugation. The product in A of an element x and its conjugate x* is written N(x) = x x* and called the norm of x.

The real numbers ${\displaystyle \mathbb {R} }$, complex 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 form that is not definite and has null vectors. However, as in the case of division algebras, when an element x has a non-zero norm, then x has a multiplicative inverse given by x*/N(x).

## Notes

1. Oxford English Dictionary, Draft Revision, June 2008
2. Nahin, O'Connor and Robertson, and functions.Wolfram.com.; for the French sense, see Littré, 1877
3. Lazare Nicolas M. Carnot, Mémoire sur la relation qui existe entre les distances respectives de cinq point quelconques pris dans l'espace, p. 105 at Google Books
4. James Mill Peirce, A Text-book of Analytic Geometry at Internet Archive. The oldest citation in the 2nd edition of the Oxford English Dictionary is from 1907. The term absolute 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. 1. ISBN   0805390219.
7. Munkres, James (1991). Analysis on Manifolds. Boulder, CO: Westview. p. 4. ISBN   0201510359.
8. Mendelson, p. 2.
9. Stewart, James B. (2001). Calculus: concepts and contexts. Australia: Brooks/Cole. ISBN   0-534-37718-1., p. A5
10. González, Mario O. (1992). Classical Complex Analysis. CRC Press. p. 19. ISBN   9780824784157.
11. Lorenz, Falko (2008), Algebra. Vol. II. Fields with structure, algebras and advanced topics, Universitext, New York: Springer, p. 39, doi:10.1007/978-0-387-72488-1, ISBN   978-0-387-72487-4, MR   2371763 .
12. Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. p. 325. ISBN   0-07-054235-X.
13. Bartel and Sherbert, p. 163
14. Peter Wriggers, Panagiotis Panatiotopoulos, eds., New Developments in Contact Problems, 1999, ISBN   3-211-83154-1, p. 31–32
15. 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).
16. Mac Lane, p. 264.
17. Shechter, p. 260. This meaning of valuation is rare. Usually, a valuation is the logarithm of the inverse of an absolute value
18. Shechter, pp. 260–261.

## Related Research Articles

In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation i2 = −1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

In mathematics, an inner product space or a Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors, often denoted using angle brackets. Inner products allow the rigorous introduction of intuitive geometrical notions, such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors. Inner product spaces generalize Euclidean spaces to vector spaces of any dimension, and are studied in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.

In mathematics, a linear map is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism.

In mathematics, real analysis is the branch of mathematical analysis that studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.

In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory.

In mathematics, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two p-adic numbers are considered to be close when their difference is divisible by a high power of p: the higher the power, the closer they are. This property enables p-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.

In mathematics, an infinite series of numbers is said to converge absolutely if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to converge absolutely if for some real number . Similarly, an improper integral of a function, , is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if

A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.

In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum.

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, the complex conjugate of is equal to The complex conjugate of is often denoted as .

In mathematics, a trace-class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and reserve "nuclear operator" for usage in more general topological vector spaces.

In mathematics, a linear form is a linear map from a vector space to its field of scalars.

In mathematics, a norm is a function from a real or complex vector space to the nonnegative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself.

In mathematics, a real coordinate space of dimension n, written Rn or , is a coordinate space over the real numbers. This means that it is the set of the n-tuples of real numbers. With component-wise addition and scalar multiplication, it is a real vector space.

In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices.

In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers is equivalent to either the usual real absolute value or a p-adic absolute value.

In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.

In computational learning theory, Rademacher complexity, named after Hans Rademacher, measures richness of a class of real-valued functions with respect to a probability distribution.

In mathematics, Minkowski's second theorem is a result in the geometry of numbers about the values taken by a norm on a lattice and the volume of its fundamental cell.