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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities: * The notation ''a'' < ''b'' means that ''a'' is less than ''b''. * The notation ''a'' > ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equivalence is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or not greater than ''b''). * The notation ''a'' ≥ ''b'' or ''a'' ⩾ ''b'' means that ''a'' is greater than or equal to ''b'' (or, equivalently, at least ''b'', or not less than ''b''). The relation not greater than can also be represented by ''a'' ≯ ''b'', the symbol for "greater than" bisected by a slash, "not". The same is true for not less than and ''a'' ≮ ''b''. The notation ''a'' ≠ ''b'' means that ''a'' is not equal to ''b''; this ''
inequation In mathematics, an inequation is a statement that an inequality holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific in ...
'' sometimes is considered a form of strict inequality. It does not say that one is greater than the other; it does not even require ''a'' and ''b'' to be member of an ordered set. In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude. * The notation ''a'' ≪ ''b'' means that ''a'' is much less than ''b''. * The notation ''a'' ≫ ''b'' means that ''a'' is much greater than ''b''. This implies that the lesser value can be neglected with little effect on the accuracy of an approximation (such as the case of ultrarelativistic limit in physics). In all of the cases above, any two symbols mirroring each other are symmetrical; ''a'' < ''b'' and ''b'' > ''a'' are equivalent, etc.


Properties on the number line

Inequalities are governed by the following properties. All of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities (< and >) and — in the case of applying a function — monotonic functions are limited to ''strictly'' monotonic functions.


Converse

The relations ≤ and ≥ are each other's converse, meaning that for any
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s ''a'' and ''b'':


Transitivity

The transitive property of inequality states that for any
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s ''a'', ''b'', ''c'': If ''either'' of the premises is a strict inequality, then the conclusion is a strict inequality:


Addition and subtraction

A common constant ''c'' may be added to or subtracted from both sides of an inequality. So, for any
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s ''a'', ''b'', ''c'': In other words, the inequality relation is preserved under addition (or subtraction) and the real numbers are an ordered group under addition.


Multiplication and division

The properties that deal with multiplication and division state that for any real numbers, ''a'', ''b'' and non-zero ''c'': In other words, the inequality relation is preserved under multiplication and division with positive constant, but is reversed when a negative constant is involved. More generally, this applies for an ordered field. For more information, see '' § Ordered fields''.


Additive inverse

The property for the additive inverse states that for any real numbers ''a'' and ''b'':


Multiplicative inverse

If both numbers are positive, then the inequality relation between the
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/' ...
s is opposite of that between the original numbers. More specifically, for any non-zero real numbers ''a'' and ''b'' that are both positive (or both negative): All of the cases for the signs of ''a'' and ''b'' can also be written in
chained notation Chained may refer to: * ''Chained'' (1934 film), starring Joan Crawford and Clark Gable * ''Chained'' (2012 film), a Canadian film directed by Jennifer Lynch * ''Chained'' (2020 film), a Canadian film directed by Titus Heckel * ''Chained'', a 2 ...
, as follows:


Applying a function to both sides

Any
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
ally increasing function, by its definition, may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in the
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
of that function). However, applying a monotonically decreasing function to both sides of an inequality means the inequality relation would be reversed. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function. If the inequality is strict (''a'' < ''b'', ''a'' > ''b'') ''and'' the function is strictly monotonic, then the inequality remains strict. If only one of these conditions is strict, then the resultant inequality is non-strict. In fact, the rules for additive and multiplicative inverses are both examples of applying a ''strictly'' monotonically decreasing function. A few examples of this rule are: * Raising both sides of an inequality to a power ''n'' > 0 (equiv., −''n'' < 0), when ''a'' and ''b'' are positive real numbers: * Taking the natural logarithm on both sides of an inequality, when ''a'' and ''b'' are positive real numbers: (this is true because the natural logarithm is a strictly increasing function.)


Formal definitions and generalizations

A (non-strict) partial order is a binary relation ≤ over a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''P'' which is reflexive, antisymmetric, and transitive. That is, for all ''a'', ''b'', and ''c'' in ''P'', it must satisfy the three following clauses: # ''a'' ≤ ''a'' ( reflexivity) # if ''a'' ≤ ''b'' and ''b'' ≤ ''a'', then ''a'' = ''b'' ( antisymmetry) # if ''a'' ≤ ''b'' and ''b'' ≤ ''c'', then ''a'' ≤ ''c'' ( transitivity) A set with a partial order is called a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
. Those are the very basic axioms that every kind of order has to satisfy. Other axioms that exist for other definitions of orders on a set ''P'' include: # For every ''a'' and ''b'' in ''P'', ''a'' ≤ ''b'' or ''b'' ≤ ''a'' (
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
). # For all ''a'' and ''b'' in ''P'' for which ''a'' < ''b'', there is a ''c'' in ''P'' such that ''a'' < ''c'' < ''b'' ( dense order). # Every non-empty
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of ''P'' with an upper bound has a ''least'' upper bound (supremum) in ''P'' (
least-upper-bound property In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if e ...
).


Ordered fields

If (''F'', +, ×) is a field and ≤ is a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
on ''F'', then (''F'', +, ×, ≤) is called an ordered field if and only if: * ''a'' ≤ ''b'' implies ''a'' + ''c'' ≤ ''b'' + ''c''; * 0 ≤ ''a'' and 0 ≤ ''b'' implies 0 ≤ ''a'' × ''b''. Both (Q, +, ×, ≤) and (R, +, ×, ≤) are ordered fields, but ≤ cannot be defined in order to make (C, +, ×, ≤) an ordered field, because −1 is the square of ''i'' and would therefore be positive. Besides from being an ordered field, R also has the
Least-upper-bound property In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if e ...
. In fact, R can be defined as the only ordered field with that quality.


Chained notation

The notation ''a'' < ''b'' < ''c'' stands for "''a'' < ''b'' and ''b'' < ''c''", from which, by the transitivity property above, it also follows that ''a'' < ''c''. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative. Hence, for example, ''a'' < ''b'' + ''e'' < ''c'' is equivalent to ''a'' − ''e'' < ''b'' < ''c'' − ''e''. This notation can be generalized to any number of terms: for instance, ''a''1 ≤ ''a''2 ≤ ... ≤ ''a''''n'' means that ''a''''i'' ≤ ''a''''i''+1 for ''i'' = 1, 2, ..., ''n'' − 1. By transitivity, this condition is equivalent to ''a''''i'' ≤ ''a''''j'' for any 1 ≤ ''i'' ≤ ''j'' ≤ ''n''. When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance, to solve the inequality 4''x'' < 2''x'' + 1 ≤ 3''x'' + 2, it is not possible to isolate ''x'' in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding ''x'' < 1/2 and ''x'' ≥ −1 respectively, which can be combined into the final solution −1 ≤ ''x'' < 1/2. Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the
logical conjunction In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents thi ...
of the inequalities between adjacent terms. For example, the defining condition of a zigzag poset is written as ''a''1 < ''a''2 > ''a''3 < ''a''4 > ''a''5 < ''a''6 > ... . Mixed chained notation is used more often with compatible relations, like <, =, ≤. For instance, ''a'' < ''b'' = ''c'' ≤ ''d'' means that ''a'' < ''b'', ''b'' = ''c'', and ''c'' ≤ ''d''. This notation exists in a few
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...
s such as Python. In contrast, in programming languages that provide an ordering on the type of comparison results, such as C, even homogeneous chains may have a completely different meaning.


Sharp inequalities

An inequality is said to be ''sharp'' if it cannot be ''relaxed'' and still be valid in general. Formally, a
universally quantified In mathematical logic, a universal quantification is a type of Quantification (logic), quantifier, a logical constant which is interpretation (logic), interpreted as "given any" or "for all". It expresses that a predicate (mathematical logic), pr ...
inequality ''φ'' is called sharp if, for every valid universally quantified inequality ''ψ'', if holds, then also holds. For instance, the inequality is sharp, whereas the inequality is not sharp.


Inequalities between means

There are many inequalities between means. For example, for any positive numbers ''a''1, ''a''2, ..., ''a''''n'' we have where they represent the following means of the sequence: ; Harmonic mean : H = \frac ; Geometric mean : G = \sqrt ;
Arithmetic mean In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
: A = \frac ;
quadratic mean In mathematics and its applications, the root mean square of a set of numbers x_i (abbreviated as RMS, or rms and denoted in formulas as either x_\mathrm or \mathrm_x) is defined as the square root of the mean square (the arithmetic mean of the ...
: Q = \sqrt


Cauchy–Schwarz inequality

The Cauchy–Schwarz inequality states that for all vectors ''u'' and ''v'' of an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
it is true that , \langle \mathbf,\mathbf\rangle, ^2 \leq \langle \mathbf,\mathbf\rangle \cdot \langle \mathbf,\mathbf\rangle, where \langle\cdot,\cdot\rangle is the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. Examples of inner products include the real and complex
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
; In
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
''R''''n'' with the standard inner product, the Cauchy–Schwarz inequality is \left(\sum_^n u_i v_i\right)^2\leq \left(\sum_^n u_i^2\right) \left(\sum_^n v_i^2\right).


Power inequalities

A "power inequality" is an inequality containing terms of the form ''a''''b'', where ''a'' and ''b'' are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises.


Examples

* For any real ''x'', e^x \ge 1+x. * If ''x'' > 0 and ''p'' > 0, then \frac \ge \ln(x) \ge \frac. In the limit of ''p'' → 0, the upper and lower bounds converge to ln(''x''). * If ''x'' > 0, then x^x \ge \left( \frac\right)^\frac. * If ''x'' > 0, then x^ \ge x. * If ''x'', ''y'', ''z'' > 0, then \left(x+y\right)^z + \left(x+z\right)^y + \left(y+z\right)^x > 2. * For any real distinct numbers ''a'' and ''b'', \frac > e^. * If ''x'', ''y'' > 0 and 0 < ''p'' < 1, then x^p+y^p > \left(x+y\right)^p. * If ''x'', ''y'', ''z'' > 0, then x^x y^y z^z \ge \left(xyz\right)^. * If ''a'', ''b'' > 0, then a^a + b^b \ge a^b + b^a. * If ''a'', ''b'' > 0, then a^ + b^ \ge a^ + b^. * If ''a'', ''b'', ''c'' > 0, then a^ + b^ + c^ \ge a^ + b^ + c^. * If ''a'', ''b'' > 0, then a^b + b^a > 1.


Well-known inequalities

Mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
s often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names: * Azuma's inequality * Bernoulli's inequality * Bell's inequality * Boole's inequality * Cauchy–Schwarz inequality *
Chebyshev's inequality In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from t ...
*
Chernoff's inequality In probability theory, the Chernoff bound gives exponentially decreasing bounds on tail distributions of sums of independent random variables. Despite being named after Herman Chernoff, the author of the paper it first appeared in, the result is d ...
* Cramér–Rao inequality * Hoeffding's inequality * Hölder's inequality * Inequality of arithmetic and geometric means * Jensen's inequality *
Kolmogorov's inequality In probability theory, Kolmogorov's inequality is a so-called "maximal inequality" that gives a bound on the probability that the partial sums of a finite collection of independent random variables exceed some specified bound. Statement of the ine ...
* Markov's inequality * Minkowski inequality * Nesbitt's inequality * Pedoe's inequality * Poincaré inequality * Samuelson's inequality * Triangle inequality


Complex numbers and inequalities

The set of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s ℂ with its operations of addition and multiplication is a field, but it is impossible to define any relation ≤ so that becomes an ordered field. To make an ordered field, it would have to satisfy the following two properties: * if , then ; * if and , then . Because ≤ is a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
, for any number ''a'', either or (in which case the first property above implies that ). In either case ; this means that and ; so and , which means (−1 + 1) > 0; contradiction. However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if , then "). Sometimes the lexicographical order definition is used: * , if ** , or ** and It can easily be proven that for this definition implies .


Vector inequalities

Inequality relationships similar to those defined above can also be defined for column vectors. If we let the vectors x, y \in \mathbb^n (meaning that x = (x_1, x_2, \ldots, x_n)^\mathsf and y = (y_1, y_2, \ldots, y_n)^\mathsf, where x_i and y_i are real numbers for i = 1, \ldots, n), we can define the following relationships: * x = y , if x_i = y_i for i = 1, \ldots, n. * x < y , if x_i < y_i for i = 1, \ldots, n. * x \leq y , if x_i \leq y_i for i = 1, \ldots, n and x \neq y. * x \leqq y , if x_i \leq y_i for i = 1, \ldots, n. Similarly, we can define relationships for x > y, x \geq y, and x \geqq y. This notation is consistent with that used by Matthias Ehrgott in ''Multicriteria Optimization'' (see References). The trichotomy property (as stated above) is not valid for vector relationships. For example, when x = (2, 5)^\mathsf and y = (3, 4)^\mathsf, there exists no valid inequality relationship between these two vectors. However, for the rest of the aforementioned properties, a parallel property for vector inequalities exists.


Systems of inequalities

Systems of
linear inequalities Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
can be simplified by
Fourier–Motzkin elimination Fourier–Motzkin elimination, also known as the FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities. It can output real solutions. The algorithm is named after Joseph Fourier who proposed the ...
. The cylindrical algebraic decomposition is an algorithm that allows testing whether a system of polynomial equations and inequalities has solutions, and, if solutions exist, describing them. The complexity of this algorithm is doubly exponential in the number of variables. It is an active research domain to design algorithms that are more efficient in specific cases.


See also

* Binary relation * Bracket (mathematics), for the use of similar ‹ and › signs as brackets * Inclusion (set theory) *
Inequation In mathematics, an inequation is a statement that an inequality holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific in ...
*
Interval (mathematics) In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
*
List of inequalities This article lists Wikipedia articles about named mathematical inequalities. Inequalities in pure mathematics Analysis * Agmon's inequality * Askey–Gasper inequality * Babenko–Beckner inequality * Bernoulli's inequality * Bernstein's ineq ...
* List of triangle inequalities *
Partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
* Relational operators, used in programming languages to denote inequality


References


Sources

* * * * * * * * * * *


External links

*
Graph of Inequalities
by
Ed Pegg, Jr. Edward Taylor Pegg Jr. (born December 7, 1963) is an expert on mathematical puzzles and is a self-described recreational mathematician. He wrote an online puzzle column called Ed Pegg Jr.'s ''Math Games'' for the Mathematical Association of Am ...

AoPS Wiki entry about Inequalities
{{Authority control Elementary algebra Mathematical terminology