In
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
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
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'' 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
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix '' ...
(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 function
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 ord ...
s.
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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...
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''/''b ...
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, 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 ord ...
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 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
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
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
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
≤ over a
set ''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 binar ...
. 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 ( reflexiv ...
).
# 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 o ...
of ''P'' with an
upper bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an elem ...
has a
''least'' upper bound (supremum) in ''P'' (
least-upper-bound property).
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 ( reflexiv ...
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 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 l ...
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 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
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired.
The harmonic mean can be expressed as the recipr ...
:
;
Geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
:
;
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 coll ...
:
;
quadratic mean :
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
where
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 Euclidean sp ...
''R''
''n'' with the standard inner product, the Cauchy–Schwarz inequality is
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
Mathematics competitions or mathematical olympiads are competitive events where participants complete a math test. These tests may require multiple choice or numeric answers, or a detailed written solution or proof.
International mathematics comp ...
exercises.
Examples
* For any real ''x'',
* If ''x'' > 0 and ''p'' > 0, then
In the limit of ''p'' → 0, the upper and lower bounds converge to ln(''x'').
* If ''x'' > 0, then
* If ''x'' > 0, then
* If ''x'', ''y'', ''z'' > 0, then
* For any real distinct numbers ''a'' and ''b'',
* If ''x'', ''y'' > 0 and 0 < ''p'' < 1, then
* If ''x'', ''y'', ''z'' > 0, then
* If ''a'', ''b'' > 0, then
* If ''a'', ''b'' > 0, then
* If ''a'', ''b'', ''c'' > 0, then
* If ''a'', ''b'' > 0, then
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, mathematical structure, structure, space, Mathematica ...
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
In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1 + ''x''. It is often employed in real analysis. It has several useful variants:
* (1 + x)^r \geq 1 + ...
*
Bell's inequality
*
Boole's inequality
In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the indivi ...
*
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics.
The inequality for sums was published by . The corresponding inequality f ...
*
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 th ...
*
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 ...
*
Cramér–Rao inequality
*
Hoeffding's inequality
*
Hölder's inequality
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of spaces.
:Theorem (Hölder's inequality). Let be a measure space and let with . ...
*
Inequality of arithmetic and geometric means
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and ...
*
Jensen's inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier ...
*
Kolmogorov's inequality
*
Markov's inequality
In probability theory, Markov's inequality gives an upper bound for the probability that a non-negative function of a random variable is greater than or equal to some positive constant. It is named after the Russian mathematician Andrey Marko ...
*
Minkowski inequality
*
Nesbitt's inequality
*
Pedoe's inequality In geometry, Pedoe's inequality (also Neuberg–Pedoe inequality), named after Daniel Pedoe (1910–1998) and Joseph Jean Baptiste Neuberg (1840–1926), states that if ''a'', ''b'', and ''c'' are the lengths of the sides of a triangle with area '' ...
*
Poincaré inequality
*
Samuelson's inequality
*
Triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, bu ...
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 for ...
s ℂ with its operations of
addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or ''sum'' of ...
and
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...
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 ( reflexiv ...
, 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
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of ...
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 vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, ...
s. If we let the vectors
(meaning that
and
, where
and
are real numbers for
), we can define the following relationships:
*
, if
for
.
*
, if
for
.
*
, if
for
and
.
*
, if
for
.
Similarly, we can define relationships for
,
, and
. This notation is consistent with that used by Matthias Ehrgott in ''Multicriteria Optimization'' (see References).
The
trichotomy property
In mathematics, the law of trichotomy states that every real number is either positive, negative, or zero.[T ...](_blank)
(as stated
above) is not valid for vector relationships. For example, when
and
, 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 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
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
*
Bracket (mathematics), for the use of similar ‹ and › signs as
bracket
A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
s
*
Inclusion (set theory)
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 ...
*
Inequation
*
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 in ...
*
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 binar ...
*
Relational operator
In computer science, a relational operator is a programming language construct or operator that tests or defines some kind of relation between two entities. These include numerical equality (''e.g.'', ) and inequalities (''e.g.'', ).
In pro ...
s, used in programming languages to denote inequality
References
Sources
*
*
*
*
*
*
*
*
*
*
*
External links
*
Graph of Inequalitiesby
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 Amer ...
AoPS Wiki entry about Inequalities
{{Authority control
Elementary algebra
Mathematical terminology