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 poin ...
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
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
Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property.
Property may also refer to:
Mathematics
* Property (mathematics)
Philosophy and science
* Property (philosophy), in philosophy and ...
. 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 numbers ''a'' and ''b'':
Transitivity
The transitive property of inequality states that for any
real numbers ''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 numbers ''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
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
states that for any real numbers ''a'' and ''b'':
Multiplicative inverse
If both numbers are positive, then the inequality relation between the
multiplicative inverses is opposite of that between the original numbers. More specifically, for any non-zero real numbers ''a'' and ''b'' that are both
positive
Positive is a property of positivity and may refer to:
Mathematics and science
* Positive formula, a logical formula not containing negation
* Positive number, a number that is greater than 0
* Plus sign, the sign "+" used to indicate a posi ...
(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 ord ...
ally increasing
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
, 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
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 sets and is a new set of ordered pairs consisting of elements in and i ...
≤ 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
In linguistics, antisymmetry is a syntactic theory presented in Richard S. Kayne's 1994 monograph ''The Antisymmetry of Syntax''. It asserts that grammatical hierarchies in natural language follow a universal order, namely specifier-head-comple ...
)
# 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).
# For all ''a'' and ''b'' in ''P'' for which ''a'' < ''b'', there is a ''c'' in ''P'' such that ''a'' < ''c'' < ''b'' (
dense order In mathematics, a partial order or total order < on a set is said to be dense if, for all and in ).
# Every non-empty
subset 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 eleme ...
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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
and ≤ is a
total order 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 this ...
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 languages such as
Python
Python may refer to:
Snakes
* Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia
** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia
* Python (mythology), a mythical serpent
Computing
* Python (pr ...
. 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 :
;
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
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 ...
:
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality states that for all vectors ''u'' and ''v'' of an
inner product space it is true that
where
is the
inner product. 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 algeb ...
; In
Euclidean space ''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 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
Mathematicians 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 In probability theory, the Azuma–Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values of martingales that have bounded differences.
Suppose \ is a martingale (or super-martingale) ...
*
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 + r ...
*
Bell's inequality
*
Boole's inequality
*
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 fo ...
*
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
*
Markov's inequality
*
Minkowski inequality
In mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are normed vector spaces. Let ''S'' be a measure space, let and let ''f'' and ''g'' be elements of L''p''(''S''). Then is in L''p''(''S''), and we have the tr ...
*
Nesbitt's inequality In mathematics, Nesbitt's inequality states that for positive real numbers ''a'', ''b'' and ''c'',
:\frac+\frac+\frac\geq\frac.
It is an elementary special case (N = 3) of the difficult and much studied Shapiro inequality, and was published at l ...
*
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
Complex numbers and inequalities
The set of
complex numbers 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 is a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, 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, 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 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, c ...
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.[ ...](_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
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.
The
cylindrical algebraic decomposition
In mathematics, cylindrical algebraic decomposition (CAD) is a notion, and an algorithm to compute it, that are fundamental for computer algebra and real algebraic geometry. Given a set ''S'' of polynomials in R''n'', a cylindrical algebraic decomp ...
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 sets and is a new set of ordered pairs consisting of elements in and i ...
*
Bracket (mathematics)
In mathematics, brackets of various typographical forms, such as parentheses ( ), square brackets nbsp; braces and angle brackets ⟨ ⟩, are frequently used in mathematical notation. Generally, such bracketing denotes some form of gro ...
, 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)
*
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. Other ...
*
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
In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions. The inequalities give an ordering of two different values: they are of the ...
*
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 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 Am ...
AoPS Wiki entry about Inequalities
{{Authority control
Elementary algebra
Mathematical terminology