Inequality

This article is about inequalities in mathematics. For other senses of this word, see inequality (disambiguation).

For the use of the < and > signs in punctuation, see Bracket. Brackets are Punctuation marks used in pairs to set apart or interject text within other text

For the not-equals statement, see Inequation. In Mathematics, an inequation is a statement that two objects or expressions are not the same or do not represent the same value

The feasible regions of linear programming are defined by a set of inequalities. In optimization (a branch of Mathematics) a candidate solution is a member of a set of possible solutions to a given problem In Mathematics, linear programming (LP is a technique for optimization of a Linear Objective function, subject to Linear equality

In mathematics, an inequality is a statement about the relative size or order of two objects. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and (See also: equality)

The notation $a < b \!\$ means that a is less than b and
The notation $a > b \!\$ means that a is greater than b. Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric

Therefore a is not equal to b. These relations are known as strict inequality; in contrast

$a \le b$ means that a is less than or equal to b (or, equivalently, not greater than b);
$a \ge b$ means that a is greater than or equal to b (or, equivalently, not smaller than b);

An additional use of the notation is to show that one quantity is much greater than another, normally by several orders of magnitude. An order of magnitude is the class of scale or magnitude of any amount where each class contains values of a fixed ratio to the class preceding it

The notation $a \ll b$ means that a is much less than b.
The notation $a \gg b$ means that a is much greater than b.

If the sense of the inequality is the same for all values of the variables for which its members are defined, then the inequality is called an "absolute" or "unconditional" inequality. If the sense of an inequality holds only for certain values of the variables involved, but is reversed or destroyed for other values of the variables, it is called a conditional inequality.

Properties

Inequalities are governed by the following properties. Note that, for the transitivity, reversal, addition and subtraction, and multiplication and division properties, the property also holds if strict inequality signs (< and >) are replaced with their corresponding non-strict inequality sign (≤ and ≥).

Trichotomy

The trichotomy property states:

For any real numbers, a and b, exactly one of the following is true:
- a b

Transitivity

The transitivity of inequalities states:

For any real numbers, a, b, c:
- If a > b and b > c; then a > c
- If a < b and b < c; then a < c

Reversal

The inequality relations are inverse relations:

For any real numbers, a and b:
- If a > b then b < a
- If a a

Addition and subtraction

The properties which deal with addition and subtraction state:

For any real numbers, a, b, c:
- If a > b, then a + c > b + c and a − c > b − c
- If a < b, then a + c < b + c and a − c < b − c

i. Generally a trichotomy is a splitting into three disjoint parts In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the inverse relation of a Binary relation is the relation taken 'backwards' as in changing the relation 'child of' to 'parent of' In Mathematics, the real numbers may be described informally in several different ways Addition is the mathematical process of putting things together Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract In Mathematics, the real numbers may be described informally in several different ways e. , the real numbers are an ordered group. In Abstract algebra, an ordered group is a group (G+ equipped with a Partial order "≤" which is translation-invariant

Multiplication and division

The properties which deal with multiplication and division state:

For any real numbers, a, b, c:
- If c is positive and a < b, then ac < bc
- If c is negative and a < b, then ac > bc

More generally this applies for an ordered field, see below. In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. A negative number is a Number that is less than zero, such as −2 A negative number is a Number that is less than zero, such as −2 In Mathematics, an ordered field is a field together with a Total ordering of its elements that agrees in a certain sense with the field operations

Additive inverse

The properties for the additive inverse state:

For any real numbers a and b
- If a −b
- If a > b then −a < −b

Multiplicative inverse

The properties for the multiplicative inverse state:

For any real numbers a and b that are both positive or both negative
- If a 1/b
- If a > b then 1/a < 1/b

Applying a function to both sides

We consider two cases of functions: monotonic and strictly monotonic. In mathematics the additive inverse, or opposite, of a Number n is the number that when added to n, yields zero. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which

Any strictly monotonically increasing function may be applied to both sides of an inequality and it will still hold. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Applying a strictly monotonically decreasing function to both sides of an inequality means the opposite inequality now holds. The rules for additive and multiplicative inverses are both examples of applying a monotonically decreasing function.

If you have a non-strict inequality (a ≤ b, a ≥ b) then:

Applying a monotonically increasing function preserves the relation (≤ remains ≤, ≥ remains ≥)
Applying a monotonically decreasing function reverses the relation (≤ becomes ≥, ≥ becomes ≤)

It will never become strictly unequal, since, for example, 3 ≤ 3 does not imply that 3 < 3.

Ordered fields

If F,+,* be a field and ≤ be a total order on F, then F,+,*,≤ is called an ordered field if and only if:

if a ≤ b then a + c ≤ b + c
if 0 ≤ a and 0 ≤ b then 0 ≤ a b

Note that both $\mathbb{Q}$ ,+,*,≤ and $\mathbb{R}$ ,+,*,≤ are ordered fields. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In Mathematics, an ordered field is a field together with a Total ordering of its elements that agrees in a certain sense with the field operations In Mathematics, an ordered field is a field together with a Total ordering of its elements that agrees in a certain sense with the field operations

≤ cannot be defined in order to make $\mathbb{C}$ ,+,*,≤ an ordered field. In Mathematics, an ordered field is a field together with a Total ordering of its elements that agrees in a certain sense with the field operations

The non-strict inequalities ≤ and ≥ on real numbers are total orders. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation The strict inequalities < and > on real numbers are strict total orders. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation

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. Obviously, by the above laws, one can add/subtract the same number to all three terms, or multiply/divide all three terms by same nonzero number and reverse all inequalities according to sign. But care must be taken so that you really use the same number in all cases, e. g. a < b + e < c is equivalent to a − e < b < c − e.

This notation can be generalized to any number of terms: for instance, a₁ ≤ a₂ ≤ . . . ≤ 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 4x < 2x + 1 ≤ 3x + 2, you won't be able to isolate x in any one part of the inequality through addition or subtraction. Instead, you can solve 4x < 2x + 1 and 2x + 1 ≤ 3x + 2 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 of the inequalities between adjacent terms. In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of For instance, a c ≤ d means that a < b, b > c, and c ≤ d. In addition to rare use in mathematics, this notation exists in a few programming languages such as Python. A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. Python is a general-purpose High-level programming language. Its design philosophy emphasizes programmer productivity and code readability

Representing inequalities on the real number line

Every inequality (except those which involve imaginary numbers) can be represented on the real number line showing darkened regions on the line. Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane In mathematics a number line is a picture of a straight line in which the Integers are shown as specially-marked points evenly spaced on the line

Inequalities between means

There are many inequalities between means. For example, for any positive numbers $a 1$ , $a 2$ , . . . , $a n$

$H \le G \le A \le Q$ , where

$H = \frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}}$ (harmonic mean),

$G = \sqrt[n]{a_1 \cdot a_2 \cdot ... \cdot a_n}$ (geometric mean),

$A = \frac{a_1 + a_2 + ... + a_n}{n}$ (arithmetic mean),

$Q = \sqrt{\frac{a_1^2 + a_2^2 + ... + a_n^2}{n}}$ (quadratic mean). In Mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of Average. The geometric mean in Mathematics, is a type of Mean or Average, which indicates the central tendency or typical value of a set of numbers In Mathematics and Statistics, the arithmetic Mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided In Mathematics, the root mean square (abbreviated RMS or rms) also known as the quadratic mean, is a statistical measure of the

Power inequalities

Sometimes with notation "power inequality" understand inequalities which contain $a b$ type expressions where $a$ and $b$ are real positive numbers or expressions of some variables. They can appear in exercises of mathematical olympiads and some calculations.

Examples

If $x > 0$ , then $x^x \ge \left( \tfrac{1}{e}\right)^ \tfrac{1}{e}$
If $x > 0$ , then $x^{x^x} \ge x$
If $x, y, z > 0$ , then $(x + y) z + (x + z) y + (y + z) x > 2$ .
For any real distinct numbers $a$ and $b$ , $\tfrac{e^b-e^a}{b-a}>e^{\frac{a+b}{2}}$
If $x, y > 0$ and $0 < p < 1$ , then $(x + y) p < x p + y p$
If $x$ , $y$ and $z$ are positive, then $x^x y^y z^z \ge (xyz)^ \frac{x+y+z}{3}$
If $a$ and $b$ are positive, then $a b + b a > 1$ . This result was generalized by R. Ozols in 2002 who proved that if $a 1$ , $a 2$ , . . . , $a n$ are any real positive numbers, then $a_1^{a_2}+a_2^{a_3}+...+a_n^{a_1}>1$ (result is published in Latvian popular-scientific quarterly The Starry Sky, see references).

Well-known inequalities

See also list of inequalities. This page lists Wikipedia articles about named mathematical inequalities.

Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. A mathematician is a person whose primary area of study and research is the field of Mathematics. Some inequalities are used so often that they have names:

Mnemonics for students

Young students sometimes confuse the less-than and greater-than signs, which are mirror images of one another. In Probability theory, the Azuma-Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a Concentration result for the In Real analysis, Bernoulli's inequality is an Inequality that approximates Exponentiations of 1 + x. In Probability theory, Boole's inequality, named after George Boole, (also known as the union bound) says that for any finite or Countable In Mathematics, the Cauchy–Schwarz inequality, also known as the Schwarz inequality, the Cauchy inequality, or the Cauchy–Schwarz–Bunyakovsky In Probability theory, Chebyshev's inequality (also known as Tchebysheff's inequality, Chebyshev's theorem, or the Bienaymé-Chebyshev inequality In Estimation theory and Statistics, the Cramér–Rao bound (CRB or Cramér–Rao lower bound (CRLB, named in honor of Harald Cramér and Hoeffding's Inequality, named after Wassily Hoeffding, is a result in Probability theory that gives an Upper bound on the Probability In Mathematical analysis Hölder's inequality, named after Otto Hölder, is a fundamental Inequality between integrals and an indispensable tool 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 In Mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a Convex function of an Integral Kolmogorov's inequality is an Inequality which gives a relation among a function and its first and second Derivatives Kolmogorov's inequality states the In Probability theory, Markov's inequality gives an Upper bound for the Probability that a non-negative function of a Random In Mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are Normed vector spaces Let S be a Measure In Mathematics, Nesbitt's Inequality is a special case of the Shapiro inequality. In Geometry, Pedoe's inequality, named after Daniel Pedoe, states that if a, b, and c are the lengths of the sides of a Triangle In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater A commonly taught mnemonic is that the sign represents the mouth of a hungry alligator that is trying to eat the larger number; thus, it opens towards 8 in both 3 < 8 and 8 > 3. for differences between alligators and crocodiles please see Crocodilia An Alligator is a Crocodilian in the Genus [1] Another method is noticing the larger quantity points to the smaller quantity and says, "ha-ha, I'm bigger than you. "

Also, on a horizontal number line, the greater than sign is the arrow that is at the larger end of the number line. Likewise, the less than symbol is the arrow at the smaller end of the number line (<---0--1--2--3--4--5--6--7--8--9--->). In mathematics a number line is a picture of a straight line in which the Integers are shown as specially-marked points evenly spaced on the line

The symbols may also be interpreted directly from their form - the side with a large vertical separation indicates a large(r) quantity, and the side which is a point indicates a small(er) quantity. In this way the inequality symbols are similar to the musical crescendo and decrescendo. In Music, dynamics normally refers to the volume of a Sound or note, but can also refer to every aspect of the execution of a given piece either stylistic The symbols for equality, less-than-or-equal-to, and greater-than-or-equal-to can also be interpreted with this perspective.

Complex numbers and inequalities

By introducing a lexicographical order on the complex numbers, it is a totally ordered set. In Mathematics, the lexicographic or lexicographical order, (also known as dictionary order, alphabetic order or lexicographic(al product Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation However, it is impossible to define ≤ so that $\mathbb{C}$ ,+,*,≤ becomes an ordered field. In Mathematics, an ordered field is a field together with a Total ordering of its elements that agrees in a certain sense with the field operations If $\mathbb{C}$ ,+,*,≤ were an ordered field, it has to satisfy the following two properties:

if a ≤ b then a + c ≤ b + c
if 0 ≤ a and 0 ≤ b then 0 ≤ a b

Because ≤ is a total order, for any number a, a ≤ 0 or 0 ≤ a. In Mathematics, an ordered field is a field together with a Total ordering of its elements that agrees in a certain sense with the field operations In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In both cases 0 ≤ a²; this means that $i 2 > 0$ and $12 > 0$ ; so $1 > 0$ and $- 1 > 0$ , contradiction.

However ≤ can be defined in order to satisfy the first property, i. e. if a ≤ b then a + c ≤ b + c. A definition which is sometimes used is the lexicographical order:

a ≤ b if $R e (a)$ < $R e (b)$ or ( $R e (a) = R e (b)$ and $I m (a)$ ≤ $I m (b)$ )

It can easily be proven that for this definition a ≤ b then a + c ≤ b + c

References

Hardy, G. In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of Brackets are Punctuation marks used in pairs to set apart or interject text within other text Fourier–Motzkin elimination is a mathematical Algorithm for eliminating variables from a System of linear inequalities. In Mathematics, an inequation is a statement that two objects or expressions are not the same or do not represent the same value In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Computer science a relational operator is a Programming language construct or operator that tests some kind of relation between two entities , Littlewood J. E. , Polya, G. (1999). Inequalities. Cambridge Mathematical Library, Cambridge University Press. ISBN 0-521-05206-8.
Beckenbach, E. F. , Bellman, R. (1975). An Introduction to Inequalities. Random House Inc. ISBN 0-394-01559-2.
Drachman, Byron C. , Cloud, Michael J. (1998). Inequalities: With Applications to Engineering. Springer-Verlag. ISBN 0-387-98404-6.
Murray S. Klamkin. ""Quickie" inequalities" (PDF).
Harold Shapiro (missingdate). Mathematical Problem Solving. The Old Problem Seminar. Kungliga Tekniska högskolan.
3rd USAMO.
. "The Starry Sky".

External links

interactive linear inequality & graph at www. mathwarehouse. com
Solving Inequalities
WebGraphing.com – Inequality Graphing Calculator.
Graph of Inequalities by Ed Pegg, Jr., The Wolfram Demonstrations Project. Ed Pegg Jr is an expert on mathematical Puzzles and is a self-described recreational mathematician

Dictionary

-noun

An unfair, not equal, state.
(mathematics) A statement that of two quantities one is specifically less than (or greater than) another. Symbol: < or ≤ or > or ≥, as appropriate.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents