Inequality

Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ...

, an inequality is a statement about the relative size or order of two objects. *The notation means that a is less than b and *The notation means that a is greater than b. These relations are known as strict inequality; in contrast * means that a is less than or equal to b and * means that a is greater than or equal to b. An additional use of the notation is to show that one quantity is much greater than another, normally by several orders of magnitude. *The notation a >> b means that a is much greater than b.

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In mathematics

Mathematics

Mathematics is the discipline that deals with concepts such as quantity [i], structure [i], space [i] a ...

, an inequality is a statement about the relative size or order of two objects.

The notation means that a is less than b and
The notation means that a is greater than b.

These relations are known as strict inequality; in contrast

means that a is less than or equal to b and
means that a is greater than or equal to b.

An additional use of the notation is to show that one quantity is much greater than another, normally by several orders of magnitude.

The notation a >> 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. The sense of an inequality is not changed if both sides are increased or decreased by the same number, or if both sides are multiplied or divided by a positive number; the sense of an inequality is reversed if both members are multiplied or divided by a negative number.

Properties

Inequalities are governed by the following properties:

Trichotomy

The trichotomy property states:

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

a < b
a = b
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 mirror images in the sense that:

For any real numbers, a and b:

If a > b then b < a
If a < b then b > a

Addition and subtraction

The properties which deal with addition

Addition

Addition is the mathematical operation [i] of increasing one amount by another. ...

and subtraction

Subtraction

Subtraction is one of the four basic arithmetic [i] operations; it is essentially the opposite of addition [i] ...

states:

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

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 a � c > b � c and a / c > b / c
If c is positive and a < b, then a � c < b � c and a / c < b / c
If c is negative and a > b, then a � c < b � c and a / c < b / c
If c is negative and a < b, then a � c > b � c and a / c > b / c

Applying a function to both sides

Any strictly monotonically increasing function may be applied to both sides of an inequality and it will still hold. Applying a strictly monotonically decreasing function to both sides of an inequality means the opposite inequality now holds.

More subtly, if you have a non-strict inequality then:

Applying a monotonically increasing function will preserve the relation
Applying a strictly increasing function will make the relation strict
Applying a monotonically decreasing function reverses the relation, but keeps it nonstrict
Applying a strictly decreasing function reverses the relation and makes it strict

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, eg. 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.

Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction

Logical conjunction

In logic [i] and mathematics [i], logical conjunction is a two-place logical operation [i] that results...

of the inequalities between adjacent terms. For instance, a < b > 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 language

Programming language

A programming language is an artificial language [i] that can be used to control [i] ...

s such as Python

Python (programming language)

Python is a programming language [i] created by Guido van Rossum [i] in 1990 [i]. ...

.

Power inequalities

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

Examples

For any real distinct numbers and
If and then
For any positive , and
For any real positive and . This result was generalized by R. Ozols in 2002 who proved that for any real positive numbers ..., is true inequality .

Well-known inequalities

See also list of inequalities.

Mathematician

Mathematician

A mathematician is a person whose primary area of study and research is the field of mathematics [i]. ...

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
Boole's inequality
Cauchy�Schwarz inequality
Chebyshev's inequality
Chernoff's inequality
Cram�r-Rao inequality
Hoeffding's inequality
H�lder's inequality
Inequality of arithmetic and geometric means
Jensen's inequality Jensen's inequality

In mathematics [i], Jensen's inequality, named after the Danish mathematician Johan Jensen [i], relates ...
Markov's inequality
Minkowski inequality
Nesbitt's inequality
Pedoe's inequality
Triangle inequality

Mnemonics for students

Young students sometimes confuse the less-than and greater-than signs, which are mirror images of one another. A commonly taught mnemonic is that the sign represents a hungry alligator

Alligator

An alligator is a crocodilian [i] in the genus [i] Alligator of the family [i] Alligatoridae [i] ...

that is trying to eat the larger number; thus, it opens towards 8 in both 3 < 8 and 8 > 3. 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 . This is actually where the greater than and less than signs came from.