Inequality

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For the socioeconomic sense, see social inequality.

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In mathematics, an inequality is a statement about the relative size or order of two objects. (See also: equality)

• The notation <math>a < b \!\ </math> means that a is less than b and
• The notation <math>a > b \!\ </math> means that a is greater than b.

These relations are known as strict inequality; in contrast

• <math>a \le b</math> means that a is less than or equal to b and
• <math>a \ge b</math> means that a is greater than or equal to 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.

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 that is trying to eat the larger number; thus, it opens towards 8 in both 3 < 8 and 8 > 3.[1] Although widely used, some believe this mnemonic is misleading because young students can overlook the sign of a number, leading to false statements such as -3 < -8, though others argue that any misconception there isn't the fault of the mnemonic but of the student for not realizing which number is really greater than which, and since -3 is greater than -8 the mnemonic analogy is still accurate.

The notation a >> b means that a is "much greater than" b (although in computer science, the symbol may denote bitwise shift operator). What this means exactly can vary, meaning anything from a factor of 100 difference to a ten order of magnitude difference. It is used in relation to equations in which a much greater value will cause the output of the equation to converge on a certain result. Often results involving "much greater than" can be formalized using limits.

Contents 1 Properties 1.1 Trichotomy 1.2 Transitivity 1.3 Reversal 1.4 Addition and subtraction 1.5 Multiplication and division 1.6 Applying a function to both sides 2 Chained notation 3 Well-known inequalities 4 See also 5 References

Properties

Inequalities are governed by the following properties:

Trichotomy

The trichotomy property states:

• For any real numbers, "a" and "b", only 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 and subtraction 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", and "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.

Chained notation

The notation a < b < c stands for a < b and b < c 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 the transitivity property, 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 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 languages such as Python.

Well-known inequalities

See also list of 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
• Bernoulli inequality
• Boole's inequality
• Cauchy-Schwarz inequality
• Chebyshev's inequality
• Chernoff's inequality
• Cramér-Rao inequality
• Hoeffding's inequality
• Hölder's inequality
• Inequality of arithmetic and geometric means
• Jensen's inequality
• Markov's inequality
• Minkowski's inequality
• Nesbitt's inequality
• Pedoe's inequality
• Triangle inequality

See also

• Binary relation
• Partially ordered set
• Inequation

References

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