inequality

in·e·qual·i·ty

I0120400 (ĭn′ĭ-kwŏl′ĭ-tē)n. pl. in·e·qual·i·ties 1. a. The condition of being unequal.b. An instance of being unequal.2. a. Lack of equality, as of opportunity, treatment, or status.b. Social or economic disparity: the growing inequality between rich and poor.3. Lack of smoothness or regularity; unevenness.4. Variability; changeability.5. Mathematics An algebraic relation showing that a quantity is greater than or less than another quantity.6. Astronomy A deviation from uniformity in the apparent motion of a celestial body.

inequality

(ˌɪnɪˈkwɒlɪtɪ) n, pl -ties1. the state or quality of being unequal; disparity2. an instance of disparity3. lack of smoothness or regularity4. social or economic disparity5. (Mathematics) maths a. a statement indicating that the value of one quantity or expression is not equal to another, as in x ≠ yb. a relationship between real numbers involving inequality: x may be greater than y, denoted by x>y, or less than y, denoted by x<y6. (Astronomy) astronomy a departure from uniform orbital motion

in•e•qual•i•ty

(ˌɪn ɪˈkwɒl ɪ ti)

n., pl. -ties. 1. the condition of being unequal; lack of equality; disparity. 2. injustice; partiality. 3. unevenness, as of surface. 4. an instance of unevenness. 5. variableness, as of climate. 6. a. any component part of the departure from uniformity in astronomical phenomena, esp. in orbital motion. b. the amount of such a departure. 7. a statement that two quantities are unequal, indicated by the symbol ≠; alternatively, by the symbol , signifying that the quantity preceding the symbol is greater than that following. [1375–1425; late Middle English < Latin]

Thesaurus

Noun

inequality - lack of equality; "the growing inequality between rich and poor"difference - the quality of being unlike or dissimilar; "there are many differences between jazz and rock"nonequivalence - not interchangeabledisparity - inequality or difference in some respectunevenness - the quality of being unbalancedequality - the quality of being the same in quantity or measure or value or status

inequality

noun disparity, prejudice, difference, bias, diversity, irregularity, unevenness, lack of balance, disproportion, imparity, preferentiality corruption and social inequalityQuotations
"All animals are equal but some animals are more equal than others" [George Orwell Animal Farm]
"Whatever may be the general endeavor of a community to render its members equal and alike, the personal pride of individuals will always seek to rise above the line, and to form somewhere an inequality to their own advantage" [Alexis de Tocqueville Democracy in America]

inequality

noun1. The condition or fact of being unequal, as in age, rank, or degree:disparity, disproportion, disproportionateness.2. Lack of smoothness or regularity:asymmetry, crookedness, irregularity, jaggedness, roughness, unevenness.

Translations

不平等

inequality

(iniˈkwoləti) noun (a case of) the existence of differences in size, value etc between two or more objects etc. There is bound to be inequality between a manager's salary and a workman's wages.不平等

inequality

inequality,

in mathematics, statement that a mathematical expression is less than or greater than some other expression; an inequality is not as specific as an equationequation,
in mathematics, a statement, usually written in symbols, that states the equality of two quantities or algebraic expressions, e.g., x+3=5. The quantity x
..... Click the link for more information. , but it does contain information about the expressions involved. The symbols < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) are used in place of the equals sign in expressions of inequalities. As in the case of equations, inequalities can be transformed in various ways. The direction of the inequality remains unchanged if some number is added to both sides or subtracted from both sides or if both sides are multiplied or divided by some positive number; e.g., subtracting 10 from both sides of the inequality x < 8 gives x − 10 < −2, and multiplying the inequality by 2 gives 2x < 16. Multiplication or division by a negative number reverses the sign of the inequality; e.g., if −2x < 8, then dividing both sides by −2 results in the inequality x > −4.

inequality

An irregularity in the orbital motion of a celestial body. The inequalities of the Moon's motion are periodic terms whose sum gives the variation of either the spherical coordinates or the osculating elements of the lunar orbit. The principal inequalities of the Moon's motion are evection and variation.

inequality

See EQUALITY.

Inequality

(mathematics), a relation between two numbers or quantities indicating which of them is greater or smaller. An inequality is denoted by the symbol 1 and 1 < 2 state the same thing, namely, 2 is greater than 1 and 1 is less than 2. Sometimes we make use of multiple inequalities as in a < b < c. In order to express the fact that of two numbers a and b, the first is greater than or equal to the second, we write a ≥ b (or b ≤ a) and read “a is greater than or equal to b” (or “a is less than or equal to b“ or simply “a is not less than b“ (or “b is not greater than a”). The notation α ≠ b signifies that the numbers a and b are unequal but does not indicate which is larger. All these relations are termed inequalities.

Inequalities have many properties in common with equalities. Thus, an inequality remains valid if the same number is added to or subtracted from both sides. We can similarly multiply both sides of an inequality by a positive number. However, if both sides are multiplied by a negative number, then the sense of the inequality is reversed, that is, the symbol > is replaced by . From the inequalities A < B and C < D follow the inequalities A + C < B + D and A —D < B —C; in other words, inequalities of the same sense (A < B and C < D) may be added term by term, and inequalities of opposite sense (A < B and D > C) may be subtracted term by term. If the numbers A, B. C, and D are positive, then the inequalities A < B and C < D also imply AC < BD and A/D < B/C; in other words, inequalities of the same sense (between positive numbers) may be multiplied term by term, and those of opposite sense may be divided term by term.

Inequalities containing quantities that can assume different numerical values may be true for some values of these quantities and false for others. Thus, the inequality x² —4x + 3 > 0 is true for x = 4 and false for x = 2. To solve such inequalities is to determine the limits within which the quantities entering into the inequalities must be taken in order for the inequalities to hold. Thus, by rewriting the inequality x² —4x + 3 > 0 in the form (x —1) (x —3) > 0, we see that the latter holds for all x satisfying either one of the inequalities x < 1, x > 3, and these yield the solution of the original inequality.

We now give a few examples of important inequalities.

(1) “Triangle” inequality. For any real or complex numbers a₁, a₂, . . . ,a_n,

ǀa₁ + a₂ + . . . a_nǀ ≤ ǀa₁ǀ + ǀa₂ǀ + . . . + ǀa_nǀ

(2) Inequality for means. The most famous inequality relates the harmonic mean, the geometric mean, the arithmetic mean, and the root-mean-square:

The numbers a₁, a₂, . . . , a_n are assumed to be positive.

(3) Linear inequalities. Consider the system of inequalities

a_i1x₁ + a_i2x₂ + . . . + a_in x_n ≥ b_i

(i = 1, 2, . .. ,m)

The totality of solutions of this system is a convex polyhedron in n-dimensional space (x₁,x₂, . . . x_n). The task of the theory of linear inequalities consists of studying the properties of this polyhedron. Certain problems in the theory of linear inequalities are closely related to the theory of best approximations, which was created by P. L. Chebyshev.

(See also, , , , and .)

Inequalities are very important in many branches of mathematics. Diophantine approximations, an entire branch of number theory, are completely based on inequalities. Analytic number theory often operates with inequalities. The axiomatic development of inequalities is given in algebra. Linear inequalities play a large role in the theory of linear programming. In geometry, inequalities are constantly encountered in the theory of convex bodies and in isoperimetric problems. In probability theory, many laws are formulated in terms of inequalities (for example, the Chebyshev inequality). Differential inequalities are used in the theory of differential equations (the Chaplygin method). In the theory of functions, various inequalities are constantly used for derivatives of polynomials and trigonometric polynomials. In functional analysis, the definition of norm in a function space requires that it satisfy the triangle inequality

ǀǀx + y ǀǀ ≤ ǀǀxǀǀ + ǀǀyǀǀ

Many classical inequalities virtually define or estimate the norm of a linear functional or a linear operator in some space.

REFERENCES

Korovkin, P. P. Neravenstva, 3rd ed. Moscow, 1966. Hardy, G. H. , S. E. Littlewood, and G. Pólya. Neravenstva. Moscow, 1948. (Translated from English.)

inequality

[‚in·i′kwäl·əd·ē] (mathematics) A statement that one quantity is less than, less than or equal to, greater than, or greater than or equal to another quantity.

inequality

1. Mathsa. a statement indicating that the value of one quantity or expression is not equal to another, as in x ≠ y b. a relationship between real numbers involving inequality: x may be greater than y, denoted by x > y, or less than y, denoted by x < y 2. Astronomy a departure from uniform orbital motion

inequality

in·e·qual·i·ty

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REFERENCES

inequality

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Synonyms for inequality

noun disparity

Synonyms

Synonyms for inequality

noun the condition or fact of being unequal, as in age, rank, or degree

Synonyms

noun lack of smoothness or regularity

Synonyms

Antonyms for inequality

noun lack of equality

Related Words

Antonyms