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)
Therefore a is not equal to b. These relations are known as strict inequality; in contrast
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
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.
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 ≥).
The trichotomy property states:
The transitivity of inequalities states:
The inequality relations are inverse relations:
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
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
The properties for the additive inverse state:
The properties for the multiplicative inverse state:
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:
It will never become strictly unequal, since, for example, 3 ≤ 3 does not imply that 3 < 3.
Note that both ,+,*,≤ and ,+,*,≤ 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 ,+,*,≤ 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
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 ≤ . . . ≤ an means that ai ≤ ai+1 for i = 1, 2, . . . , n − 1. By transitivity, this condition is equivalent to ai ≤ aj 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 < 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. 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
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
There are many inequalities between means. For example, for any positive numbers a1, a2, . . . , an
Sometimes with notation "power inequality" understand inequalities which contain ab 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.
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:
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  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.
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 ,+,*,≤ 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 ,+,*,≤ were an ordered field, it has to satisfy the following two properties:
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 i2 > 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:
It can easily be proven that for this definition a ≤ b then a + c ≤ b + c