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In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. [1] It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than and greater than.
There are several different notations used to represent different kinds of inequalities:
In either case, a is not equal to b. These relations are known as strict inequalities, [1] meaning that a is strictly less than or strictly greater than b. Equality is excluded.
In contrast to strict inequalities, there are two types of inequality relations that are not strict:
In the 17th and 18th centuries, personal notations or typewriting signs were used to signal inequalities. [2] For example, In 1670, John Wallis used a single horizontal bar above rather than below the < and >. Later in 1734, ≦ and ≧, known as "less than (greater-than) over equal to" or "less than (greater than) or equal to with double horizontal bars", first appeared in Pierre Bouguer's work . [3] After that, mathematicians simplified Bouguer's symbol to "less than (greater than) or equal to with one horizontal bar" (≤), or "less than (greater than) or slanted equal to" (⩽).
The relation not greater than can also be represented by the symbol for "greater than" bisected by a slash, "not". The same is true for not less than,
The notation a ≠ b means that a is not equal to b; this inequation sometimes is considered a form of strict inequality. [4] It does not say that one is greater than the other; it does not even require a and b to be member of an ordered set.
In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another, [5] normally by several orders of magnitude.
This implies that the lesser value can be neglected with little effect on the accuracy of an approximation (such as the case of ultrarelativistic limit in physics).
In all of the cases above, any two symbols mirroring each other are symmetrical; a < b and b > a are equivalent, etc.
Inequalities are governed by the following properties. All of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities (< and >) and — in the case of applying a function — monotonic functions are limited to strictly monotonic functions.
The relations ≤ and ≥ are each other's converse, meaning that for any real numbers a and b:
The transitive property of inequality states that for any real numbers a, b, c: [8]
If either of the premises is a strict inequality, then the conclusion is a strict inequality:
A common constant c may be added to or subtracted from both sides of an inequality. [4] So, for any real numbers a, b, c:
In other words, the inequality relation is preserved under addition (or subtraction) and the real numbers are an ordered group under addition.
The properties that deal with multiplication and division state that for any real numbers, a, b and non-zero c:
In other words, the inequality relation is preserved under multiplication and division with positive constant, but is reversed when a negative constant is involved. More generally, this applies for an ordered field. For more information, see § Ordered fields .
The property for the additive inverse states that for any real numbers a and b:
If both numbers are positive, then the inequality relation between the multiplicative inverses is opposite of that between the original numbers. More specifically, for any non-zero real numbers a and b that are both positive (or both negative):
All of the cases for the signs of a and b can also be written in chained notation, as follows:
Any monotonically increasing function, by its definition, [9] may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in the domain of that function). However, applying a monotonically decreasing function to both sides of an inequality means the inequality relation would be reversed. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function.
If the inequality is strict (a < b, a > b) and the function is strictly monotonic, then the inequality remains strict. If only one of these conditions is strict, then the resultant inequality is non-strict. In fact, the rules for additive and multiplicative inverses are both examples of applying a strictly monotonically decreasing function.
A few examples of this rule are:
A (non-strict) partial order is a binary relation ≤ over a set P which is reflexive, antisymmetric, and transitive. [10] That is, for all a, b, and c in P, it must satisfy the three following clauses:
A set with a partial order is called a partially ordered set . [11] Those are the very basic axioms that every kind of order has to satisfy. Other axioms that exist for other definitions of orders on a set P include:
If (F, +, ×) is a field and ≤ is a total order on F, then (F, +, ×, ≤) is called an ordered field if and only if:
Both and are ordered fields, but ≤ cannot be defined in order to make an ordered field, [12] because −1 is the square of i and would therefore be positive.
Besides being an ordered field, R also has the Least-upper-bound property. In fact, R can be defined as the only ordered field with that quality. [13]
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. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative. Hence, for example, 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, it is not possible to isolate x in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved 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. For example, the defining condition of a zigzag poset is written as a1 < a2 > a3 < a4 > a5 < a6 > ... . Mixed chained notation is used more often with compatible relations, like <, =, ≤. For instance, a < b = c ≤ d means that a < b, b = c, and c ≤ d. This notation exists in a few programming languages such as Python. In contrast, in programming languages that provide an ordering on the type of comparison results, such as C, even homogeneous chains may have a completely different meaning. [14]
An inequality is said to be sharp if it cannot be relaxed and still be valid in general. Formally, a universally quantified inequality φ is called sharp if, for every valid universally quantified inequality ψ, if ψ ⇒ φ holds, then ψ ⇔ φ also holds. For instance, the inequality ∀ a ∈ R. a2 ≥ 0 is sharp, whereas the inequality ∀a ∈ R. a2 ≥ −1 is not sharp.[ citation needed ]
There are many inequalities between means. For example, for any positive numbers a1, a2, ..., an we have H ≤ G ≤ A ≤ Q, where they represent the following means of the sequence:
The Cauchy–Schwarz inequality states that for all vectors u and v of an inner product space it is true that where is the inner product. Examples of inner products include the real and complex dot product; In Euclidean space Rn with the standard inner product, the Cauchy–Schwarz inequality is
A power inequality is an inequality containing terms of the form ab, where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises.
Examples:
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:
The set of complex numbers with its operations of addition and multiplication is a field, but it is impossible to define any relation ≤ so that becomes an ordered field. To make an ordered field, it would have to satisfy the following two properties:
Because ≤ is a total order, for any number a, either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ −a). In either case 0 ≤ a2; this means that i2 > 0 and 12 > 0; so −1 > 0 and 1 > 0, which means (−1 + 1) > 0; contradiction.
However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if a ≤ b, then a + c ≤ b + c"). Sometimes the lexicographical order definition is used:
It can easily be proven that for this definition a ≤ b implies a + c ≤ b + c.
Inequality relationships similar to those defined above can also be defined for column vectors. If we let the vectors (meaning that and , where and are real numbers for ), we can define the following relationships:
Similarly, we can define relationships for , , and . This notation is consistent with that used by Matthias Ehrgott in Multicriteria Optimization (see References).
The trichotomy property (as stated above) is not valid for vector relationships. For example, when and , there exists no valid inequality relationship between these two vectors. However, for the rest of the aforementioned properties, a parallel property for vector inequalities exists.
Systems of linear inequalities can be simplified by Fourier–Motzkin elimination. [17]
The cylindrical algebraic decomposition is an algorithm that allows testing whether a system of polynomial equations and inequalities has solutions, and, if solutions exist, describing them. The complexity of this algorithm is doubly exponential in the number of variables. It is an active research domain to design algorithms that are more efficient in specific cases.
In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of without regard to its sign. Namely, if is a positive number, and if is negative, and . For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n; this coefficient can be computed by the multiplicative formula
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word partial is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable.
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann.
In mathematics, a monotonic function is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
In mathematics, a (real) interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. An interval can contain neither endpoint, either endpoint, or both endpoints.
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.
In probability theory, Chebyshev's inequality provides an upper bound on the probability of deviation of a random variable from its mean. More specifically, the probability that a random variable deviates from its mean by more than is at most , where is any positive constant and is the standard deviation.
In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces.
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph is a convex set. In simple terms, a convex function graph is shaped like a cup , while a concave function's graph is shaped like a cap .
In mathematics, a concave function is one for which the value at any convex combination of elements in the domain is greater than or equal to the convex combination of the values at the endpoints. Equivalently, a concave function is any function for which the hypograph is convex. The class of concave functions is in a sense the opposite of the class of convex functions. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations.
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 real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same.
In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means.
In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. This calculation is applicable in engineering problems involving heat and mass transfer.
Carleman's inequality is an inequality in mathematics, named after Torsten Carleman, who proved it in 1923 and used it to prove the Denjoy–Carleman theorem on quasi-analytic classes.
In probability theory and statistics, a stochastic order quantifies the concept of one random variable being "bigger" than another. These are usually partial orders, so that one random variable may be neither stochastically greater than, less than, nor equal to another random variable . Many different orders exist, which have different applications.
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called inradius and circumradius, and incenter and circumcenter respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and circumscribed quadrilateral. It has also rarely been called a double circle quadrilateral and double scribed quadrilateral.
In mathematics, Young's inequality for products is a mathematical inequality about the product of two numbers. The inequality is named after William Henry Young and should not be confused with Young's convolution inequality.
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