Inequality (mathematics)
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities:
- The notation a < b means that a is less than b.
- The notation a > b means that a is greater than b.
In contrast to strict inequalities, there are two types of inequality relations that are not strict:
- The notation a ≤ b or a ⩽ b means that a is less than or equal to b.
- The notation a ≥ b or a ⩾ b means that a is greater than or equal to b.
The notation a ≠ b means that a is not equal to b, and is sometimes considered a form of strict inequality. 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, normally by several orders of magnitude. This implies that the lesser value can be neglected with little effect on the accuracy of an approximation.
- The notation a ≪ b means that a is much less than b.
- The notation a ≫ b means that a is much greater than b.
Properties on the number line
Inequalities are governed by the following properties. All of these properties also hold if all of the non-strict inequalities are replaced by their corresponding strict inequalities and — in the case of applying a function — monotonic functions are limited to strictly monotonic functions.Converse
The relations ≤ and ≥ are each other's converse, meaning that for any real numbers a and b:Transitivity
The transitive property of inequality states that for any real numbers a, b, c:If either of the premises is a strict inequality, then the conclusion is a strict inequality:
Addition and subtraction
A common constant c may be added to or subtracted from both sides of an inequality. So, for any real numbers a, b, c:In other words, the inequality relation is preserved under addition and the real numbers are an ordered group under addition.
Multiplication and division
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.
Additive inverse
The property for the additive inverse states that for any real numbers a and b:Multiplicative inverse
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 :All of the cases for the signs of a and b can also be written in [|chained notation], as follows:
Applying a function to both sides
Any monotonically increasing function, by its definition, may be applied to both sides of an inequality without breaking the inequality relation. 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 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:
- Raising both sides of an inequality to a power n > 0, when a and b are positive real numbers:
- Taking the natural logarithm on both sides of an inequality, when a and b are positive real numbers:
Formal definitions and generalizations
- a ≤ a
- if a ≤ b and b ≤ a, then a = b
- if a ≤ b and b ≤ c, then a ≤ c
- For every a and b in P, a ≤ b or b ≤ a.
- For all a and b in P for which a < b, there is a c in P such that a < c < b.
- Every non-empty subset of P with an upper bound has a least upper bound in P.
Ordered fields
- a ≤ b implies a + c ≤ b + c;
- 0 ≤ a and 0 ≤ b implies 0 ≤ a × b.
Besides from 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.
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. 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, a_{1} ≤ a_{2} ≤... ≤ 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.
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 condiction of a zigzag poset is written as a_{1} < a_{2} > a_{3} < a_{4} > a_{5} < a_{6} >.... 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.
Sharp inequalities
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 holds, then also holds. For instance, the inequality is sharp, whereas the inequality is not sharp.Inequalities between means
There are many inequalities between means. For example, for any positive numbers a_{1}, a_{2}, …, a_{n} we have whereCauchy–Schwarz inequality
The Cauchy–Schwarz inequality states that for all vectors u and v of an inner product space it is true thatwhere is the inner product. Examples of inner products include the real and complex dot product; In Euclidean space R^{n} with the standard inner product, the Cauchy–Schwarz inequality is
Power inequalities
A "power inequality" is an inequality containing terms of the form a^{b}, where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises.Examples
- For any real x,
- If x > 0 and p > 0, then
- If x > 0, then
- If x ≥ 1, then
- If x, y, z > 0, then
- For any real distinct numbers a and b,
- If x, y > 0 and 0 < p < 1, then
- If x, y, z > 0, then
- If a, b > 0, then
- If a, b > 0, then
- If a, b, c > 0, then
- If a, b > 0, then
Well-known inequalities
- Azuma's inequality
- Bernoulli's inequality
- Bell's 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
- Kolmogorov's inequality
- Markov's inequality
- Minkowski inequality
- Nesbitt's inequality
- Pedoe's inequality
- Poincaré inequality
- Samuelson's inequality
- Triangle inequality
Complex numbers and inequalities
- if, then ;
- if and, then.
However, an operation ≤ can be defined so as to satisfy only the first property. Sometimes the lexicographical order definition is used:
- , if
- *, or
- * and
Vector inequalities
Inequality relationships similar to those defined above can also be defined for column vectors. If we let the vectors , we can define the following relationships:- , if for.
- , if for.
- , if for and.
- , if for.
The trichotomy property is not valid for vector relationships. For example, when and, there exists no valid inequality relationship between these two vectors. Also, a multiplicative inverse would need to be defined on a vector before this property could be considered. However, for the rest of the aforementioned properties, a parallel property for vector inequalities exists.
Systems of inequalities
Systems of linear inequalities can be simplified by Fourier–Motzkin elimination.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.