Inequality (mathematics)

There are several different notations used to represent different kinds of inequalities: In either case, a is not equal to b.

[2] For example, In 1670, John Wallis used a single horizontal bar above rather than below the < and >.

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.

[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 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).

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.

A strict partial order is a relation < that satisfies where ≮ means that < does not hold.

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.

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.

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 example, for any positive numbers a1, a2, ..., an we have 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

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

[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.