Inequality (mathematics)

In mathematics, an inequality is a relation that compares two expressions to show they are not equal. While an equation states that two sides are exactly the same, an inequality describes a relationship of size. It is most commonly used to compare two numbers on a number line.

This comparison allows mathematicians to define ranges of possible values rather than just a single, exact point.

There are two primary categories of inequalities: strict and non-strict. A strict inequality uses the symbols < (less than) or > (greater than). In these cases, the two values cannot be equal. For example, if a < b, then a is strictly smaller than b. Non-strict inequalities include the possibility of equality. These use the symbols ≤ (less than or equal to) or ≥ (greater than or equal to). A value that is ≤ b is often described as being "at most" b, while a value that is ≥ b is described as "at least" b.

Mathematical notation for these symbols has evolved over centuries. In 1670, John Wallis used a single horizontal bar placed above the < or > signs. Later, in 1734, Pierre Bouguer introduced symbols that featured double horizontal bars, known as "less than or equal to with double horizontal bars." Over time, mathematicians simplified Bouguer's work into the single-bar symbols we recognize today, such as ≤ or ⩽. In engineering, scientists sometimes use specialized notation to show massive differences in scale. The symbol ≪ indicates that one value is "much less than" another, while ≫ means it is "much greater than." This allows researchers to neglect tiny values when making approximations.

Inequalities follow specific logical properties when you perform arithmetic. The transitive property states that if a < b and b < c, then it must follow that a < c. When adding or subtracting a constant to both sides, the inequality remains unchanged.

However, multiplication and division require more care. Multiplying or dividing by a positive constant preserves the inequality.

If you multiply or divide by a negative constant, the inequality sign must be reversed.

Functions can also be applied to inequalities, but the result depends on the function's behavior. A monotonically increasing function, such as the natural logarithm, preserves the inequality direction.

If you apply a monotonically decreasing function to both sides, the inequality relation is reversed. This is why the rules for additive and multiplicative inverses work the way they do. These rules are specific examples of applying strictly monotonically decreasing functions to an expression.

In more advanced studies, inequalities are used to define ordered fields and partial orders. A partial order is a relation that is reflexive, antisymmetric, and transitive. A total order is a specific type of partial order where any two elements in a set can be compared. Mathematicians use these structures to organize complex sets of data. In fields like linear programming, inequalities are used to define a "feasible region."

This region represents all the possible solutions that satisfy a specific set of constraints.

Many famous inequalities exist that help bound quantities when exact answers are difficult to find. These include the Cauchy-Schwarz inequality, which relates vectors in an inner product space, and the Inequality of Arithmetic and Geometric Means. Other notable examples include Bernoulli's inequality and the Triangle inequality. These tools act as mathematical fences, providing limits that a value cannot cross. By using these bounds, mathematicians can study the behavior of complex systems even when they cannot calculate a precise number.