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. The main types of inequality are less than and greater than (denoted by < and >, respectively the less-than and greater-than signs).

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, 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. 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 . 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 ${\displaystyle a\ngtr b,}$ the symbol for "greater than" bisected by a slash, "not". The same is true for not less than, ${\displaystyle a\nless b.}$

The notation a ≠ b means that a is not equal to b; this inequation sometimes is 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 a 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.