Interval (mathematics)

In mathematics, a real interval is a specific subset of the real number line. It consists of all real numbers that lie between two fixed points, known as endpoints. A key characteristic of an interval is that it contains no gaps or missing values. If you pick any two numbers within an interval, every number between them is also part of that interval.

This property makes intervals fundamental to mathematical analysis. They are used to define continuity and are essential in the study of integrals. They even appear in the intermediate value theorem, which states that the image of an interval under a continuous function is itself an interval.

To understand how intervals work, we must look at the endpoints. These endpoints are defined by two mathematical concepts: the infimum and the supremum. The infimum is the greatest lower bound, or the largest number that is less than or equal to every element in the set. The supremum is the least upper bound, or the smallest number that is greater than or equal to every element in the set. If an interval is not empty and has no lower bound, we say the endpoint is negative infinity. If it has no upper bound, the endpoint is positive infinity.

Intervals are classified by whether they include their endpoints. An open interval, denoted with parentheses like (a, b), does not include its endpoints. This means the set contains all numbers greater than a and less than b, but not a or b themselves. A closed interval, denoted with square brackets like [a, b], includes both endpoints. This set contains all numbers from a to b, including the values of a and b.

You can also have a half-open interval. These include one endpoint but exclude the other, such as [a, b) or (a, b]. These are described as being left-open or right-open depending on which side is excluded.

We can also categorize intervals by their size and limits. A bounded interval has both a left-bound and a right-bound. These are often called finite intervals because they have a measurable length. The diameter of a bounded interval is the absolute difference between its two endpoints. You can find the center or midpoint by calculating the average of the endpoints. In contrast, an unbounded interval lacks a bound on at least one side. An interval can be left-unbounded, right-unbounded, or unbounded at both ends, such as the set of all real numbers.

There are special cases in interval theory, such as degenerate and empty intervals. A degenerate interval consists of only a single real number, written as [a, a]. Some mathematicians also include the empty set in this category. A proper interval is one that is neither empty nor degenerate, meaning it contains infinitely many elements. The entire set of real numbers is considered an interval, as is the empty set. These distinctions help mathematicians precisely define the scope of their calculations.

Historically, there has been some confusion regarding terminology. In older mathematical literature, the terms "segment" and "interval" were often used differently. Some sources used "interval" to mean an open set and "segment" to mean a closed set. Others, like Rudin in "Principles of Mathematical Analysis," used "interval" for closed sets and "segment" for open sets. Modern mathematics has largely moved past this ambiguity. Today, mathematicians prefer to use the specific terms open, closed, or half-open interval to ensure total clarity.

Intervals are not limited to real numbers; they can be defined on any totally ordered set. For example, integer intervals consist only of whole numbers between two points. In computer programming, specifically in languages like Pascal, integer intervals are used to define subrange types. This is very helpful for setting the valid bounds of an array index.

Additionally, interval arithmetic is a vital tool for handling uncertainty. Instead of using single numbers, computers can perform calculations using intervals. This provides a guaranteed enclosure for results, which helps account for rounding errors and uncertain input data.