Interval (mathematics)

Imagine a long line of numbers.

You can pick two numbers to mark a space. This space holds all the numbers in between. It has no gaps. This is called an interval.

Some intervals have two ends. We call these endpoints. You can choose to include the ends. You can also leave them out. This is like choosing if a fence includes the posts.

An interval can be very small. It can even be just one single number. Some intervals can be very large. They can go on forever without an end.

These groups of numbers help us in math. They help us measure and group things together.

Imagine a long line of numbers. You can pick two numbers to mark a space. This space holds every number in between. It has no gaps. In math, we call this an interval.

The two numbers you pick are called endpoints. You can choose to include these ends or leave them out. If you include both ends, it is a closed interval. If you leave both ends out, it is an open interval. You can also have a half-open interval. This means you include one end but not the other.

Some intervals are very small. An interval can be just one single number. These are called degenerate intervals. Other intervals can be very large. Some go on forever. We say these are unbounded. They might go toward positive infinity or negative infinity.

Intervals are used in many ways. They help us with math rules. They also help us work with numbers that are not exact. This can help us find a safe range for an answer. Even whole numbers can form their own kind of interval.

Imagine a long, straight line filled with every possible number. An interval is a specific section of this line. It contains every single number sitting between two chosen points. There are no gaps or missing pieces in an interval. These two chosen points are called endpoints. An interval can be very small or very large. It can even be a single number.

How we treat the endpoints changes the name of the interval. If you include both endpoints, it is a closed interval. You might use square brackets to show this. If you leave both endpoints out, it is an open interval. We use parentheses for this type of interval. You can also have a half-open interval. This means you include one endpoint but not the other.

Some intervals have limits on both sides. We call these bounded intervals. They have a set size or length. You can find the middle of these by looking at the center point. Other intervals go on forever in one or both directions. These are called unbounded intervals. One end might point toward positive infinity. The other end might point toward negative infinity.

Mathematicians use intervals to solve many different kinds of problems. They use them in a way called interval arithmetic. This helps when numbers are not perfectly exact. It provides a safe range for a result. This is helpful when there are rounding errors. Intervals also appear in many important math rules. For example, they are used to define how functions behave.

You can see the idea of an interval in your daily life. Think about a ruler measuring a piece of string. The string starts at one mark and ends at another. Every tiny point on that string is part of the interval. You can also use intervals with whole numbers. These are called integer intervals. They include only the whole numbers between two points. This is very common in computer programming today.

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.