Interval (mathematics)

In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. Intervals are very useful in many areas of math because they help describe ranges of values.

For example, the set of real numbers from 0 to 1, including both ends, is written as [0, 1] and is called the unit interval. The set of all positive real numbers is written as (0, ∞), and the set of all real numbers is written as (−∞, ∞). Even a single number, like 5, can be thought of as a very small interval.

Intervals are important in mathematical analysis. They appear in ideas like the epsilon-delta definition of continuity and the intermediate value theorem, which tells us that a continuous function will take on every value between two points. They are also used when we calculate integrals of real functions.

We can also use intervals with other types of numbers, like integers or rational numbers, by using similar ideas and notations.

Definitions and terminology

An interval is a group of numbers that includes every number between two specific numbers. For example, the numbers from 1 to 2, or all numbers greater than 10, are intervals. Sometimes, the entire group of real numbers or no numbers at all can also be considered intervals.

Intervals can have endpoints, which are the smallest and largest numbers in the interval. These endpoints can be real numbers or can be thought of as stretching forever in one direction, using symbols like positive infinity (+) or negative infinity (−). Whether the endpoints are included in the interval depends on the type of interval. For example, an open interval does not include its endpoints, while a closed interval does. These ideas help us describe intervals clearly using special notations.

Properties

Intervals are special sets of numbers that are connected, meaning there are no gaps. If you use a continuous function—like drawing a smooth curve without lifting your pencil—the result will still be an interval. This idea is part of the intermediate value theorem.

Intervals are also convex, which means if you pick any two numbers in the interval, every number between them is also in the interval. When you include the endpoints of an interval, it stays an interval. For example, adding the endpoints to an open interval like (a, b) makes it a closed interval [a, b]. The overlap of intervals is always an interval, and combining two intervals can also form a new interval under certain conditions.

Applications

Intervals are important in many areas of mathematics. In real analysis, intervals help us understand how functions behave. For example, the integral of a function — a way to add up tiny pieces of its values — is always calculated over an interval.

Intervals also play a key role in understanding how smooth a function is, using ideas like the epsilon-delta definition of continuity. They help us describe ranges of values clearly and precisely, which is useful in many mathematical proofs and applications.

Generalizations

Intervals can be thought of in many ways beyond simple numbers between two points. In one dimension, an open interval like (a, b) acts like a small ball with a center halfway between a and b, and a radius that reaches from the center to either end. A closed interval [a, b] is like a solid ball that includes its endpoints.

When we move to more dimensions, intervals can form shapes like rectangles in two dimensions or boxes in three dimensions. These are created by taking intervals in each direction and combining them. Even more complex shapes, like convex polytopes, can be built by combining many half-spaces, similar to how intervals are built from half-bounded ranges.

Topological algebra

Intervals in mathematics can be linked to points on a plane. They are connected to pairs of real numbers, where one number is usually smaller than the other. When we look at all possible intervals together, they form a special kind of mathematical structure called a topological ring.

This structure has special properties, like having an identity element and being able to reverse certain intervals. Intervals can also be seen as symmetric around a central point. A different way to understand intervals was introduced in 1956, linking them to hyperbolic numbers, which have some similarities to complex numbers.