Interval (mathematics)

Interval (mathematics)

In mathematics, a real interval is the set of all real numbers between two fixed points. These points can be real numbers or plus or minus infinity. Intervals help us describe ranges of values.

For example, the numbers from 0 to 1, including both ends, are written as [0, 1]. This is called the unit interval. All positive real numbers are written as (0, ∞). All real numbers are written as (−∞, ∞). Even a single number, like 5, can be seen as a very small interval.

Intervals are important in mathematical analysis. They help us understand ideas like the epsilon-delta definition of continuity and the intermediate value theorem. They are also used when we calculate integrals of real functions.

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

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 with 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, meaning if you pick any two numbers in the interval, every number between them is also in the interval. 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.

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 help us describe ranges of values clearly and precisely. This 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.

Topological algebra

Intervals in mathematics connect to points on a plane. They are linked to pairs of real numbers, where one number is 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 new way to understand intervals started in 1956, linking them to hyperbolic numbers, which are a bit like complex numbers.