Interval (mathematics)

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In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying 0 \le x \le 1 is an interval which contains 0 and 1, as well as all numbers between them. Other examples of intervals are the set of all real numbers \R, the set of all negative real numbers, and the empty set.

Real intervals play an important role in the theory of integration, because they are the simplest sets whose "size" or "measure" or "length" is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure.

Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical approximations, and arithmetic roundoff.

Contents

Notations for intervals

The interval of numbers between a and b, including a and b, is often denoted [a,b]. The two numbers are called the endpoints of the interval.

Excluding the endpoints

To indicate that one of the endpoints is to be excluded from the set, many writers substitute a parenthesis for the corresponding square bracket. Thus, in set builder notation,

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