Nested intervals

For example, the ancient Babylonians discovered a method for computing square roots of numbers.

As stated in the introduction, historic users of mathematics discovered the nesting of intervals and closely related algorithms as methods for specific calculations.

Some variations and modern interpretations of these ancient techniques will be introduced here: When trying to find the square root of a number

If the midpoint is smaller, one can set it as the lower bound of the next interval

, and if the midpoint is larger, one can set it as the upper bound of the next interval.

and calculating the reciprocal after the desired level of precision has been acquired.

Thus, using this interval, one can continue to the next step of the algorithm by calculating the midpoint of the interval, determining whether the square of the midpoint is greater than or less than 19, and setting the boundaries of the next interval accordingly before repeating the process: The Babylonian method uses an even more efficient algorithm that yields accurate approximations of

As shown in the image, lower and upper bounds for the circumference of a circle can be obtained with inscribed and circumscribed regular polygons.

Around 250 BCE Archimedes of Syracuse started with regular hexagons, whose side lengths (and therefore circumference) can be directly calculated from the circle diameter.

Around the year 1600 CE, Archimedes' method was still the gold standard for calculating Pi and was used by Dutch mathematician Ludolph van Ceulen, to compute more than thirty digits of

Early uses of sequences of nested intervals (or can be described as such with modern mathematics), can be found in the predecessors of calculus (differentiation and integration).

E.g. the bisection method can be used for calculating the roots of continuous functions.

In contrast to mathematically infinite sequences, an applied computational algorithm terminates at some point, when the desired zero has been found or sufficiently well approximated.

In mathematical analysis, nested intervals provide one method of axiomatically introducing the real numbers as the completion of the rational numbers, being a necessity for discussing the concepts of continuity and differentiability.

Historically, Isaac Newton's and Gottfried Wilhelm Leibniz's discovery of differential and integral calculus from the late 1600s has posed a huge challenge for mathematicians trying to prove their methods rigorously; despite their success in physics, engineering and other sciences.

The axiomatic description of nested intervals (or an equivalent axiom) has become an important foundation for the modern understanding of calculus.

The second property formalizes the notion, that interval sizes get arbitrarily small; meaning, that for an arbitrary constant

It is also worth noting that property 1 immediately implies that every interval with an index

Note that some authors refer to such interval-sequences, satisfying both properties above, as shrinking nested intervals.

In formal notation this axiom guarantees, that The intersection of each sequence

This contradicts property 2 from the definition of nested intervals; therefore, the intersection can contain at most one number

By generalizing the algorithm shown above for square roots, one can prove that in the real numbers, the equation

Comparing to the section above, one achieves a sequence of nested intervals for the

As was seen, the existence of suprema and infima of bounded sets is a consequence of the completeness of

This in turn allows for a proof of the completeness property above, showing their equivalence.

The possibility of an empty intersection can be illustrated by looking at a sequence of open intervals

By De Morgan's laws, the complement of the intersection is a union of two disjoint open sets.

This shows that the intersection of (even an uncountable number of) nested, closed, and bounded intervals is nonempty.

In two dimensions there is a similar result: nested closed disks in the plane must have a common intersection.

This result was shown by Hermann Weyl to classify the singular behaviour of certain differential equations.