![Illustration nested intervals](https://upload.wikimedia.org/wikipedia/commons/4/48/Illustration_nested_intervals.svg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a sequence of nested intervals can be intuitively understood as an ordered collection of
intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets
* A statistical level of measurement
* Interval e ...
on the
real number line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
with
natural numbers
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met:
# Every interval in the sequence is contained in the previous one (
is always a subset of
).
# The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold
after a certain index
).
In other words, the left bound of the interval
can only increase (
), and the right bound can only decrease (
).
Historically - long before anyone defined nested intervals in a textbook - people implicitly constructed such nestings for concrete calculation purposes. For example, the ancient
Babylonians
Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. ...
discovered a
method for computing square roots of numbers. In contrast, the famed
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
constructed sequences of polygons, that inscribed and surcumscribed a unit
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
, in order to get a lower and upper bound for the circles circumference - which is the
circle number Pi (
).
The central question to be posed is the nature of the
intersection over all the natural numbers, or, put differently, the set of numbers, that are found in every Interval
(thus, for all
). In modern mathematics, nested intervals are used as a construction method for the real numbers (in order to
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of rational numbers).
Historic motivation
As stated in the introduction, historic users of mathematics discovered the nesting of intervals and closely related
algorithms
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing c ...
as methods for specific calculations. Some variations and modern interpretations of these ancient techniques will be introduced here:
Computation of square roots
One intuitive algorithm is so easy to understand, that it could well be found by engaged high school students. When trying to find the square root of a number
, one can be certain that
, which gives the first interval