Lesson plan
Nested intervals
In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals I_n on the real number line with natural numbers n=1,2,3,\dots 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 ( I_{n+1} is always a subset of I_n ). In other words, the left bound of the interval I_n can only increase ( a_{n+1}\geq a_n ), and the right bound can only decrease ( b_{n+1}\leq b_n ). # The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold \varepsilon after a certain index N ). Historically - long before anyone defined nested intervals in a textbook - people implicitly constructed such nestings for concrete calculation purposes. For example, the ancient Babylonians discovered a method for computing square roots of numbers. In contrast, the famed Archimedes constructed sequences of polygons, that inscribed and circumscribed a unit circle, in order to get a lower and upper bound for the circles circumference - which is the circle number Pi ( \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 I_n (thus, for all n\in\mathbb{N} ).…
Article summary
In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals I_n on the real number line with natural numbers n=1,2,3,\dots 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 ( I_{n+1} is always a subset of I_n ). In other words, the left bound of the interval I_n can only increase ( a_{n+1}\geq a_n ), and the right bound can only decrease ( b_{n+1}\leq b_n ). # The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold \varepsilon after a certain index N ). Historically - long before anyone defined nested intervals in a textbook - people implicitly constructed such nestings for concrete calculation purposes. For example, the ancient Babylonians discovered a method for computing square roots of numbers. In contrast, the famed Archimedes constructed sequences of polygons, that inscribed and circumscribed a unit circle, in order to get a lower and upper bound for the circles circumference - which is the circle number Pi ( \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 I_n (thus, for all n\in\mathbb{N} ).…
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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