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Classroom Capsules and Notes

Capsules By Courses. We are organizing the capsules into courses, when possible using the same topics as are used in Course Communities. So far we have organized capsules for the following courses:

You may select topics within each course.


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This capsule applied the concept of cross ratio in complex analysis to derive the Pythagorean Theorem. It appears to be the first in the literature to use this approach to prove the famous theorem.

In this note, the Cauchy-Schwarz inequality is derived from Pearson's inequality. It follows that the two are equivalent.

Geometric construction of Pythagorean Triples through the Double Angle Formula

An alternative way to find the improper integral \(\int_{0}^{\infty} e^{-x^2} dx\)

Elementary derivation for probability of relative primality

Generalizations of the problem: A goat is tethered to the edge of a disc shaped field of radius \(r\). The goat's rope is of length \(kr\). If the field is \(n\)-dimensional, what fraction of it can the goat reach, and what happens as \(n\) approaches infinity?