# 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.

## Featured Items

##### Pythagoras by the Cross Ratio

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.

##### The Pearson and Cauchy-Schwarz Inequalities

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

##### Proof without Words: Pythagorean Triples via Double Angle Formulas

Geometric construction of Pythagorean Triples through the Double Angle Formula

##### A Note on the Gaussian Integral

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

##### The Probability that $(a,b) = 1$

Elementary derivation for probability of relative primality

##### Return of the Grazing Goat in '$n$' Dimensions

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?