# Browse Classroom Capsules and Notes

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Displaying 1071 - 1080 of 1214

The author offers two examples that illustrate important central ideas in introductory linear algebra (independent or dependent vectors; invertible or singular matrices) which may aid students in...

A series of rational numbers whose absolute values converge to a rational number while the series itself converges to an irrational number is presented.

The author shows that a square inscribed in a semicircle has $$2/5$$ the area of a square inscribed in a circle of the same radius.

The author proposes two extensions of the Monte Hall problem, with solutions involving the numbers $$\pi$$ and $$e$$, respectively.
A combinatorial proof of the sum of the cubes of the first $$n$$ integers is presented, by counting edges in complete bipartite graphs.
The author proves visually that if $$a$$ and $$b$$ are the legs and h the altitude to the hypotenuse $$c$$ of a right triangle, then $$(1/a)^2 + (1/b)^2 = (1/h)^2$$.
Arguments for certain sums of sines and cosines are given.
This article gives an explanation why the solution to a number partitioning problem would be $$3$$. The answer is that $$3$$ is close to $$e$$.
The article attempts to explain why exponential functions were chosen to define hyperbolic functions.
An elementary proof for the generalized Schwarz inequality.