# Browse Classroom Capsules and Notes

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Displaying 1061 - 1070 of 1209

The author states a "Master Theorem" from which, exploiting concavity, he deduces several classical inequalities.

The creation of 'extriangles' based on a given triangle is iterated, giving rise to four quadrilaterals each with area five times the area of the original triangle.

This note places a primality test developed by Dennis P. Walsh ("A curious test for primes," this Magazine 80(4), October 2007, 302-303) in context by providing increasingly simpler test...
The author shows that two ostensibly different generalizations of the Fundamental Theorem of Algebra are equivalent.
This paper offers a visual illustration of the fact that every octagonal number is the difference of two squares.

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.