# Browse Classroom Capsules and Notes

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Displaying 31 - 40 of 1209

The author gives a visual proof of the Pythagorean Theorem.

The author presents a geometric interpretation of Leibnizs rule for differentiating under the integral sign, and gives an informal visual derivation of the rule.

Using basic linear algebra, the authors obtain a complete, easy to compute, winning strategy for the $$4 \times 4$$ mini Lights Out game.

The Theorem “An infinite group is cyclic when each of its nonidentity subgroups have finite index” is proved and discussed, and a test to show groups are not cyclic is ...

The author gives a proof of the title as an application of Lagranges Theorem, allowing the theorem to be presented in the first semester of an undergraduate abstract algebra course.

A visual representation of the title is presented.

The author provides a combinatorial explanation of a strikingresult from Herbert Wilf`s book “generatingfunctionology” , equating the likelyhood of certain permutations with the...

Every integer can be expressed in base $$2$$ using the set $$\{-1, 0, 1\}$$ as coefficients. Does one need to use this set, or might another set of numbers do as well? The author investigates...

The authors study period-$$3$$ orbits of the logistic function $$f_r(x)=rx(1-x)$$, and provide another derivation of the fact that $$r_0=1+\sqrt {8}$$, where $$r_0$$ is the smallest positive...

Andrew Wiles proves that Fermat's Last Theorem is false for integer exponents larger than $$2$$.  Using the Gelfond-Schneider Theorem on transcendental numbers, the author generalizes...