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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.

Notes: Sequences and Series is part of One-Variable Calculus. We felt that since this topic had so many capsules associated with it, we wanted to introduce sub-topics. Also, the Number Theory collection of capsules does not correspond to a course in Course Communities, but has topics selected by the Editorial Board for Classroom Capsules and Notes.

 

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The author considers two infinite decimals, where the \(n\)th digit is the last non-zero digit of \(n!\), creating the number \(F\), and using \(n^n\), creating the number \(P\). The author shows that both \(F\) and \(P\) are irrational.

The author clarifies the wording of Cramer's rule, sidestepping a common misconception.  The Kronecker-Capelli theorem is introduced to help see Cramer's rule in a more complete context.

A typical first course on linear algebra is usually restricted to vector spaces over the real numbers and the usual positive-definite inner product.  Hence, the proof that \(\dim(\mathcal{S}) + \dim(\mathcal{S^\perp}) = \dim(\mathcal{V})\) is not presented in a way that generalizes to non-positive-definite inner products or to vector spaces over other fields.  In this note the author gives such a proof.

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.

The author gives a rule which approximates the integral representing the Gini coefficient, and which is exact for 5th degree poynomials.

 

The author gives an expression for \(\pi\) involving an infinite sequence of determinants, each representing the area of a triangle.