# 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.

## Featured Items

##### Proof Without Words: Logarithm of a Number and its Reciprocal.

A visual representation of the title is presented.

##### Digit Reversal Without Apology

The number 1089 has the curious property that when one reverses its digits, an integer multiple of itself is obtained. The author gives a structure theorem for certain sets of reversible numbers.

##### Coefficients of the Characteristic Polynomial

The author derives formulas for the coefficients of the characteristic polynomial of a square matrix in terms of the partial derivatives of its determinant, which is viewed as a function of the entries of the matrix.

##### On Some Symmetric Sets of Unit Vectors

Given symmetric unit vectors $$u_i$$, conditions on real numbers $$x_i$$ are considered to be able to conclude that $$\sum x_i u_i \Rightarrow x_i = 0$$ for all $$i$$. Different kinds of “symmetry” lead to different conclusions.