# Browse Classroom Capsules and Notes

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Displaying 51 - 60 of 1213

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...

Like Pascal’s triangle, Faulhaber’s triangle is easy to draw: all you need is a little recursion.  The author demonstrates that the rows are the coefficients of polynomials...

The author considers the effects of inflation and interest rates on computing the present value of \$100 three years from now.

It is well known that maximal ideals in the ring of continuous functions on the closed interval $$[0, 1]$$ are not finitely generated. Less well known is the fact that these ideals are not...

The author relates a personal story of a post-WWII assignment in the U.S. Navy.  His analysis of data from a test of a harbor mine-detecting algorithm leads to a novel construction of the...

When is the average of sums of powers of integers itself a sum of the ﬁrst $$n$$ integers raised to a power? We provide all solutions when averaging two sums, and provide some conditions...

The Cantor subset of the unit interval $$[0,1)$$ is large in cardinality and also large algebraically, that is, the smallest subgroup of $$[0,1)$$ generated by the Cantor set (using addition mod...

The author presents a variety of double and triple integrals designed to exercise a student's ability to visualize the interplay between an integrand and the region over which it is...

Using two well known criteria for the diagonalizability of a square matrix plus an extended form of Sylvester's Rank Inequality, the author presents a new condition for the diagonalization...

Here is a technique for proving the fundamental theorems of analysis that provides a unified way to pass from local properties to global properties on the real line, just as ordinary induction...