Browse Classroom Capsules and Notes

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Spherical Triangles of Area \(\pi\) and Isosceles Tetrahedra

The tiling of a sphere by spherical triangles is discussed.

A Butterfly Theorem for Quadrilaterals

The Butterfly Theorem states, "Through the midpoint I of a chord AC of a circle, two other chords EF and HG are drawn. If EG and HF intersect AC at M and N, respectively, then IM = IN."

Row Rank Equals Column Rank

A short elementary proof of the equality of row rank and column rank is given. The proof requires only the definition of matrix multiplication and the fact that a minimal spanning set is a...

Proof without Words: Alternating Sums of Odd Numbers

The author gives a visual proof of the fact that the sum of (2k-1)(-1)^n-k from k = 1 to n is n.

A Short Proof of Chebychev's Upper Bound

The authors provide a proof, accessible to beginning number theory students, of Chebychev`s upper bound on the number of primes no greater than n.

Emil Post and his Anticipation of Goedel and Turing

The author discusses Emil Post`s life and ideas about incompletness and insolvability which later became famous with Goedel and Turing`s work, and the differences between Post and Goedel.

Enter, Stage Center: The Early Drama of the Hyperbolic Functions

A history of the hyperbolic functions is presented.

Upper Bounds on the Sum of Principal Divisors of an Integer

After a brief historical discussion of the aliquot parts of an integer the authors use elementary arguments to prove a theorem about the sums of its principal divisors.

Centroids Constructed Graphically

The authors "describe two different methods for locating the centroid of a finite number of given points ... without using coordinates or numerical calculations". Some examples are...

There Are Only 9 Finite Groups of Fractional Linear Transformations with Integer Coefficients

The material in the article should be accessible to undergraduates. Some open question are posed.