Browse Classroom Capsules and Notes

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Convergence of Complex Continued Fractions

Convergence of a complex continued fraction can be analyzed using analysis, algebra, number theory, topology or complex analysis.

Math Bite: A Quick Counting Proof of the Irrationality of \(\sqrt[n]{k}\)

The proof of the irrationality of the \(n\)-th root of \(k\) uses congruences.

Trapezoidal Numbers

Which numbers can be written as a sum of consecutive positive integers, such as 12 = 3 + 4 + 5? The answer: all positive integers except the powers of 2. The authors prove this result and...

Proof of a Conjecture of Lewis Carroll

Can one find three distinct right triangles, each having integral sides, all having the same area? Carroll conjectured an answer of yes; he was right.

Counting Squares in \(Z_n\)

How many numbers in the integer mod \(n\) ring will be perfect squares?

Tetrahedral Numbers as Sums of Square Numbers

The authors find those arithmetic progressions such that the sums of the squares of their terms can be represented by a binomial coefficient.

Regular Polygons with Rational Area or Perimeter

Two different approaches are given to determine which regular polygons, inscribed within or circumscribed about a unit circle, possess rational area or rational perimeter.

A Nonlinear Recurrence Yielding Binary Digits

The authors obtain a recurrence relation that yields the \(n\)th digit in the binary expansion of any real number.

The Catalan Numbers and Pi

The author gives an elementary justification for the representation of \(\pi\) involving Catalan numbers.

Almost Pythagorean Triples

The author discusses the solutions to the two Diophantine equations \(x^2 + y^2 = z^2 + 1\) and their relations.