In 1997, Bailey and Bannister showed that a + b greater than c + h holds for all triangles with [gamma] less than arctan (22/7)where a, b, and c are the sides of the triangle, "h" is the altitude to side "c", and [gamma] is the angle opposite c. In this paper, we show that a + b greater than c + h holds approximately 92% of the time for all…

The usual definition of the Riemann integral as a limit of Riemann sums can be strengthened to demand more of the function to be integrated. This super-Riemann integrability has interesting properties and provides an easy proof of a simple change of variables formula and a novel characterization of derivatives. This theory offers teachers and…

We summarize the most important facts about the Steiner point of a triangle and find formulas for its distance to each vertex in terms of the side-lengths of the triangle.

Using techniques that show that e and pi are transcendental, we give a short, elementary proof that pi is irrational based on Euler's identity. The proof involves evaluations of a polynomial using repeated applications of Leibniz formula as organized in a Leibniz table.

Like Pascal's triangle, Faulhaber's triangle is easy to draw: all you need is a little recursion. The rows are the coefficients of polynomials representing sums of integer powers. Such polynomials are often called Faulhaber formulae, after Johann Faulhaber (1580-1635); hence we dub the triangle Faulhaber's triangle.

Finding the sum of a series in the form of a closed expression has always been a challenging problem in analysis. The paper presents an elementary method for summation of series with terms generated by functions satisfying subtraction identities.

This student research project explores the properties of a family of matrices of zeros and ones that arises from the study of the diagonal lengths in a regular polygon. There is one family for each n greater than 2. A series of exercises guides the student to discover the eigenvalues and eigenvectors of the matrices, which leads in turn to…

The notion of an island has surfaced in recent algebra and coding theory research. Discrete versions provide interesting combinatorial problems. This paper presents the one-dimensional case with finitely many heights, a topic convenient for student research.

If you want your students to graph a cubic polynomial, it is best to give them one with rational roots and critical points. In this paper, we describe completely all such cubics and explain how to generate them.

Purely combinatorial proofs are given for the sum of squares formula, 1[superscript 2] + 2[superscript 2] + ... + n[superscript 2] = n(n + 1) (2n + 1) / 6, and the sum of sums of squares formula, 1[superscript 2] + (1[superscript 2] + 2[superscript 2]) + ... + (1[superscript 2] + 2[superscript 2] + ... + n[superscript 2]) = n(n + 1)[superscript 2]…

It seems rather surprising that any given polynomial p(x) with nonnegative integer coefficients can be determined by just the two values p(1) and p(a), where a is any integer greater than p(1). This result has become known as the "perplexing polynomial puzzle." Here, we address the natural question of what might be required to determine a…

The inter-derivability of the Pythagorean Theorem and Heron's area formula is explained by applying Al-Karkhi's factorization to Heron's formula.

Pompeiu's theorem states that if ABC is an "equilateral" triangle and M a point in its plane, then MA, MB, and MC form a new triangle. In this article, we have a new look at this theorem in the realm of arbitrary triangles. We discover what we call Pompeiu's Area Formula, a neat equality relating areas of triangles determined by the points A, B,…

A. Lobb discovered an interesting generalization of Catalan's parenthesization problem, namely: Find the number L(n, m) of arrangements of n + m positive ones and n - m negative ones such that every partial sum is nonnegative, where 0 = m = n. This article uses Lobb's formula, L(n, m) = (2m + 1)/(n + m + 1) C(2n, n + m), where C is the usual…

Consider a circular segment (the smaller portion of a circle cut off by one of its chords) with chord length c and height h (the greatest distance from a point on the arc of the circle to the chord). Is there a simple formula involving c and h that can be used to closely approximate the area of this circular segment? Ancient Chinese and Egyptian…