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.

The number Pi (approximately 3.14159) is defined to be the ratio C/d of the circumference (C) to the diameter (d) of any given circle. In particular, Pi measures the circumference of a circle of diameter d = 1. Historically, the Greek mathematician Archimedes found good approximations for Pi by inscribing and circumscribing many-sided polygons…

For at least 2000 years people have been trying to calculate the value of [pi], the ratio of the circumference to the diameter of a circle. People know that [pi] is an irrational number; its decimal representation goes on forever. Early methods were geometric, involving the use of inscribed and circumscribed polygons of a circle. However, real…

"Lord Brouncker's continued fraction for pi" is a well-known result. In this article, we show that Brouncker found not only this one continued fraction, but an entire infinite sequence of related continued fractions for pi. These were recorded in the "Arithmetica Infinitorum" by John Wallis, but appear to have been ignored and forgotten by modern…

Using continued fraction expansions, we can approximate constants, such as pi and e, using an appropriate integer n raised to the power x[superscript 1/x], x a suitable rational. We review continued fractions and give an algorithm for producing these approximations.

If students are in an advanced mathematics class, then at some point they enjoyed mathematics and looked forward to learning and practicing it. There is no reason that this passion and enjoyment should ever be lost because the subject becomes more difficult or rigorous. This author, who teaches advanced precalculus to high school juniors,…

Most high school mathematics teachers completed a mathematics history course in college, and many of them likely found it intriguing. Unfortunately, very few of them find the time to allow much, if any, mathematics history to trickle into their instruction. However, if mathematics history is taught effectively, students can see the connections…

In this article we consider an example suitable for investigation in many mid and upper level undergraduate mathematics courses. Fourier series provide an excellent example of the differences between uniform and non-uniform convergence. We use Dirichlet's test to investigate the convergence of the Fourier series for a simple periodic saw tooth…

An anharmonic solution to the differential equation describing the oscillations of a simple pendulum at large angles is discussed. The solution is expressed in terms of functions not involving the Jacobi elliptic functions. In the derivation, a sinusoidal expression, including a linear and a Fourier sine series in the argument, has been applied.…

A variety of approaches to the concept of oxidation and reduction appear in organic textbooks. The method proposed here is different than most published approaches. The oxidation state is calculated by totaling the number of heterogeneous atoms, [pi]-bonds, and rings. A comparison of the oxidation states of reactant and product determine what type…

An analytic approximation of the solution to the differential equation describing the oscillations of a simple pendulum at large angles and with initial velocity is discussed. In the derivation, a sinusoidal approximation has been applied, and an analytic formula for the large-angle period of the simple pendulum is obtained, which also includes…

The well-known result for the frequency of a simple spring-mass system may be combined with elementary concepts like speed = wavelength x frequency to obtain wave propagation speeds for an infinite chain of springs and masses (masses "m" held apart at equilibrium distance "a" by springs of stiffness "gamma"). These propagation speeds are dependent…

Bertrand's paradox (Bertrand 1889 "Calcul des Probabilites" (Paris: Gauthier-Villars)) can be considered as a cautionary memento, to practitioners and students of probability calculus alike, of the possible ambiguous meaning of the term "at random" when the sample space of events is continuous. It deals with the existence of different possible…

Several Excel applications are presented which are part of the syllabus in the first semester of engineering studies at Haugesund College. The aim of the applications is for the students to acquire both computing skills and mathematical understanding at the same time. The applications cover numerical solution of equations, differentiation,…

Through numerical investigations, various examples of the Duffing type forced spring equation with epsilon positive, are studied. Since [epsilon] is positive, all solutions to the associated homogeneous equation are periodic and the same is true with the forcing applied. The damped equation exhibits steady state trajectories with the interesting…