Update: Scroll to the bottom of this post to see the solution to Smullyan’s logic puzzle discussed below. Raymond Smullyan has written many books. What is the Name of This Book?, published in 1978, is a collection of logic puzzles and paradoxes that culminate in a development of Gödel‘s incompleteness theorem. The first page of Chapter … Read more
My friend Mary sent me this link to a video by Vi Hart about why $\pi$ is wrong. The video is funny, and the rest of her site is wonderful, engaging, and worth checking out. There is a nice article about her in the New York Times. The originator of the “$\pi$ is wrong” idea … Read more
The two most commonly used measures for angles are degrees and radians. There are 360 degrees in a full circle (a right angle is 90 degrees), and $2\pi$ radians in a full circle (there are $\pi/2$ radians in a right angle), so there are about 57 degrees in a radian. Students typically learn about degrees … Read more
One of the important skills that a mathematics or science student graduating from high school ought to have is the ability to quickly visualize the graphs of basic functions. The repertoire should include power functions, polynomial functions, trigonometric functions, exponential functions, and logarithmic functions. Having an instant and intimate knowledge of such graphs is a … Read more
In the two previous posts in this series we explored a method for solving second order linear differential equations with constant coefficients that is different from the standard textbook methods taught nowadays. I found the method in a 1941 book (or see here) by the Sokolnikoffs. The key point of the method, as we learned, … Read more
In a previous post we discussed an operator method for solving certain second order ordinary differential equations. In this post I’ll explore this operator method a little further. I first learned about this method from an old book, Higher Mathematics for Engineers and Physicists, by Ivan S. Sokolnikoff and Elizabeth S. Sokolnikoff, McGraw-Hill, 1941. I … Read more
One of the most important skills in “numerical literacy” is proportional reasoning. I was reminded of this earlier today when I went out to buy a couple of pies for this evening’s family gathering. The pies came in many flavours, and two sizes, 8 inch diameter and 10 inch diameter. There was never any question … Read more
Over at Freakonometrics, here is a pair of nice probability problems. The first problem also appears here as Problem #5. A number of solutions for your consideration are here. A very elegant solution by Ted Hwa is here. (This post first appeared at my other (now deleted) blog, and was transferred to this blog on … Read more
In talking about power series in a previous post, I mentioned one of their uses: as an aid in solving differential equations. This reminds me of a neat trick for solving some differential equations, which I will discuss in this post. A standard method for solving linear differential equations with constant coefficients is to assume … Read more
We’ve been talking about reductionism in the past couple of posts, and we’ll continue the story by discussing power series in this post. The idea behind reductionism in mathematics is to identify some elementary “objects” and to express a complicated “thing” in terms of the elementary things. The intention is either to make it easier … Read more