Trigonometric functions and rational multiples of pi

Recall that a real number is algebraic if it is the root of a polynomial with integer coefficients and that it is transcendental otherwise. For example is algebraic because it is a root of the polynomial , but is transcendental because it is not the root of any such equation. (On a recent blog post…

Happy tau day!

It is 6/28! Happy tau day—a day that is twice as fun a pi day! Let’s celebrate our new favorite mathematical constant:  Remember, as Bob Palais told us, is wrong! I’m a fan of tau.

More about the neat calculator trick

Yesterday I wrote about a neat calculator trick that I had just learned. We saw that if the calculator was set to degree mode, then times a high enough power of 10 is approximately . A commenter named Robert suggested looking at the difference between this approximation for and itself. He remarked that the error…

The math behind a neat calculator trick

[Update: after you read this post, read my follow-up post.] I received an interesting comment on yesterday’s blog post from Nemo. It was a cool calculator trick that I’d never seen before. Nemo wrote: Reminds me of my favorite calculator trick. Set your calculator to degree mode (NOT radians). Type in a bunch of 5’s:…

Interesting approximations using trigonometry

Today on Twitter John D. Cook, writing as @AlgebraFact, posted the following tweet: In radians, sin(11) is very nearly -1. (It happens to be -0.9999902…) I thought that was awesome! So, I (@divbyzero) replied that cos(333) is approximately 1. (It is 0.999961…) Then @michiexile chimed in, pointing out that cos(355) is closer to -1 than…

Proof that pi is irrational

Have you ever seen the proof that is irrational? If not, I highly recommend heading over to The Math Less Traveled. Blogger Brent Yorgey just posted the last of his six part series in which he gives Ivan Niven’s easy-to-follow 1947 proof of that famous fact. The proof uses only basic calculus.

Mathematical holiday cookies at Starbucks

When I was in Starbucks the other day my eyes were drawn to the cookie display, for what did I see, but a row of cookies all decorated with ‘s! Upon closer inspection I discovered, to my disappointment, that it was not the Greek letter, but the red scarf of a polar bear. (As Homer…

Ambigram: pie≠3.14

I thought the first comment on this article was funny. It says that pie and 3.14 are mirror images, if written in a certain way. Regular readers may recall that I had fun creating ambigrams a few months ago. This blog comment inspired me to whip up a quick ambigram exhibiting this symmetry. I like…

Fallacies, Flaws, and Flimflam it’s been good to know ya

At the end of 2008 the College Mathematics Journal will stop running its 20-year column “Fallacies, Flaws, and Flimflam.” The column was devoted to “mistakes, fallacies, howlers, anomalies, and the like”—usually found on student work. In honor of the departure of this entertaining column I submit the following FFF which was submitted as a solution…

Is π the right constant?

In the November 3, 2001 issue of the Mathematical Intelligencer Bob Palais wrote an article called “ is wrong!” In it Palais does not assert that we have miscalculated the value of , just that many mathematical formulas would be more elegant if we had chosen a different value for our named constant—he thinks that…