This evening something reminded me of the following line from Rudyard Kipling’s famous poem If: … If all men count with you, but none too much … It would be good career advice for a mathematician to say “Let all…

This evening something reminded me of the following line from Rudyard Kipling’s famous poem If: … If all men count with you, but none too much … It would be good career advice for a mathematician to say “Let all…

Here’s a curious result I ran across the other day. Suppose you have a quintic equation of the form z x5 – x – 1 = 0. (It’s possible to reduce a general quintic equation to this form, known as…

Here’s an approximation to e by Richard Sabey that uses the digits 1 through 9 and is accurate to over a septillion digits. (A septillion is 1024.) MathWorld says that this approximation is accurate to 18457734525360901453873570 decimal digits. How could…

Here’s a strange way to do arithmetic on the real numbers. First, we’ll need to include +∞ and -∞ with the reals. We define the new addition of two elements x and y to be -log (exp(-x) + exp(-y) ).…

The previous post gave a relationship between the imaginary unit i and the golden ratio. This post highlights a comment to that post explaining that the relationship generalizes to generalizations of the golden ratio. GlennF pointed out that taking the…

This morning Andrew Stacey posted a beautiful identity I’d never seen before relating the golden ratio ϕ and the imaginary unit i: Here’s a proof: By De Moivre’s formula, and so Related posts: Golden ratio and special angles Golden strings…

“Much as beavers, who as a species hate the sound of running water, plaster a creek with mud and sticks until alas that cursed tinkle stops, so do category theorists derive elaborate and obscure definitions in an attempt to capture…

When does a rational portion of a circle have a rational cosine? If r is a rational number, cos(2πr) rational if and only if the denominator of r is 1, 2, 3, 4, or 6. This means that the special…

Yesterday Jack Kennedy told me about a graph he’d made as part of a project he’s working on and I asked if I could post it here. The Galois group of a quadratic polynomial x2 + bx + c is…

Today’s a prime day. Whether you write the date in American (MMDDYY), European (DDMMYY), or ISO (YYYYMMDD) format, you get a prime. That is, 112913 and 291113 and 20131129 are all prime numbers. We’ll call a date an American prime…

When I was in college, my advisor and I published a paper in a journal called “Applicable Analysis.” At the time, I thought that was a good name for a journal. It suggested research that was toward the applied end…

From Michael Atiyah’s essay Trends in Pure Mathematics. In any given field of mathematics there are always some very fine points which present great technical challenges to the specialist but not are usually of interest to the general mathematician. To…

I posted a couple prime-generating fractions on Google+ this weekend and said that I’d post an explanation later. Here it goes. The decimal expansion of 18966017 / 997002999 is .019 023 029 037 047 059 073 089 107 127 149…

This morning I ran across a list of 23 math problems compiled by DARPA. I assume their choice of exactly 23 problems was meant to be an allusion to Hilbert’s famous list of 23 math problems from 1900. Some of…

Binomial coefficients can be defined by Edouard Lucas defined the Fibonomial coefficients analogously by where Fj is the jth Fibonacci number. Binomial coefficients satisfy and Fibonomial coefficients satisfy an analogous identity Incidentally, this identity can be used to show that…