From Concrete Mathematics: Incidentally, when we’re faced with a “prove or disprove,” we’re usually better off trying first to disprove with a counterexample, for two reasons: A disproof is potentially easier (we just need one counterexample); and nit-picking arouses our…
Blog Archives
Synchronizing cicadas with Python
Suppose you want to know when your great-grandmother was born. You can’t find the year recorded anywhere. But you did discover an undated letter from her father that mentions her birth and one curious detail: the 13-year and 17-year cicadas…
Searching for Perrin pseudoprimes
A week ago I wrote about Perrin numbers, numbers Pn defined by a recurrence relation similar to Fibonacci numbers. If n is prime, Pn mod n = 0, and the converse is nearly always true. That is, if Pn mod…
Looking in both directions
From David Mumford’s May 2013 interview in SIAM News: The applied mathematician has the difficult job of looking at a problem in context with no explicit mathematics and trying to see what kinds of mathematical ideas are under the surface…
NYT Book of Mathematics
Today’s newspaper may be interesting because it reports new information. Newspapers from decades ago may be interesting for different reasons, not for the explicit content but for the implicit content. What were the contemporary reactions to what’s now well known?…
Posted in Math
Mutually odd functions
The floor of a real number x is the largest integer n ≤ x, written ⌊x⌋. The ceiling of a real number x is the smallest integer n ≥ x, written ⌈x⌉. The floor and ceiling have the following symmetric…
Almost if and only if
The Perrin numbers have a definition analogous to Fibonacci numbers. Define P0 = 3, P1 = 0, and P2 = 2. Then for n > 2, define Pn+3 = Pn+1 + Pn+0. The Concrete Tetrahedron says It appears that n…
Ramanujan approximation for circumference of an ellipse
There’s no elementary formula for the circumference of an ellipse, but there is an elementary approximation that is extremely accurate. An ellipse has equation (x/a)² + (y/b)² = 1. If a = b, the ellipse reduces to a circle and…
Recognizing numbers
I was playing around with SymPy, a symbolic math package for Python, and ran across nsimplify. It takes a floating point number and tries to simplify it: as a fraction with a small denominator, square root of a small integer,…
More sides or more dice?
My previous post looked at rolling 5 six-sided dice as an approximation of a normal distribution. If you wanted a better approximation, you could roll dice with more sides, or you could roll more dice. Which helps more? Whether you…
Rolling dice for normal samples: Python version
A handful of dice can make a decent normal random number generator, good enough for classroom demonstrations. I wrote about this a while ago. My original post included Mathematica code for calculating how close to normal the distribution of the sum…
Why j for imaginary unit?
Electrical engineers use j for the square root of -1 while nearly everyone else uses i. The usual explanation is that EE’s do this because they use i for current. But here’s one advantage to using j that has nothing…
Social networks in fact and fiction
SIAM News arrived this afternoon and had an interesting story on the front page: Applying math to myth helps separate fact from fiction. In a nutshell, the authors hope to get some insight into whether a myth is based on…
Physical versus medical modeling
Modeling is more fun when you have some confidence in your modeling assumptions. I’ve been working with models of physical systems lately and it’s been more enjoyable than the medical modeling I’ve done over the last few years. I have…
Abelian consulting and Lévy consulting
Eric Jonas once asked me on Twitter whether I was an Abelian consultant. The pun is an allusion to Abelian groups, groups in which the group operation commutes. No, I’m not an Abelian consultant. I don’t have a regular commute.…
