Here are three things we noticed this month which didn’t get a proper write-up, due to thesis/Edinburgh fringe/holidays: a big proof, a fun maths book club, and a ridiculous bit of pi-related madhattery.
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Cushing your luck: properties of randomly chosen numbers
Long-time Aperiodical muse David Cushing has made a bet with us that he can give us an interesting post every Friday for the next ten weeks. Every week that he sends a post, we buy him a bar of chocolate. Every week that he doesn’t send us a post, he buys us a bar of chocolate. For his first trick, David is going to do some unnatural things with the natural numbers.
The greatest common divisor (gcd) of two or more integers is the greatest integer that evenly divides those integers. For example, the gcd of $8$ and $12$ is $4$ (usually written as $\gcd(8,12)=4$). Two integers are called coprime (or “relatively prime”) if their gcd is equal to $1$.
A reasonable question to ask is,
Given two randomly chosen integers $a$ and $b$, what is the probability that $\gcd(a,b)=1$?
Pi day live, starring Marcus du Sautoy
Since it’s Pi Day in March, and ridiculously there are still people who haven’t spurned π completely in favour of τ, maths/media pixie Marcus Du Sautoy is running a free online event, called Pi Day Live (hashtag: #pidaylive). Since everything is more exciting when it’s happening LIVE, actually and IN REALITY, they’ll be conducting a live experiment, using circle measuring, marble arranging, Buffon-y needle dropping and in some extreme cases, river length approximation to calculate, LIVE, in REAL LIFE, approximations to the value of Pi.
π vs τ: FOTSN/Tau Day special
Since it’s $\tau$ Day, we thought we’d give Festival of the Spoken Nerd constant-fans Matt Parker and Steve Mould a chance to air their respective viewpoints in the $\tau$ vs $\pi$ debate. It’s a maths showdown!
WLTM real number. Must be normal and enjoy long walks on the plane
Something that whipped round Twitter over the weekend is an early version of a paper by Francisco Aragón Artacho, David Bailey, Jonathan Borwein and Peter Borwein, investigating the usefulness of planar walks on the digits of real numbers as a way of measuring their randomness.
A million step walk on the concatenation of the base 10 digits of the prime numbers, converted to base 4
A problem with real numbers is to decide whether their digits (in whatever base) are “random” or not. As always, a strict definition of randomness is up to either the individual or the enlightened metaphysicist, but one definition of randomness is normality – every finite string of digits occurs with uniform asymptotic frequency in the decimal (or octal or whatever) representation of the number. Not many results on this subject exist, so people try visual tools to see what randomness looks like, comparing potentially normal numbers like $\pi$ with pseudorandom and non-random numbers. In fact, the (very old) question of whether $\pi$ is normal was one of the main motivators for this study.
Happy π day
People celebrate today as “π day“, an official designation in the US*
3.141592653589793238462643383279502
88419716939937510582097494459230781
64062862089986280348253421170679821
48086513282306647093844609550582231
72535940812848111745028410270193852
110555964462294895493038196...5...
Do you know how we know those digits? If you’d like to find out, I’ve written a blog post over at Second-Rate Minds: “How to calculate π“.
* This is not necessarily the best day. “3.14” is a very crude approximation for π and translating 14th March into this number requires a month/day dating system. 22/7 gives 22nd July in day/month format and gives a slightly closer approximation for π: 3.142857… (3.14 is π-0.00159…, while 22/7 is π+0.00126…). Having said that, 14th March has the tantalising prospect of 1:59 and 2.65359… seconds giving an actual value of π… Perhaps we need a fourteenth month…
How to calculate π
A new post is available over at Second-Rate Minds by Peter Rowlett.
How would you calculate ? I remember being sent as a boy of ten into the schoolyard to find circular objects to measure. Our attempts to wrap a tape around a dustbin lid clearly did not represent the optimal method. Perhaps you know a few digits of . Starting from scratch, how would you find …
Read the full post: “How to calculate π“