PI Day 3.14.16 – Some Eerie Pi coincidences
Another year, another PI Day, which as we all know is March 14, using the U.S. system of calendar date nomenclature. In fact, this year is 3/14/16, which is pi rounded to 4 decimal places (3.1416, rounding of 3.14159). That's a nice pi-related coincidence. Here are a few more.
First off, let's recall the most interesting point about transcendental and irrational numbers: they never end and never repeat. Or, as Spock says about pi in “Wolf in the Fold”: "the value of pi is a transcendental figure without resolution".
A pi related calculation, eerily close to an integer
This one is courtesy of the XKCD internet comic.
epi – PI = 20.00 (actually it is 19.9991)
This one is seemingly strangely significant, being so close to the integer 20. But, it can't really have a deep meaning, can it?. Since e and pi are both transcendental numbers, e raised to the power pi must also be transcendental, or at least irrational. Then, subtracting an infinite non-repeating number (pi) from another infinite non-repeating number (e to the pi), you should never be able to come up with an integer, unless the first number was the same as the second (i.e. pi), with some integer added to it.
Which seems to be impossible, though it might be difficult to prove – or not, but there's no time left before Pi Day, so I'll leave it at that. Besides, I am a data analyst and this is a job for a pure math type.
I suppose there must be an infinite number of similar examples, where transcendental valued functions like this almost result in integers. Anyway, a better mathematician than me can probably prove all that. :)
Some interesting coincidences in expansions of transcendentals and/or irrationals
We can look at the decimal expansions of some famous transcendental or irrational numbers, to see where approximations of other numbers show up. There are a number of internet sites that allow you to plug in a number, and see where the first occurrence of that number is within a long decimal expansion of the transcendental or irrational number.
Here are some examples, using pi, e (the base of the natural logarithms), the square root of 2, and phi (the golden mean 1.61803):
Pi = 3.14159. If we look for the first occurrence of the string 314159 in the decimal expansions of the numbers listed above, we get:
For pi itself: 176,451 (i.e. if you go 176451 places out in pi, you will come to the string 314159).
For e: 1,436,935
For square root of 2: 199,409
For phi, 607,276
e = 2.71828. If we look for the first occurrence of the string 271828 in those numbers, we get:
For pi: 33,789.
For e: 252,474.
For square root of 2: 1,827,315.
For phi, 708. 385.
root 2 = 1.41421. If we look for the first occurrence of the string 141421 in those numbers, we get:
For pi: 52,638.
For e: 325,839.
For square root of 2: 110,269.
For phi: 360,709.
Phi = 1.61803. If we look for the first occurrence of the string 161803 in those numbers, we get:
For pi: 144,979.
For e: 389,765.
For square root of 2: 944,257.
For phi: 2,200,371.
The interesting thing about these numbers, is that pi always “wins”. You come across the desired string more quickly in the expansion of pi than in the expansion of the other numbers. Does that make pi somehow a better, or more “complete” infinite number than the others? After all, there are 256 (4X4X4X4) ways to order the sets containing the first occurrence of each string in these numbers. It somehow seems significant that pi always wins, doesn't it. One chance in 256, that's better than a one percent p-value.
Not really, though. Since there are 256 ways to order these numbers, any given ordering is equally unlikely. It is our minds that enforce the significance of pi always coming out on top first – it seems important to us. It would have seemed equally significant if root 2 would have won, or if pi always came in third. In fact, all sorts of combinations would have appealed to the pattern seeking instincts of our minds.
These sorts of post-hoc (after the fact) analyses often seem significant when they really are not. In big data sets you often see unusual runs of numbers, or a correlation between two random variables will pass as statistical significance test, at some given probability level. In a thourogh statistical analysis, you correct for these effects via Bonferroni adjustments and the like, though lots of papers in applied science areas miss that subtley. That's one reason a lot of results are not reproducable, in nutrition studies and the like.
Anyway, if that happens in large but finite datasets, how much more scope then, in infinite numbers, for these uncanny coincidences? Well, infinite scope, I suppose.
Here's a paper by a professor at Florida State University on a related theme, testing the randomness of pi, e, and root 2:
Finally, here's a nice pie chart about Pi Day.
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It's mid-March, and the sun is beginning to come on noticeably stronger in the more temperate regions. Spring is around the corner now, and that brings on thoughts of ROAD TRIP. Sure, it is still a bit early, but you can still start making plans for your next road trip with help of “On the Road with Bronco Billy”. Sit back and go on a ten day trucking trip in a big rig, through western North America, from Alberta to Texas, and back again. Explore the countryside, learn some trucking lingo, and observe the shifting cultural norms across this great continent. Then, come spring, try it out for yourself.
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Amazon Germany: http://www.amazon.de/gp/product/B00X2IRHSK
Amazon Canada: http://www.amazon.ca/gp/product/B00X2IRHSK