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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