Pi Day! 10 Fun Pi-Related Facts!

Hey, hey! So what’s so cool about pi? Probably an infinite number of things! Here I want to give you a few fun facts about pi that aren’t directly related to geometry and circles.

So let’s begin!

#1 Buffon!

One of the first recorded uses of the Monte Carlo method (before it was even called that!) in a geoemetric setting is now known as “Buffon’s Needle Problem”. Take a rectangular surface with evenly spaced stripes and throw a bunch of needles on it at random. The needles ought to be a specific size. Do this with enough needles and you can approximate $$\pi$$! Here’s one sample:

For further reading, check out: Buffon.

#2 The “Most Beautiful Equation”

Beauty is in the eye of the beholder. However, a 2014 paper concluded that $$e^{i\pi} + 1 = 0$$ (Euler’s identity) was indeed the most beautiful equation. This equation uses the mathematical constant $$e$$, $$\pi$$, the imaginary number $$i$$, the number $$1$$, and the number $$0$$. You decide, most beautiful?

If you want to read about the research, check out this paper.

#3 Infinite Series

Well, we’ve got Euler’s identity, but there are some cool looking infinite series that add up a quantity involving $$\pi$$. Here is one particular one that is often a standard problem to prove in a Fourier series course or by use of Parseval’s theorem (or a bunch of other ways).

$$\sum_{k=0}^{\infty}\frac{1}{k^{2}} = \frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \cdots = \frac{\pi^{2}}{6}$$

#4 Actually, my version of beauty

$$\int_{-\infty}^{\infty}e^{-x^{2}}\ dx = \sqrt{\pi}$$

’nuff said.

#5 OEIS Sequence A283247

Guess who is credited with this $$\pi$$-prime sequence? Yours truly!

It begins like this: $$3, 31, 13147, 73141, 314159, 314159, 131415923, \ldots$$ — the description of this sequence is “$$a(n)$$ is the smallest prime number whose representation contains as a substring the first $$n$$ digits of $$\pi$$ in base 10.” It’s a $$\pi$$ containing sequence of prime numbers!

#6 Speaking of prime numbers

David Radcliffe finds this beautiful (probably) prime number using only the digits 3, 1, and 4!

I had found a measly 463-digit number …
3141143341443144133134133414411314111313431343331433433334311443134444443334411143134144133143343311113144131431144414314114434341444114411144331444113313414341411141314344441344331131143134133343341134333144311444434333413131114414313313441431313131344413411111434331441333134333414431331433444344114341333331334341411434444341431334444114141441131343133314333333334414344411444443111443114141434133414314444133143431111343111313314141314114143331143434131411441

Scavenger hunt question for you! Reading left to right only, how many “314” sequences can you find?

#7 Continued Fraction!

$$\frac4{\pi}=1+\cfrac{1^2}{2+\cfrac{3^2}{2+\cfrac{5^2}{2+\cfrac{7^2}{2+\cdots}}}}$$

#8 A Fun Comparison

Which is larger? $$e^{\pi}$$ or $$\pi^{e}$$. But let’s add a twist!

Order these four numbers from least to greatest:
$$e^{\pi}, \pi^{e}, e^{\frac{22}{7}}, \Big(\frac{22}{7}\Big)^{e}$$

#9 It’s a little complex

$$i^{i} = e^{-\frac{\pi}{2}}$$
Maybe this the most beauitful equation!

#10 It’s not just irrational, it’s transcendental!

$$\pi$$ is a transcendental number! And it is unknown if $$\pi + e$$ is an irrational number! But I’m putting my money on that that sum is irrational. Learn more about transcendental numbers.

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