# Happy Pi Day! A Short History of Pi!

Today is π day, so I thought I should give a brief history on both.

### Origins of π:

π is quite the most elusive and yet famous number of all. It's a simple number that even the ancients thought about it. And yet it's concept is quite easy to explain. It is the ratio between the circumference and the diameter. The reason why the diameter was chosen rather than the radius was because it was the only thing they could directly measure [though I argue that the circle constant should Tau. It's understandable once you get to angle measurements]. Some well known approximations are as follow: π pops up in the Bible, with a value of 3. The Bablylonians wrote that π was approximately 25/8 which is 3.125.

But things got interesting with the ancient Greeks. Archimedes attempted to use what is best described as an early for on integration called a method of exhaustion, and it's actually rather simple.

Have you ever wondered as a kid about regular polygons? Notice that they seem to get closer and closer to a circle the more sides you add onto them. By using this property, Archimedes created an upper and lower bound for π. 223/71<π<22/7.

Eventually, the Chinese mathematicians would take this and attempt to get a much more accurate approximation, setting the record at 7 digits, which lasted for 800 years.

### Rebirth of Greco-Roman culture and the Scientific Revolution:

The study of mathematics was less relevant during the 1000 years of darkness in Europe. The attitude towards mathematics began to change during the Renaissance age, which led to the scientific revolution. Many major development happened after the Renaissance such as Cartesian coordinates, which showed the connection between Algebra and Geometry. Newton and Leibniz created calculus for the first time, and with it came a new age of discovering π. Tools such as limits and infinite series allowed for faster rates of convergence onto the value of π. We start to see a different way of thinking from the Greek, where problems with infinity can be much easier to solve. Many mathematicians in Europe raced to see who can create the fastest converging series for π. In 1789, a Slovene mathematician, Jurij Vega, calculated π correctly to 126 digits.

### Things are getting interesting:

However, not all studies into π was about trying to get better approximations. Leonhard Euler is probably a name many know. He wrote many things about π, such as Euler's identity or solving the Basel problem. With solving the Basel problem, he created basis for Riemann Zeta function, which is still being studied today in the field of analytic continuation. The Riemann Zeta function also became important for number theory when talking about probability of coprimality.

Carl Fredrick Gauss worked on π as well. He used π in Gaussian integration, which led to what we now know as the error function. We also see Lambert's proof of π irrationality using a continued fraction of the tangent function. And in 1882, Ferdinand von Lindemann proved π was not a solution to any algebraic equation, thus establishing π transcendental property. This also showed that it is impossible to square a circle, a problem from antiquity. [And for anyone who wants a funny story, look up Indiana Pi Bill]

### Modern Era:

In 1910, Srinivasa Ramanujan found a formula that rapidly converges to π. His formula would be the basis for the fastest current algorithm to calculate π. By the mid-20th centuries, computers started to be used to calculate π, with the current world record being set at 31.4 trillion digits of π by Emma Haruka Iwao in March 2019.

So, is this the end of π's story? Since only 30 digits of π is required to hypothetically calculate the circumference of the universe down to an error of a hydrogen atom, would we ever need to investigate π anymore? The study of π, in my view was never really about getting more and more precise formula. I do think we have more than enough digits of π. Ironically, I think I am more alike to the Greeks, but in a different way. The Pythagoreans admire the relation between integers and ratios of integers. They viewed it as numbers sharing a connection. In a sense, π started off as a simple concept and a simply conquest to find the ratio of 2 lengths. And the journey that mathematicians took down the stream of mathematics reveals things about the nature of geometry, algebra, calculus, probability, number theory, complex numbers, and the universe itself. And I don't believe in one myTake, I could do justice for the nature of π.

## So happy π day!

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• 6d

loving these invites :))

tbh the problem of squaring the circle looks like geometry or algebra (of the constructible lengths) but it turned out to be way beyond the powerlevel of the people from Gauss's era and the likes (let alone the Greeks 😂). I think it could only be solved be post Weierstrass era, where they have the technology of epsilon and delta, remember the calculus exercise I wrote awhile ago about proving e to be transcendental? that was the key to showing pi is transcendental as well: basically it's has a step of "... and here the miracle happens..." where you have to conjure up a formula related to the algebraic equation for e. Then you use lengthy algebra to show that this magic formula (which is a linear combination of polynomial - exponential integrals) has to be some non zero integer, then you use the epsilon technology of (post 1900s) aka first year calculus to show that it must be between -1/2 and 1/2 as well, then you have a contradiction and you're done. Then you prove that e^x is transcendental if x is nonzero non transcendental (like before, you also need to conjure up some integral of polynomial - exponential hybrid with a lot of bells and whistles and try to get an explicit relation with the epsilons), then from e^(ipi) = -1 you can conclude that pi is transcendental and thus is not a constructible length

Whew that summed up few hundred years of analysis on this problem (and the thousand years of failed attempts on it with algebra, geometry and number theory 😂) also the thing with this problem is apparently there's no more modern way of doing it other than hard core 19th century analysis, I wrote it's calculus but proper calculus is actually very sadistic, it takes a special type of personality to be able to sit down, solve each of these integral inequalities and still enjoy life. Of course there's a pattern (some analysts can bound these integrals of series with exponential from above and below by reflex), but it's one of the patterns that you will probably never encounter again if you have a well paid job

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• 6d

I've been trying to prove Pi's irrationality with little success. I tried using the same method I used to prove e is irrational, which was much easier.

• 6d

yup pi's irrationality is also much harder than e (showing analytical properties of pi is ofc harder because e was almost born from Calculus of series, you have nice fast asymptotic right away, look at the series for pi that can be derived from elementary functions, they converge very slowly) So for pi you have to conjure up unnatural polynomials

• 6d

Even though I'm not really talented in math and physics, which also means I don't really love these subjects because of this, I find your take very readable and I admire your excitement for such things 😊

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## What Girls & Guys Said

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• 6d

It's interest that the Indiana General Assembly (at a time we'd expect them to be focusing on developing the state) made time to introduce a legislative bill regarding mathematics! :)

It's exciting to be living in a time when history is being made! I didn't hear about Emma Haruka Iwao on the news today. Thank you for mentioning her. ✨

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• 4d

Except you know what the Indiana Pi bill was about. It was about squaring a circle, and I kid you not, assign the value of pi to be 3.2. Luckily a mathematician was walking by and he heard this. He decided to teach the Senate about math and why this bill is ridiculous, and it was quite funny.

• 4d

Thank you for explaining it. :)

• 6d

Hmm... you wrote all this or is it from some website?
because you must be one smart 17y. o.
Its all Greek to me lol
I'd rather write a songpoem or bake a Baked Alaska...

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• 6d

I wrote this. Pi is quite the elusive number. Trying to prove it's irrationality is much harder than proving e's irrationality.

• 6d

@Deathraider Very impressive! :)

• 6d

I had my share of problems with π at school so I am not a big fan of it.

However, it is a great take and despite my dislike for the Greek P, it is nevertheless impacting our daily life just as α, λ, δ and the others that gave me sleepless nights.

So, I will close an eye and wish π the credit it deserves!

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• 6d

Love this myTake! Happy pi day to you too!! I just found out about 10 minutes ago, haha!

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• 6d

Pi day is a made up holiday by mathematicians to sell more maths.

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• 4d

😂 Gotta watch those mathematicians! 😂

I was wondering how long will it be before Hallmark starts selling Pi Day cards. 😄 Guys: "Ya mean there's another day I have to buy my girlfriend a gift? How many days of the years is that now?" 😂😂

• 4d

@ConnieS Too many days!

• 3d

😂😅 Not for your girlfriend. 😂😅

• 5d

Outside the USA it's celebrated on the 22nd of July.

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• 5d

Why July 22nd?

• 5d

I should have thought that was obvious, but 22/7 is the approximate fraction we were taught to use.

• 5d

Wait, they taught you to use that approximation? That's only a 3 digit accuracy.

• 6d

Happy Pi day :D

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• 6d

What will you be doing today?

• 6d

Assignments (Like usual)

• 5d

It's the Ides of March now :p

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• 5d

Cool. I did not know that.

• 4h

Damn now I want some pie 🥧 but math is cool too 😎

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• 5d

Your enthusiasm and effort astounds me.

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• 6d

Great ode to pi!

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• 6d

Fascinating!!

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• 6d

lets eat pai in the sake of of Pi

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• 6d

Pie*

• 6d

I second that! 😋

• 5d

Good take.

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• 6d

Coolies.

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• 4d

Keep that shit at your high school not on gag pls

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• 6d

Loser

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• 6d

If I knew you in real life, I’d beat you up and stuff you in a locker

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• 5d

"Why are you people so mad" Oh, I don't know. Maybe it was that loser comment! 😄
You were joking? Oh, ok. I'll put laughing on my to do list.{Now that's a joke - not that funny, but it's my attempt at humor.
Insulting someone isn't funny. And when texting, we need to let the reader know we're joking, when we're joking. Otherwise people can't be blamed for thinking you're serious.}

• 5d

If you let a simple ass comment like that get you angry, then you have no business on the internet 😂