The mystery of pi.

Now that is impressive. 

50 digits is quite an achievement.

According to the Guinness Book of Records, Rajveer Meena memorised pi to 70,000 decimal places. It took him nearly ten hours to recite it.

https://www.guinnessworldrecords.com/world-records/most-pi-places-memorised

My nephew used to be able to recite 50 digits of pi from memory. It always amazed me. 

I was thinking about what a circle really is and isn't it fair to say any circle is essentially made up of innumerable straight lines?!

So, there can never be a "perfect" circle since you can always refine it endlessly. Therefore,  on the one hand you have the diameter, which can be represented as a whole number, but then you're trying to divide something that is not fixed (the circumference) by something that is (the diameter), and maybe this is the fundamental problem?

It may be coincidence at one level, since there are infinite irrational numbers.

The connections are more at the conceptual level, and some of the simplest proofs for each are by contradiction.

Pi and e do have some connections on important formulas, but not sure whether their irrationality proofs are similar.

Just adding a bit more context to the last sentence:

"Hippasus of Metapontum (flourished c. 500 bc) was a philosopher and an early follower of Pythagoras. He was coupled by Aristotle with Heraclitus in identifying fire as the first element in the universe. Some traditions say that he was drowned after revealing a mathematical secret of the Pythagorean brotherhood."

(The secret was the irrationality of 2, of which they were embarrased.)

This was derived or understood from Pythagoras's theorem.

www.britannica.com/.../Hippasus-of-Metapontum

Is the fact that pi and numbers like sqrt 2 or sqrt 5 irrational for the same reason or purely coincidence?

Historically people have been obsessed for thousands of years trying to find the exact value of pi.  Until it was proved irrational in the 18th century.  Very few people can follow the proof.  Part of the fascination is that it is so easy to define. But the exact value is unknown. 

A similar problem is that the square root of 2 is irrational.   The ancient Greeks found a proof and it became a closely guarded secret, punishable by death if disclosed to outsiders.

Apologies if it contains too much detail.

The question is great, and the last link is an example of how hard it is to prove, even for a simpler case (square root of two, which is also irrational).

I tried to justify simpler cases, as examples, before getting to the link.

If I can help further or there are parts unclear feel free to ask.

ooh ooh ooh another play on words - (I had to look up what pi and to have had my breakfast is to grind this one out tho') "solving the mystery of pi - transcendental meditation for autistic people" :-)

I appreciate your effort, ArchaeC, thanks,

Thanks, emanningis, I think I'm out of my depth here among such bright people.

Yes, I know it's the Circumference/Diameter but why should it be irrational?

Thanks for the Wiki link but it's way above my head, I'm afaid.

Thank you, Phased.

A circle is curved, so no matter its size, it’s constant, so the ratio between its circumference and diameter is represented by the same irrational number that has an indefinite number of digits. 

I’m afraid I’m at the limit of my maths knowledge with this one. All I know is that it is easier to express its value as pi rather than 3.14159 …

Good answers on here and thanks for asking the question lostmyway  the possibility of a never ending pie is also one worth considering!  hmmm... reckon I need my breakfast :-)

Here is the answer.  en.wikipedia.org/.../Proof_that_pi_is_irrational

And if people can't follow the maths.  Tough. You need to go back to school.

The geometric interpretation is easy.  You take a perfect circle,  measure the circumference and the diameter.

PI = Circumference/ Diameter

Since all perfect circles are similar.  This ratio is always the same for all different sized circles.

I don't know the answer, but here are some comments. Thanks for the great question.

Exact division

10 / 2 means distributing 10 items in 2 groups, such as 10 candies between 2 people.

Division with reminder

11 / 2 is similar, but we get 5 candies each, and 1 candy is left as reminder. When we have more people than candies left, we perform a trick. We imagine multiplying it by ten: 1*10 = 10 candies again, yay!

This trick is specified as a number after decimal point. We can do the example above, in two steps:

1. With 11 / 2 we distribute 10 (5 each), and are left with 1.
2. So we do our trick, multiply by 10, and re-distribute between two: that is 10 / 2 = 5.

So the result of 11 / 2 = 5.5, the first 5 comes from the first step, the second five (after the decimal point) from the second step.

Periodic Reminder

This can be continued forever when there is a reminder. For example, 1/3 is:
1. 0 from our first step (because 1 is less than 3),
2. With the trick, we get 10 / 3, distribute 3 to each, and get 1 left. (As a partial result we then have 0.3),
3. With the trick, we get 10 / 3 again! And again distribute 3 to each, and get 1 left. (We have 0.33),

This process continues forever. (We have 0.333...). Some numbers, such as the beautiful 1/7 repeat not one digit but a set of them. In this case, if we perform the steps above, every six steps we go back to the same numbers.

Rational Numbers

Those are rational numbers because we can express them as a ratio of integers (a fraction). As we saw, a rational number can still repeat forever (1/3, 1/7,..).

• If a number terminates, it is rational, since it can always be represented as a fraction (multiply by many thousands until we get an integer).

Imagine we have 0.124134643250982432159 (my random typing). Then we can just do 124134643250982432159 / 1000...

And we got the fraction.

We could ask a few questions: Do all infinite numbers with a repeating motif (such as 0.333...) have a corresponding fraction? (I assume so, but can't prove it)

Irrational Numbers

What if it does not repeat and continues forever (does not have an end)? How do we even know if those don't have an end? (Usually by some proof).

• So we may hypothesise here (I may be wrong) that, if a number does terminates (previous bullet) or repeats a motif it's rational. Irrational numbers then repeat forever without a motif.

Examples of such numbers are the square root of 2 or of 5, continue forever without a repeating pattern. Pi and Euler's number e are cousins of those.

The proof for "the square root of 2 being irrational" is more accessible than the proof for pi (I don't understand the one for pi).

How do we prove the case for the square root of 2 being irrational?

The simplest way is by contradiction, and can be found here.

Ah, I see what you mean!

Yes, so with a straight line, for example, you have consistency, but a circle is kind of always getting away from you, so to speak.

Thank you, ArchaeC, you've summed it beautifully in one sentence. When I looked it up all I got was several mathematical proofs by various famous mathematicians which might as well have been in Chinese. Brilliant!

Hem … pi goes on forever because it’s an irrational number and can’t be expressed as a simple fraction. A circle is curved so it can’t be expressed in a finite number.

My maths is basic so I’m sure there will be better answers.