Happy Pi Day (2 Viewers)

iRuler · Mar 14, 2012

hehe I've had Pi as my location for such a long time

mirakon · Mar 14, 2012

Carrotsticks said:
Oh I understand your confusion.

They are not called indefinite integrals in the sense that the limits themselves are indefinite (taking the definition of 'indefinite' to be 'going on forever')

There is a lot of very *deep* mathematics that goes into the relationship between indefinite and the definite integral, but the essential thing is that indefinite does not imply infinite in this case.

Ah ok. Another thing to look up

mnmaa · Mar 14, 2012

Carrotsticks said:
Pi shows its face in even the most unexpected of problems seemingly totally related to pi.

For example, suppose I have a stick of some unit length.

In front of me, I have horizontal intervals (imagine floorboards) of the same unit length.

When I throw the stick, it can either cross one of the intervals or lie in between two intervals.

The probability of it CROSSING a line is.... you guessed it!

$\frac{2}{\pi}$

So just by throwing sticks, I can approximate pi!

This is more commonly known as Buffon's Needle Problem.

thats a good problem. In the cambridge 3unit book if I recall correctly

mirakon · Mar 14, 2012

Probs under the devilish "extension" section. I always felt like a badass when I solved those XD

mnmaa · Mar 14, 2012

mirakon said:
Probs under the devilish "extension" section. I always felt like a badass when I solved those XD

certainly was. And dont we all

mirakon · Mar 14, 2012

mnmaa said:
certainly was. And dont we all

Yeah I kinda miss maths, has been 2 years for me. I must confess to lurking in the maths threads and having the odd go at an interesting question but I often fail at the more complex solutions nowadays lol

Carrotsticks · Mar 14, 2012

mirakon said:
Yeah I kinda miss maths, has been 2 years for me. I must confess to lurking in the maths threads and having the odd go at an interesting question but I often fail at the more complex solutions nowadays lol

If you have the interest, then why not a career in it? It is a beautiful subject.

mnmaa · Mar 14, 2012

Carrotsticks said:
If you have the interest, then why not a career in it? It is a beautiful subject.

too hard. Plus my professors are putting me off maths

Carrotsticks · Mar 14, 2012

mnmaa said:
too hard. Plus my professors are putting me off maths

This makes me feel sad.

mirakon · Mar 14, 2012

Carrotsticks said:
If you have the interest, then why not a career in it? It is a beautiful subject.

I always saw maths as a hobby rather than a passion so I never really considered it as a career. But yes it is actually a very elegant subject, particularly pure maths

Plus my maths teachers fanatical obsession with prime numbers still sometimes inspires me to google the latest developments sonetimes , but its got so technical I only understand a bit of it

mnmaa · Mar 14, 2012

Carrotsticks said:
This makes me feel sad.

well, if you become a maths lecturer make sure you make it interesting.

Carrotsticks · Mar 14, 2012

mirakon said:
I always saw maths as a hobby rather than a passion so I never really considered it as a career. But yes it is actually a very elegant subject, particularly pure maths

Plus my maths teachers fanatical obsession with prime numbers still sometimes inspires me to google the latest developments sonetimes , but its got so technical I only understand a bit of it

Prime numbers also have a relationship with pi!

There is a very famous problem called the "Basel Problem", which was originally solved by Euler.

The series is as follows:

$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...= \frac{\pi ^2}{6}$

Does anybody recognise this from any particular HSC exams?

Consider the following series:

$\prod_{p}^{\infty} \frac{p^2}{p^2 -1} $ where $ p $ is a Prime Number $ \geq 2$

The series, when expanded, is:

$\frac{2^2}{2^2 - 1} \times \frac{3^2}{3^2 - 1} \times \frac{5^2}{5^2 - 1} \times \frac{7^2}{7^2 - 1} \times ...$

Guess to what value it converges!

The answer is...

$\frac{\pi ^2}{6}$

So there exists a strong relationship between pi and the Prime Numbers! Who would have thought of such a thing?? (except Euler).

SpiralFlex · Mar 14, 2012

^2010 lol

mirakon · Mar 15, 2012

Carrotsticks said:
Prime numbers also have a relationship with pi!

There is a very famous problem called the "Basel Problem", which was originally solved by Euler.

The series is as follows:

$\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...=\frac{{\pi}^2}{6}$

Does anybody recognise this from any particular HSC exams?

Consider the following series:

$\prod_{p}^{\infty} \frac{p^2}{p^2 -1} $ where $ p $ is a Prime Number $ \geq 2$

The series, when expanded, is:

$\frac{2^2}{2^2 - 1} \times \frac{3^2}{3^2 - 1} \times \frac{5^2}{5^2 - 1} \times \frac{7^2}{7^2 - 1} \times ...$

Guess to what value it converges!

The answer is...

$\frac{\pi ^2}{6}$

So there exists a strong relationship between pi and the Prime Numbers! Who would have thought of such a thing?? (except Euler).

Yep

Was over the moon when I got that one

PaterzAttack · Mar 15, 2012

a day late, but whatever

(stolen from american pie fb page)

seanieg89 · Mar 15, 2012

Primes are beautiful, right now I am writing an article about the Prime Number theorem. (Which at least involves the greek letter pi I guess...)

$Let $\pi(x)$ denote the number of primes less than or equal to $x$. Then\\ $\frac{\pi(x)\log(x)}{x}\rightarrow 1$ as $x\rightarrow \infty$. So speaking roughly, $\pi(x)$ is approximately equal to $\frac{x}{\log(x)}$ for large $x$.$

Most proofs of it are pretty difficult.

mirakon · Mar 15, 2012

seanieg89 said:
Primes are beautiful, right now I am writing an article about the Prime Number theorem. (Which at least involves the greek letter pi I guess...)

$Let $\pi(x)$ denote the number of primes less than or equal to $x$. Then\\ $\frac{\pi(x)\log(x)}{x}\rightarrow 1$ as $x\rightarrow \infty$. So speaking roughly, $\pi(x)$ is approximately equal to $\frac{x}{\log(x)}$ for large $x$.$

Most proofs of it are pretty difficult.

Is there any reason you used to pi(x) rather than say b(x) or something?

Also on a slightly unrelated note the maths teacher who loved primes tutors at usyd now I think, do u know a mr price?

seanieg89 · Mar 15, 2012

Well that's just the standard notation for the function. Has been used for at least a hundred years now. A standard notation is needed because more can be said about the distribution of the primes than just the above asymptotic relation above.

Nope, I don't think I ever met him...

SpiralFlex · Mar 15, 2012

Happy day after pi day!

Carrotsticks · Mar 15, 2012

mirakon said:
Is there any reason you used to pi(x) rather than say b(x) or something?

Also on a slightly unrelated note the maths teacher who loved primes tutors at usyd now I think, do u know a mr price?

Pi and Prime both begin with P. Same reason why a giant Pi symbol is used for Product notation and why a Sigma is used for Sum notation.

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