Happy Pi Day!

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Re: Happy Pi Day!

#21  Postby jamest » Mar 15, 2016 9:43 am

Evolving wrote:
jamest wrote:Blimey. I put 999999999 into the 2nd link for a laff and was shocked to see it come up after about 500 million digits. I just don't get that. I mean, how can any number divided by 7 produce such a lengthy string of 9s? :scratch:


22/7 is only an approximation.

Really? So what's the real deal?
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Re: Happy Pi Day!

#22  Postby Evolving » Mar 15, 2016 9:46 am

Pi is an irrational number, meaning that it cannot be expressed as a fraction. (Any attempt to do so leads to a contradiction, as the Greeks established long ago.)
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Re: Happy Pi Day!

#23  Postby jamest » Mar 15, 2016 9:48 am

Evolving wrote:Pi is an irrational number, meaning that it cannot be expressed as a fraction. (Any attempt to do so leads to a contradiction, as the Greeks established long ago.)

Okay, so what method/numbers does one use to produce an accurate portrayal of pi?
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Re: Happy Pi Day!

#24  Postby Evolving » Mar 15, 2016 9:53 am

Pi is defined as the ratio between the circumference and the diameter of a circle, and is as such "accurate" by definition. That is the number that we use in trigonometry and in areas of physics that make use of the concept ("angular" concepts like angular momentum). If we want a decimal representation, we can get arbitrarily close, but it's impossible to represent any irrational number exactly as a decimal.

The same applies to any irrational number, such as the square root of two, square root of five etc.
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Re: Happy Pi Day!

#25  Postby VazScep » Mar 15, 2016 10:06 am

jamest wrote:
Evolving wrote:Pi is an irrational number, meaning that it cannot be expressed as a fraction. (Any attempt to do so leads to a contradiction, as the Greeks established long ago.)

Okay, so what method/numbers does one use to produce an accurate portrayal of pi?
The early geometers would do it by drawing two convex polygons with a large number of sides whose vertices each touch the circumference of a circle. One polygon is drawn to lie inside the circle (inscribing) and the other drawn to lie outside (circumscribing). You can calculate the length of the perimeters of these polygons, divide by the diameter of the circle, and then you have an upper and lower bound for the value of π. Archimedes proved that this method could get you within an arbitrary distance of π.

An "analytical" definition of π is often given by saying it's the sum of the series:

4(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...)

If you compute the bracketed sum out to a few thousand terms, you'll get a somewhat crappy approximation of π. More efficient methods can be found here.
Last edited by VazScep on Mar 15, 2016 10:39 am, edited 1 time in total.
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Re: Happy Pi Day!

#26  Postby VazScep » Mar 15, 2016 10:09 am

Evolving wrote:Pi is an irrational number, meaning that it cannot be expressed as a fraction. (Any attempt to do so leads to a contradiction, as the Greeks established long ago.)
Are you sure the Greeks established this? I'm not sure they had the means.
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Re: Happy Pi Day!

#27  Postby Evolving » Mar 15, 2016 10:11 am

I was quoting from memory, which may be faulty. Possible that I was getting it mixed up with the square root of 2.
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Re: Happy Pi Day!

#28  Postby VazScep » Mar 15, 2016 10:43 am

Evolving wrote:I was quoting from memory, which may be faulty. Possible that I was getting it mixed up with the square root of 2.
Yeah, they definitely proved that. I think Plato went so far as to say the proof should be common knowledge for all citizens.
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Re: Happy Pi Day!

#29  Postby logical bob » Mar 15, 2016 4:05 pm

Plato was correct, obviously.

According to Wikipedia pi was proven to be irrational in 1761 using a bizarre looking method of expressing trigonometric functions as continued fractions which could somehow be demonstrated to be irrational. There's also a one page proof that could be done with A level calculus - so no, none of this would have been available to the Greeks.

Pi is also transcendental, meaning that it's not a solution to

a0 + a1x + a2x2 + ... + anxn = 0

for any rational numbers a0, a1, ... , an.

I remember that I used to be able to prove that, but looking at the proof now I can't tell what's going on. That makes me sad. :(
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Re: Happy Pi Day!

#30  Postby logical bob » Mar 15, 2016 4:29 pm

VazScep wrote:More efficient methods can be found here.


Euler wrote:pi = 20 arctan 1/7 + 8 arctan 3/79

WTF?

Ramanujan wrote:1/pi = (2√2/9801)Σk=1(-1)k(6k)!(1103+26390k)/(k!)43964k

:ahrr:
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Re: Happy Pi Day!

#31  Postby Evolving » Mar 15, 2016 5:02 pm

Those formulae look utterly ridiculous, don't they? How can they possibly be right? (one thinks). Somehow very pleasing that they actually are.
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Re: Happy Pi Day!

#32  Postby Evolving » Mar 15, 2016 5:04 pm

I was indeed thinking of the square root of two, and the reason I got them mixed up is because, for the Greeks, the square root of two was about triangles and Pythagoras's formula. So in both cases it's about geometry.
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Re: Happy Pi Day!

#33  Postby jamest » Mar 15, 2016 5:08 pm

logical bob wrote:
VazScep wrote:More efficient methods can be found here.


Euler wrote:pi = 20 arctan 1/7 + 8 arctan 3/79

WTF?

Ramanujan wrote:1/pi = (2√2/9801)Σk=1(-1)k(6k)!(1103+26390k)/(k!)43964k

:ahrr:

I'll settle for 22/7. :shifty:
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Re: Happy Pi Day!

#34  Postby tuco » Mar 15, 2016 5:24 pm

Better late than never, they (dont ask me who) say

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Re: Happy Pi Day!

#35  Postby Cito di Pense » Mar 14, 2019 6:31 am

Oops, skipped a year or two.
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Re: Happy Pi Day!

#36  Postby newolder » Mar 14, 2019 8:54 am

Evolving wrote:Pi is defined as the ratio between the circumference and the diameter of a circle, and is as such "accurate" by definition. That is the number that we use in trigonometry and in areas of physics that make use of the concept ("angular" concepts like angular momentum). If we want a decimal representation, we can get arbitrarily close, but it's impossible to represent any irrational number exactly as a decimal.

The same applies to any irrational number, such as the square root of two, square root of five etc.


Is the square root of each prime number irrational? I don't know a) the answer (but I think it's a yes) and b) how to prove such a thing. :dunno:

Happy π day! (It's like a caek day but irrational.)
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Re: Happy Pi Day!

#37  Postby Cito di Pense » Mar 14, 2019 9:20 am

newolder wrote:
Is the square root of each prime number irrational?


Do the vectors (π, e) and (-π, e) form a basis for R2? That is, can we express any vector in R2 as a linear combination of these two vectors?

What about a vector in R2 whose 2-norm is a prime integer?
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Re: Happy Pi Day!

#38  Postby Evolving » Mar 14, 2019 11:08 am

newolder wrote:
Evolving wrote:Pi is defined as the ratio between the circumference and the diameter of a circle, and is as such "accurate" by definition. That is the number that we use in trigonometry and in areas of physics that make use of the concept ("angular" concepts like angular momentum). If we want a decimal representation, we can get arbitrarily close, but it's impossible to represent any irrational number exactly as a decimal.

The same applies to any irrational number, such as the square root of two, square root of five etc.


Is the square root of each prime number irrational? I don't know a) the answer (but I think it's a yes) and b) how to prove such a thing. :dunno:

Happy π day! (It's like a caek day but irrational.)


I suppose you would have to prove it by contradiction: let there be a fraction a/b whose square is the prime x, then square it and see where it all goes wrong.

I can’t see how to do it atm, though.
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Re: Happy Pi Day!

#39  Postby Evolving » Mar 14, 2019 11:11 am

Cito di Pense wrote:
newolder wrote:
Is the square root of each prime number irrational?


Do the vectors (π, e) and (-π, e) form a basis for R2? That is, can we express any vector in R2 as a linear combination of these two vectors?

What about a vector in R2 whose 2-norm is a prime integer?


Those two vectors are not linearly dependent, so yeah.

What about the vector whose norm is a prime? What’s the question?
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Re: Happy Pi Day!

#40  Postby scott1328 » Mar 14, 2019 12:06 pm

newolder wrote:
Evolving wrote:Pi is defined as the ratio between the circumference and the diameter of a circle, and is as such "accurate" by definition. That is the number that we use in trigonometry and in areas of physics that make use of the concept ("angular" concepts like angular momentum). If we want a decimal representation, we can get arbitrarily close, but it's impossible to represent any irrational number exactly as a decimal.

The same applies to any irrational number, such as the square root of two, square root of five etc.


Is the square root of each prime number irrational? I don't know a) the answer (but I think it's a yes) and b) how to prove such a thing. :dunno:

Happy π day! (It's like a caek day but irrational.)

From the math forum
Code: Select all

Date: 10/08/97 at 17:00:17
From: Terry Dobbins
Subject: Irrational numbers

My question is: Will all square roots of positive numbers that
are not perfect squares be irrational numbers?

I am a new teacher and this was asked of me by another teacher.
I think that it is a true statement but I can't prove it.

Thanks for the help.


Date: 10/08/97 at 18:18:08
From: Doctor Tom
Subject: Re: Irrational numbers

Yes. They are all irrational. The proof is similar to the proof that
sqrt(2) is irrational.

In case you haven't seen that, here's how it goes:

Suppose sqrt(2) is rational. Then you can write sqrt(2) as a/b, where
a and b are integers, and the fraction is reduced to lowest terms.

So a^2/b^2 = 2 so a^2 = 2*b^2. So a is even. Since it's even, write
a = 2*c.  (2c)^2 = 2*b^2 or 4c^2 = 2b^2 or 2c^2 = b^2, so b is also
even. But then you didn't reduce a/b to lowest terms since they both
have a factor of 2.

To show that sqrt(p) is irrational where p is a prime number, the same
approach works, except instead of saying "a is even," you'll be saying
"a is a multiple of p."  The proof goes the same way, except that you
find that a and b are both multiples of p, and hence your original
fraction wasn't reduced as you said it was.

For an arbitrary number n that's not a perfect square, you can factor
it as follows:

n = p1^n1*p2^n2*p3^n3*... for a finite number of terms. At least one
of the n1, n2, n3, ... must be odd, or n is a perfect square.
Suppose n1 is the one that's odd. If n1 is 1, just go through the same
proof above and show that the a and b in your a/b are multiples of p1. 
If n is odd and bigger than one, write your a/b as a*p1^((n1-1)/2)/b. 
That'll get rid of the part of the product of primes that's a perfect
factor of p1.

To make this concrete, suppose I want to show that 216 does not have a
rational square root.

  216 = 2^3*3^3.

If 216 has a rational square root, it will be  2*sqrt(216/2^2)
= 2*sqrt(54), so let sqrt(216) = 2*a/b, reduced to lowest terms. 
Then 4a^2/b^2 = 54, so 2a^2/b^2 = 27, or 2a^2 = 27b^2, so b must be
even. There's already a contradiction.

-Doctor Tom,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/   


Date: 10/08/97 at 18:31:06
From: Doctor Wallace
Subject: Re: Irrational numbers

Dear Terry,

The answer is yes, all non-perfect square square roots are irrational. 
Remember that a rational number is one that can be expressed as the
ratio of 2 integers. If you look in our archives, you'll find a proof
for the fact that the square root of 2 is irrational. Search on the
terms irrational and square root of 2. I won't repeat the details
here, except to say that the proof involves assuming that the square
root of 2  IS  rational, and working to a contradiction. The proof is
simple and elegant. If I remember correctly, there is also a proof in
the archives for the square root of 3.

As to a general proof that ALL non-perfect square square roots are
irrational, I'm not sure. I know that one exists, though. Perhaps it
is accomplished through extension of the two proofs I mentioned. 

I hope this helps.  Don't hesitate to write back if you have more
questions.

-Doctor Wallace,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/   
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