An argument of infinite proportions (pi)
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@unknownuser said:
That makes sense, however there is nothing to say that a ratio cannot work properly without an end, only infinitely increasing levels of accuracy (which on its own is a paradox, because you cant do better than perfect right?)
Sorry, I dont get you.
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So who was the genius that coined the phrase "easy as pi"?
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A number isn't just something made up it's the answer to an equation. Under that definition if you take that answer to equations can sometimes be quantified as between a real & imaginary number and infinity then numbers can go on forever. Numbers can be limitless.
You should tell him 2 isn't really 2 and see what he says.
Pi is limitless, Pie is not... especially around guys from the south. -
Okay, maths is very much my weakness, but I'm confused by your stance. In the other forum you wrote:
"I say they never end as pi is an irrational (which by definition never ends) and that 1/3 also never ends as it is a non terminating repeating decimal."
But in your post here you state:
"Now, anyone who paid attention in math class in middle school knows that he is wrong, that pi is called an irrational number for a reason, and that 1/3 does in fact end, thats why its called a repeating, terminating decimal."
For the record I agree that neither number ends- especially 1/3 as a decimal. Your "opponent" ain't worth the argument if he seriously reasons that 1/3 expressed as a decimal logically ends in a 3!
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What are you guys smoking? :egrin: Pi r not Sq...Pi art round!
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@masta squidge said:
someone in the profession of calculating stupid numbers...
Recently it was reported that some university professor here in Canada completed a massive seven or eight year long calculation using a supercomputer. He was able to absolutely prove that a game of checkers will always end in a draw provided neither player ever makes a mistake or cheats. It is only the mistakes and cheats that swing the outcome away from a draw.
Its great to see my tax dollars at work.
I can't understand why anyone would think 1/3 would ever end. The repeating character of the ratio seems easy enough to comprehend as unending. I hope that university professor doesn't decide to tackle the 1/3 number next.
Regards Ross
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@unknownuser said:
...and that 1/3 does in fact end, thats why its called a repeating, terminating decimal."
I believe that is what we call a typo
I believe it should read "does in fact NOT end, thats why its called a repeating, NON-terminating decimal.Yeah, I slipped, it was past my bedtime.
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@masta squidge said:
Please by all means message this joke to me! Unless its the one about the Special Olympics of course lol, because that is all too familiar.
Yep, that's the one.
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Not that it will help, but here is a mathematical proof that pi is irrational:
@unknownuser said:
Niven's proof
The proof uses the characterization of π as the the smallest positive zero of the sine function. Like all proofs of irrationality, the argument proceeds by reductio ad absurdum.
Preparation: Suppose that π is rational, i.e. π = a / b for some integers a and b ≠ 0, which may be taken without loss of generality to be positive. Given any positive integer n, we define the polynomial function
f(x) = {x^n(a - bx)^n \over n!},\quad x\in\mathbb{R},and denote by
F(x) = f(x) + \cdots + (-1)^j f^{(2j)}(x) + \cdots + (-1)^n f^{(2n)}(x). \,the alternating sum of f and its first n even derivatives.
Claim 1: F(0) = F(π)
Proof: Since
f(x)=b^n{x^n(\pi - x)^n \over n!}=f(\pi-x),\quad x\in\mathbb{R},the chain rule and mathematical induction imply
f^{(j)}(x) = (-1)^j f^{(j)}(\pi - x),\quad x\in\mathbb{R},for all the derivatives, in particular
f^{(2j)}(0)=f^{(2j)}(\pi)\,for j = 1, 2, ...,n and Claim 1 follows from the definition of F.
Claim 2: F(0) is an integer.
Proof: Using the binomial formula to expand (a – bx)n and the index transformation j = k + n, we get the representation
f(x)={1\over n!}\sum_{j=n}^{2n}{n \choose j-n}a^{2n-j}(-b)^{j-n}x^{j}.\,Since the coefficients of x0, x1, ..., xn − 1 are zero and the degree of the polynomial f is at most 2n, we have f (j)(0) = 0 for j < n and j > 2n. Furthermore,
f^{(j)}(0)={j!\over n!}{n \choose j-n}a^{2n-j}(-b)^{j-n}\quad\mbox{for } n\le j\le 2n.Since j ≥ n, the fraction of the two factorials is an integer. The same holds for the binomial coefficient, as can be seen from its combinatorical interpretation or Pascal's triangle. Hence f and every derivative of f at 0 is an integer and so is F(0).
Claim 3:
\frac12 \int_0^\pi f(x)\sin(x)\,dx=F(0)Proof: Since f (2n + 2) is the zero polynomial, we have
F'' + F = f.\,The derivatives of the sine and cosine function are given by (sin x)' = cos x and (cos x)' = −sin x, hence the product rule implies
(F'\cdot\sin - F\cdot\cos)' = f\cdot\sin.By the fundamental theorem of calculus
\frac12 \int_0^\pi f(x)\sin(x)\,dx= \frac12 \bigl(F'(x)\sin x - F(x)\cos x\bigr)\Big|_{x=0}^{x=\pi}.Since sin 0 = sin π = 0 and cos 0 = –cos π = 1 (here we use the abovementioned characterization of π as a zero of the sine function), Claim 3 follows from Claim 1.
Conclusion: Since f(x) > 0 and sin x > 0 for 0 < x < π (because π is the smallest positive zero of the sine function), Claims 1 and 3 show that F(0) is a positive integer. Since
x(\pi -x) = \Bigl(\frac\pi2\Bigr)^2-\Bigl(x-\frac\pi2\Bigr)^2\le\Bigl(\frac\pi2\Bigr)^2,\quad x\in\mathbb{R},and 0 ≤ sin x ≤ 1 for 0 ≤ x ≤ π, we have
\frac12 \int_0^\pi f(x)\sin(x)\,dx\le \frac{b^n}{n!}\Bigl(\frac\pi2\Bigr)^{2n+1},which is smaller than 1 for large n, hence F(0) < 1 by Claim 3 for these n. This is impossible for the positive integer F(0).
Not that I understand it, but there it is.
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Aha...
Well, thanks - at least for that very last line (now I can hope I'm not alone in the universe who does not understand it
).
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