Prove pi doesn't equal 4
>>7862534
p*i = 4 iff p = 4 and i = 1 OR p = 1 and i = 4 OR p = i = 2 or p = i = -2
>>7862534
Define the transcendental number pi := 3.14159....
Then pi != 4. QED
>>7862534
Because nobody spells qt4.0 instead of qt3.14
Clearly, a circle of radius 1 can be properly inscribed in a square of radius 2
The circle has area pi and the square has radius 4, so pi < 4
>>7862559
>square of radius 2
whoops, I meant square with side of length 2
>>7862534
What's interesting is finding a method of expressing 4-pi as a series of squares (or a series of series of squares)
A while back I tried this but could only figure out how to recursively compute the squares along the large square's perimeter. I wonder if it's possible without having to start computing from the 1st one onwards...
>>7862534
the hypotenuse of each perpendicular edge around the square is a better approximation to pi
assuming we are dividing equally they will be some sigma n*sqrt(2) and the would the sum converge to pi?
You could use the same method to show that the square root of 2 is 2.
>>7862534
You just proved it. no matter how much you try to bound it into squares it never reaches 3.14, it will always remain 4
From the Barnett identity:
[eqn]\sum_{n \,=\, 0}^\infty n \,=\, \frac{{\rm e}^{\mathrm i \,\pi}}{11.999\dots}[/eqn]
By multiplying by the 12th bait number, we get
[eqn]11.999\dots \,\times\, \sum_{n \,=\, 0}^\infty n \,=\, {\rm e}^{\mathrm i \,\pi}[/eqn]
Apply the natural log and you get
[eqn]\ln\,11.999\dots \,+\, \ln\,\sum_{n \,=\, 0}^\infty n \,=\, \mathrm i \,\pi[/eqn]
Finally, by squaring both sides and multiplying by −1, we get the Archimedes-Barnett identity:
[eqn]\pi \,=\, -\ln^2\,11.999\dots \,-\, 2\ln\,11.999\dots \,\times\, \ln\,\sum_{n \,=\, 0}^\infty n \,-\, \ln^2\,\sum_{n \,=\, 0}^\infty n[/eqn]
Which, after applying a first order Wildberger transform, gives us:
[eqn]\pi \,=\, -0.9 \,-\, 2 \,\times\, 2 \,\times\, \infty \,-\, \infty \,=\, -0.9 \,-\, 5 \,\infty \,\neq\, 4[/eqn]
Q.E.D.
>>7862534
>What is pointwise convergence
>What is uniform convergence
>>7862534
>spouting meme arrow
>not having a clue why the argument is wrong
There is uniform convergence dipshit.
>>7862625
beautiful
>thanks /sci/ for doing my homework, hehe
>dumb faggots
Do your own Real Anal homework you effeminite homo
>>7862534
https://en.wikipedia.org/wiki/Method_of_exhaustion#Archimedes
>>7862581
Can someone please verify if it's possible to express the terms of that series of squares non-recursively?
>>7862706
>tfw no qt math nerd girl to have rigorous real anal sessions w/
>>7862697
Google.
>>7862691
> No. No there isn't.
Of course there is.
The reason it does not work is because the limit curve isn't so smooth.
>>7862628
\thread
>>7862543
I bet you're the P=NP guy in every thread
>>7862625
keked heartily
Consider the upper right quadrant of the circle.
The curve you end up with after iterating through this process is non-differentiable at every dyadic rational value of x, which is very much not a property of a circle. Therefore the limit of this sequence is not a circle, but it does indeed have arclength 1. If you were approximating area, this process would likely give a much more positive result.
>>7862625
First order wildberger transform omg where have u been all my life
>>7862625
Bodacious
>>7862980
No that's me, he's just an imitator plebshit
>>7863625
https://www.youtube.com/watch?v=68JQtxTzjqc
>>7862534
Okay, so you are trying to tell us:
>The area of a circle is A = pi * (r^2)
>The radius of the circle is 1
>A = pi * (1^2) = pi
>The surrounding square has L = 2
>The area of a square is A = L^2
>A = 2^2 = 4
>There are four regions which don't belong to the circle around it
>PIEQUALSTHEAREAOFTHESQUARE.jpg
You can proof pi doesn't equal 4 by doing a circular integer with the radius of the circle.
>>7862980
P=0 or N=1
BOOM SOLVED
>>7862534
In a different topology, your proof actually is correct.
The same way that you can use that prove that the hypotenuse of a triangle is equal to the sum of its other 2 sides, not squared. This is true in a different topology.
But PI is actually defined as a number. So even though in such a topology you can prove that the circumference of your circle is 4 or whatever you want, you haven't proved PI is 4. Because PI is a specific real number and that number is not 4.
I.e. this: >>7862548
That's not actually pi though: >>7862559
That's just a thing which is usually equal to PI. You found a case where it isn't equal to PI. Doesn't change PI.
[math]length( lim_{n \to \infty} A_n ) \neq lim_{n \to \infty} length( A_n)[/math]
>>7864637
no one gives a shit
I know it's a persian knitting imageboard here, but I kind of wasn't expecting shitposting when discussing actual math/science.
What was I thinking.
>>7864880
Why? It's a lot more interesting than the main topic of this thread, about which there is nothing to discuss.
>>7864871
a different topology? what does topology have to do with this? give an example of one such topology or accept you're spewing bullshit
>>7864944
Are you fucking serious? The trivial topology of course.
>>7864944
https://en.wikipedia.org/wiki/Taxicab_geometry
>>7862534
>Prove pi doesn't equal 4
pi can be defined as the area of a unit circle, 1/4 of the area of a unit circle is given by [math]\int^1_0\sqrt{1-x^2}\ dx=0.785[/math] to 3 significant figures.
Hence [math]\pi=4*0.785=3.14\neq{4}[/math] QED
>>7862691
>No. No there isn't.
are you implying there is a point that stay far from the circle at any iteration of the sequence ?
Or maybe you don't even know what uniform convergence is?
>>7865008
>What the fuck does Pi mean in the trivial topology?
>Pi
Are you even following the discussion?
>>7862625
Someone, give this nigga an official 4chan seal of approval.
>>7862534
my maths teacher said so
>draw circle
>measure circumference with string
>measure string
>measure diameter
>divide numbers
not 4
>>7862625
10/10
