Why is Pi infinite, sci? Why can't you calculate the area

>>8803608
We can prove that pi is irrational.
I don't think there's a proof that the ratio of circumference to diameter HAS to be irrational, if that makes sense. It just turns out that is is.

Here's a list of proofs that pi is irrational:
https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational

>>8803658
That's interesting, thank you. Sadly, I am no mathematician.

My idea was that Pi has to be infinite, because you can only calculate something that has corners. So if you want to calculate the area of a circle, you think of the circle as a shape with infinitely small corners. And to do that, you need an infinite number, that happens to be Pi. Would that be somewhat correct?

>>8803671
Many cornered shapes need infinite numbers.
Like a right triangle with legs length 1.

>>8803671
Not really, sorry man.
Take a right-angled triangle. Make the two shorter sides have length 1 each. The hypotenuse then has a length of root(2). root(2) is also irrational (cannot be expressed as a fraction, and has an infinite number of non-zero decimals).

>>8803671
Also I think you're confusing yourself by using the term "infinite" number.

Pi isn't really an "infinite" number. It's digits go on forever and don't repeat. Still though, pi lies between 3 and 4. It's a finite number, as in there are numbers larger than it.

We call this property "irrational" - pi cannot be expressed as a fraction (like 1/2 or 3/4). You can get pretty close with some fractions (22/7 is a famous example), but even if you change both numbers to make it more accurate, you'll never have a fraction a/b where a and b are integers (1,2,3,4...) which exactly equals pi.

Pi is also transcendental. This means that you can't make an integer with pi simply by adding and multiplying factors of pi together (like pi squared + pi cubed + 2*pi)

>>8803608
No circle has ever been calculated using pi. Any time you think you've used it you've really just substituted an ersatz facsimile using a rational approximation.

>>8803608
Maybe pi is normal? http://www.pi314.at/math/normal.html

>>8803706
What about pi +(-pi)?

>>8803726
Good spot, but I was simplifying the actual definition of a transcendental number, which is a number that is not a root of any non-zero polynomial with integer coefficients.

You've described x=pi as the solution to x-x=0, which is just 0=0.

>>8803689
>>8803679
Isn't that kind of a special case though? If i remember correctly Trigonometry is based on a circle with radius 1, and a right triangle with two legs being 1 is basically a quarter of that circle.

>>8803755
Nah man, other irrational numbers appear in trig too. You can get root(2) and root(3) from triangles too.

https://en.wikipedia.org/wiki/Trigonometric_constants_expressed_in_real_radicals

>>8803706
Well, what i meant is you calculate the are of the circle by thinking about it as smaller sqaures with the length of the radius. So obviously 3 squares are smaller than the circle, and 4 are bigger. So the 4th square gets broken into infinetely more smaller squares "to cover" to remaining area of the circle. These smaller squares are describes by pi minus 3.

>>8803762
But right triangles are just pieces of a circle.

Shapes are fucking my head man.

>>8803765
What you're trying to do is say that 4 squares of side length r (for radius) - an infinite number of smaller squares equals pi?

>>8803608
>Why can't you calculate the area of a circle using a normal number?
but you can. the result will be wrong though

>>8803776
No they're not.
You'd need an infinite number of infinitely small triangles to fill a circle directly.

Don't think of a circle as being "made up" of smaller shapes. That's where this confusion can come from.

Good old geometry can find irrational numbers like pi or root(2) from the ratios between side lengths of shapes.

The ratio of a circle's diameter to it's circumference is irrational.
The length of the hypotenuse of a triangle with short side length x is x*root(2).

>>8803780
But isn't any calculation using pi wrong by the degree that we don't know pi to?

>>8803786
It's inaccurate depending on how precise a value for pi you use.

The thing is, we can calculate pi for as long as we need it to, pretty easily. There are formulae that can calculate any decimal place of pi you like. You don't even need to know the decimals before it.

>>8803786
welcome to the wonderfully frustrating world of partial sums, anon

>>8803778
Nah, 3 squares with the length r plus infinetely small brackets of one of those squares equal... well, not pi, but the area of the circle with the radius r. Since you now somehow have to calculate those infinietely small brackets you end up with pi sqaures. So basically, i think about Pi as 3 + an irrational number (with this irrational number obviously being pi minus 3). So this irrational number describes how many "parts" of a whole square you would need to describe the remaining area. And since you would need infinite infinietely small brackets of that square you end up with that irrational number. Hope that makes sense.

>>8803799
I guess, but I wouldn't recommend thinking about irrational numbers that way.

You shouldn't break up "irrational" shapes into infinitely many smaller shape.
Well, you can if you like. But it's not really that useful.

>>8803799
Could you draw what you mean?

File: 140503839566.jpg (37KB, 500x503px) Image search: [Google]

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>Why is Pi infinite
[math]|\pi | < 4 < \infty[/math]

File: YmN6wVi.jpg (57KB, 715x810px) Image search: [Google]

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>>8803869
they're a bit confused about irrationality, don't bite their head off

>>8803828
Well, I could, but I don't know if that would make it less confusing.

So let me rephrase it. You try to describe the area of a circle using squares. You take squares the length of the radius of the circle. You end up with it being a little more than 3 of these squares. This "a little more" is the percentage of a 4th square you need to fully cover the whole area of the square. So let's say in a parallel universe Pi is exactly 3,1. That would mean you need exactly 10% of the 4th square. In our universe though this "a little more" is an irrational number. And that is because what you need of the 4th square are infinitely small pieces.

So yeah my though basically was this. And the question that made me open this thread is could you not calculate that remaining area in some way that doesn't involve irrational numbers. Thank you.

>>8803883
*fully cover the whole area of the CIRCLE

File: chrome_2017-04-04_21-53-44.png (3KB, 256x41px) Image search: [Google]

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>>8803883
Sounds like you're almost defining the way a limit works, or a convergent series.

We can actually do this. Image related is a formula that does this.

The problem is, this series only equals pi exactly if you let it run forever. If you stop it at any point, it's not equal to pi, just closer and closer to it.

>>8803900
So would you say it is correct if I imagine Pi to be 3 squares and the "percentage" of the 4th square?

And are there other ways to calculate the area of a circle?

Because so far I don't believe that circle really exist.

[math]3 < \pi < 4[/math]

>>8803909
Yeah, if you like. But the percentage of that 4th square is not a "nice" percentage to calculate or write down, if that makes sense. That percentage would probably still also be irrational (have an infinite number of digits).

There are other methods to calculate the area of a circle, but you always end up back at the same point: Area = pi*(Radius)^2. That is an invariant result.

You're identifying that a "perfect" circle cannot exist. You're correct. However, we can use enough digits of pi that we literally cannot measure the difference (I think it's around 50 digits is more than enough for any calculation).

>>8803909
>Because so far I don't believe that circle really exist.
Babby's first philosophy.

What does it mean to "perfectly exist"? Atoms can only get so close to a perfect geometrical shape. By that, ANY mathematically defined shape "doesn't really exist".

>>8803608
>DUDE PI IS INFINITE LMAO

>>8803999
I respect your trips, but i meant it doesnt even exist theoretically. Even theoretically you can only describe a shape with a lot of corners, how many depends on how many digits of pi you use. This is not true for the other shapes. Those have precise identities theoretically.

>>8804035
nah man, you can describe a circle perfectly using the ratio of diameter to circumference (pi).

Pi is an exact figure. it's decimal representation is infinitely long

>>8804064
Infinite =/= exact

The circle is a weird shape, maybe one day i will understand it.

>>8804080
Pi is an exact figure man. it's decimals are infinitely long, but we know what we are and can calculate as many as we like (around 100 million at least IIRC)

you can define Pi exactly as the ratio between the circumference and diameter of a shape drawn by the equation x^2+y^2=r^2

>>8803608
Because a true circle doesn't exist. Best you can do is calculate the surface area of an n-gon (perfect polygon with n sides) where n goes towards infinity. As you add more sides, they get smaller and the shape resembles a circle more and more. If Pi was finite, it would mean that a cricle is just an n-gon with a fixed amount of sides. But in reality, Pi is infinite because a circle is an abstract shape. You can always add more sides to an n-gon and you will never get a circle. Just like infinity - it's not a number, because you can simply increment any given number, resulting in a bigger number, meaning that the original number isn't infinity.

>>8803999
>ANY mathematically defined shape "doesn't really exist"
That's almost true, but...
An equipotential line around a point particle is perfectly spherical, and really exists.

>>8804335
Assuming point particles exist.

>>8804352
non-point particles still have point particles at their center

>>8803608
Because the area of a circle of radius 1 is infinity.

>>8804367
1-d particles, maybe, but if that resolution of the QM/GR conflict is accurate I'm not seeing the infinite subdivisions as being valid. As in, no point particles allowed.

>>8804473
that's not quite what i meant
if you wanted to describe the gravitational acceleration generated by an object, you would model that object's center of mass as a point particle. The object is not pointlike, but its center of mass is. And the equipotential surface around that object's center of mass in 3 dimensional space would be a perfect sphere regardless of the actual geometry of the object.

>>8803608
What's the probability of pi existing?

It comes down to perfect circles, an imaginary object, having infinite points.

In reality you would never need to use more than 3.14159.

>>8803608
Because circles dont actually exist

>>8804652
why don't you drop some matchsticks to find out

Makes you think that wildberger is on to something