What is the mathematical proof behind the value of Pi?

[Stephanus]

Dear PF FOrum,
I have a question about Pi.
In some Youtube videos, they explain why the volume of a sphere is so,...
Why the area of a sphere is so,...
Why the area of a circle is so,...
All are good explanations, with or without differential.
But I can't find the proof why Pi is 3.14.
I have searched google (and Youtube).
The best that I have is Archimedes tried to divide a circle into 96 slices.
Is there any math proof why Pi is Pi?
Perhaps like regression number?

Thanks for any idea.

 

[mfb]

Pi is pi by definition.
There are hundreds of ways to find approximate values for it, the 96-sided polygon is one of them. In decimal notation, it happens to be approximately 3.14159. That's just how it is.

 

[Stephanus]

Hi, Mfb. I didn't expect to find you here, I tought you were in Cosmology forum
Perhaps I can state like this.
Euler number is: 0! + 1! + 2! + 3! + 4! + 5! + ...
This is the case where Achilles tries to catch turtle in Zeno paradox, where Achilles runs ten times as fast as the turtle, and their distance is separated by 100 metres.
100 + 10 + 1 + 0.1 + 0.01 + ..., actually Achilles will catch the turtle at... 1000/9 = 111.1111 metres
So, Pi is just Pi, no regresion number like that?

 

[mfb]

There are many series that equal pi (or some related number like pi squared).
As an example, pi = 4/1 - 4/3 + 4/5 - 4/7 + ...
many more formulas

Those are just ways to express pi, in the same way 0!+1!+2!+... equals e, but this is just one way to calculate it.

 

[Stephanus]

There are many series that equal pi (or some related number like pi squared).
As an example, pi = 4/1 - 4/3 + 4/5 - 4/7 + ...
many more formulas

Those are just ways to express pi, in the same way 0!+1!+2!+... equals e, but this is just one way to calculate it.

That's what I'm looking for! Thanks.
And thanks for the link, too.

 

[micromass]

Euler number is: 0! + 1! + 2! + 3! + 4! + 5! + ...

That is not the "Euler number". The "Euler number" is
$$e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + ...$$

Anyway, yes $\pi$ can be defined as the area of a disk with radius $1$. But that of course doesn't explain why the volume/area of a sphere are the way they are. In my opinion, the easiest derivations of such formulas involve integral calculus. Any decent calculus book should explain:
1) The area of a disk
2) The circumference of a circle
3) The volume of a sphere
4) The area of a sphere
And many more like cones and perhaps the torus. The great book by Lang "A first course in calculus" covers all of these. Furthermore, in his chapter on "sine" and "cosine", he has a brilliant exposition which relates the area and the circumference of a circle without using integrals.

 

[SteamKing]

Pi is pi by definition.
There are hundreds of ways to find approximate values for it, the 96-sided triangle is one of them. In decimal notation, it happens to be approximately 3.14159. That's just how it is.

I'm not sure what a "96-sided triangle" looks like. Perhaps, you mean a 96-sided polygon.

 

[SteamKing]

Hi, Mfb. I didn't expect to find you here, I tought you were in Cosmology forum

The Mentors at PF occasionally let us wander around a bit. It keeps the other members on their toes.

 

[Stephanus]

I'm not sure what a "96-sided triangle" looks like. Perhaps, you mean a 96-sided polygon.

Yes and no. Well you have 96 triangles and put it side by side to form a near circle. But of course it has to have angle like 3.75⁰ and 88.12⁵ and 88.125⁰. Of course you have to put the 3.75⁰ angle inside. Yeah, actually it's a polygon.

 

[mfb]

I'm not sure what a "96-sided triangle" looks like. Perhaps, you mean a 96-sided polygon.

I think we should overcome the discriminating idea that all triangles must have three sides.
Yes, I meant 96-sided polygon (or 96 triangles).

I'm not that frequent in the cosmology section.

 

[aikismos]

@Stephanus, I see a lot of good explanations here, but something even simpler.

Take two lengths of string and wrap one around a pop can and trim it so it's the circumference (C).
Take another and lay it across and trim it so it's the diameter (D).

Now measure the lengths of both. You'll find that $D\pi = C$.

If you want more decimal places, find a ruler with finer and finer gradients.

 

[Stephanus]

@Stephanus, I see a lot of good explanations here, but something even simpler.

Take two lengths of string and wrap one around a pop can and trim it so it's the circumference (C).
Take another and lay it across and trim it so it's the diameter (D).

Now measure the lengths of both. You'll find that $D\pi = C$.

If you want more decimal places, find a ruler with finer and finer gradients.

Come on aikismos, it's like you measure the volume of a shpere by puncture it and fill it with water.

 

[Stephanus]

Something just hit me.
The volume of a sphere: $\frac{4}{3} \pi r^3$ is the integral of $4 \pi r^2$ which is the area of a sphere. Is this true? Is this related?
The area of a sphere: $4 \pi r^2$ is the integral of $8 \pi r$ which is four times the of a circle. Is this true? Is this related?

 

[mfb]

The first relation can be seen if you split a sphere (approximately) into many cones, all with their base at the outer shell and with the top in the center.
The second one doesn't make much sense I think.

 

[aikismos]

Come on aikismos, it's like you measure the volume of a shpere by puncture it and fill it with water.

Hahaha, okay. How bout this. In https://en.wikipedia.org/wiki/Pi, Karl Weirstrass is cited as having defined $\pi$ by finding the integral of the top half of the unit circle. In this way, the integral of $\frac{\pi r^2}{2}$

Something just hit me.
The volume of a sphere: $\frac{4}{3} \pi r^3$ is the integral of $4 \pi r^2$ which is the area of a sphere. Is this true? Is this related?
The area of a sphere: $4 \pi r^2$ is the integral of $8 \pi r$ which is four times the of a circle. Is this true? Is this related?

If you're interested in the relation of $\pi$ to derivatives and integrals, Wikipedia (https://en.wikipedia.org/wiki/Pi) has Karl Weierstrass's definition as the integral of the unit circle ($x^2 + y^2 = 1$) from -1 to 1 where x is on or above the x-axis.

 

[VKnopp]

Using the zeta function:

$$\sum_{n=0}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$$Multiply by six and take the positive square root. You can approximate by choosing the number of terms in the partial sum.

 

[HallsofIvy]

Going back to the original post, you won't find a "proof that $\pi$ is 3.14" because it isn't!

There is a proof, going back to the ancient Greeks, that the ratio of the circumference of a circle to its diameter is the same for all circles. That ratio is called "$\pi$". Approximations to the correct value of that number have been done in a variety of ways. For example, Archimedes, using polygons of increasing number of sides, showed that $\pi$ must be between 21/7= 3 and 22/7= 3.1428. For many years 22/7 was the "standard" approximation for $\pi$.

$\pi$ is, in fact, an "irrational" number so cannot be written as a fraction or terminating or repeating decimal. One "test" for new computers is to calculate $\pi$ to many decimal places. I believe the latest "record" is five trillion digits.

 

[Stephanus]

That is not the "Euler number". The "Euler number" is
$$e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + ...$$

Anyway, yes $\pi$ can be defined as the area of a disk with radius $1$. But that of course doesn't explain why the volume/area of a sphere are the way they are. In my opinion, the easiest derivations of such formulas involve integral calculus. Any decent calculus book should explain:
1) The area of a disk
2) The circumference of a circle
3) The volume of a sphere
4) The area of a sphere
And many more like cones and perhaps the torus. The great book by Lang "A first course in calculus" covers all of these. Furthermore, in his chapter on "sine" and "cosine", he has a brilliant exposition which relates the area and the circumference of a circle without using integrals.

Ahh, I just saw your answer now. Yes, sorry. It's a typo.
It can't be 0!+1!+2!+... then it would be greater than 2.71...
Sorry.

 

[Stephanus]

Going back to the original post, you won't find a "proof that $\pi$ is 3.14" because it isn't!

Yes, I know. Thanks for point it out.
Pi is not 3.14, there's no proof about it.
Anyway, it's not the proof that I'm looking for. Perhaps my title is misleading. What I want to know, why Pi is 3.14... There's a youtube video that shows Archimedes divide the circle by 96 pies. But it's like feeling a sphere with water then measure the water.
There is a video about why the volume of sphere is thus. It explains by dividing the sphere into several cones, etc...
There is a video about why the area of a sphere is thus. I forget how the explanation is. But once I see it, I'll remember it all.
But no proof about why Pi is 3.14...
@mfb and @VKnopp have given me the sequence which lead to Pi. It's good enough for me.

There is a proof, going back to the ancient Greeks, that the ratio of the circumference of a circle to its diameter is the same for all circles. That ratio is called "$\pi$".

Yes, by drawing two circle. Small and big which centres coincide. I have to watch the video again to remember it.

Approximations to the correct value of that number have been done in a variety of ways. For example, Archimedes, using polygons of increasing number of sides, showed that $\pi$ must be between 21/7= 3 and 22/7= 3.1428. For many years 22/7 was the "standard" approximation for $\pi$.

Yes, the approximation. But why the sequence?

$\pi$ is, in fact, an "irrational" number so cannot be written as a fraction or terminating or repeating decimal.

Yes, it's an irrational number. In movie "Contact", Jodie Foster uses the frequency of Hydrogen vibration (the supposedly universal frequency which alien civilization might use, we can't use decimal because there's no certainity that alien have ten fingers) times Pi, so that the frequency wouldn't resonance with any other object.
But is there a proof that Pi is irrational? I think this belong to a new thread. Which I don't want to ask right now. Still struggling with the proof why Pi is Pi.
I mean proof like this.
Why the volume of a pyramide is 1/3 height times area. Because if we cross section the box in so and so,...
or
Why the hypotenuse is c^2 = a^2 + b^2, because if we cross a right triangle from the hypotenuse the right angle, then we can find why c^2 = a^2 + b^2.
But no proof of pi.
Number sequence (leading to Pi) is enough for me. I think the proof is in there.

One "test" for new computers is to calculate $\pi$ to many decimal places. I believe the latest "record" is five trillion digits.

That much? Computer harddisk now is about 2 tera bytes. Amazing!

 

[HallsofIvy]

Okay, Stephanus, exactly what are you asking? You asked, first, "what is the proof that $\pi$ is 3.14". But you knew that was not exactly true. Now you are asking "what is the proof that $\pi$ is 3.14...". What do you mean by that? Grammatically, this means "how do we know the first three digits are "3.14?" If you are asking, "How do we know all the digits for $\pi$?" we don't! $\pi$ is an irrational number so has an infinite number of digits. I note that below, on "Similar discussions for, there is "Easy Proof of Irrationality of PI" https://www.physicsforums.com/threads/easy-proof-of-irrationality-of-pi.1570/. Is that what you are asking about- "how do we prove $\pi$ is irrational?"

 

[HallsofIvy]

Okay, Stephanus, exactly what are you asking? You asked, first, "what is the proof that $\pi$ is 3.14". But you knew that was not exactly true. Now you are asking "what is the proof that $\pi$ is 3.14...". What do you mean by that? Grammatically, this means "how do we know the first three digits are 3.14?" If you are asking, "How do we know all the digits for $\pi$?" we don't! $\pi$ is an irrational number so has an infinite number of digits. I note that below, on "Similar discussions for, there is "Easy Proof of Irrationality of PI". Is that what you are asking about- "how do we prove $\pi$ is irrational".

(I got that "5 trillion" figure from http://www.numberworld.org/misc_runs/pi-5t/details.html.)

 

[Mirero]

The reason why $\pi$ is 3.14…… is simply because we have defined $\pi$ to be that constant. It is defined to be the ratio of the circumference of a circle to its diameter, and in our world it just happens to be 3.14….

This happens to be the case because of the way we have defined length in the Euclidean plane. The unit circle in the usual sense is the set of all $(x,y)$ such that $\sqrt{x^2+y^2} =1$. However, one could define a different way to measure length in the plane, and $\pi$ (along with the unit circle) would be different depending on how we define length.

For example, in the context of taxicab geometry, we define the distance of two points $(x_1,y_1)$ and $(x_2,y_2)$ by the formula $d=|x_2-x_1|+|y_2-y_1|$. The distance of any point from the origin in this case is simply the sum of the magnitude of each coordinate. For example, $(3,4)$ would be 7 units from the origin, and $(-4,-8)$ would be 12 units away from the origin. Our unit circle would be the set of all $(x,y)$ such that $|x|+|y|=1$. The unit circle in taxicab geometry ends up looking like the following figure:

The diameter is the greatest length between two points on our circle, so in this case the diameter is 2. Some quick calculations show that $\pi$ in this kind of world is simply 4.

There are, of course, other ways to define a notion of distance between two points in a space: this is called the $\textit{metric}$ of our space. It just so happens that in the metric we usually use, $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} =d$, $\pi$ is the irrational number 3.14...

 

[mfb]

But is there a proof that Pi is irrational?

Sure. There are many proofs, actually.

Still struggling with the proof why Pi is Pi.

By definition.

There are proofs that it is approximately 3.141592653589793238462643383279502884197169399375105820974944... using some of the methods discussed here (the 96-sided polygon is one of them, but you need many more sides to get better bounds).

 

[Stephanus]

Okay, Stephanus, exactly what are you asking? [..]Is that what you are asking about- "how do we prove $\pi$ is irrational?"

"Pi is irrational"? No. That's not what I ask. But that's just crossed my mind. Another idea. I have never thought "Why Pi is irrational?" I'd like to know that . But not in this thread. It's rather off topic. Perhaps a week later.
Emmm... how can I express my question?

Perhaps this:
How can we proof that the diagonal of a square is $\sqrt{2}$ its side?
- Because it's the hyponetuse of symmetry right triangle. My "math English" can't be right. But those who can answer my question, will understand this.

Why the hypotenuse of a symmetry right triangle is $\sqrt{2}$ its side?
- Because $\text{Hypotenuse}^2 = a^2 + b^2$ In this case a = b = 1

Why the hypotenuse of a triangle(C) is $c^2 = a^2 + b^2$?
- Because

I don't have to tell you this. Because you must have already known this outright. But I upload it, so we both can have the same impression.
The angle between a and d is A.
Why? Because <ad = 180 - 90 - B
While 180 - 90 - B = A.
So $c^2 = a^2+b^2$ can only happen in right triangle.
Okay, so <ad = A and <bd = B
$c = e+f$
$\frac{a}{c} = \frac{e}{a}; ce = a^2$

and consequently.

$\frac{b}{c} = \frac{f}{b}; cf = b^2$

$ce+cf = a^2 + b^2; c(e+f) = a^2 + b^2$
$c = e+f$, so $c(c) = a^2 + b^2$
So the ratio of the diagonal of a square by its side is $\sqrt{2}$, because of that.

So the ratio of the circumference of a circle by its diameter is 3.14, because...?

If we divide the circle into 96 slice, it's like we measure the length of the hypotenuse with a rule and say. This is the proof.

But I appreciate for those who gave me the number sequence because that is the answer of my (other) question, how can the computer derive pi. And I read there are sequence of trigonometric. At least for cosine. And we can derive sine from $\sqrt{1-\cos ^2}$. We also tangent, cotangent.
Thans for the answer.

 

[Stephanus]

The reason why $\pi$ is 3.14…… is simply because we have defined $\pi$ to be that constant. It is defined to be the ratio of the circumference of a circle to its diameter, and in our world it just happens to be 3.14….

But it just like saying.
Supposed I have a constants, let's say D, $\sqrt{2} = 1.414$
Mirero, I would never dreamt of challenging a mathematician with my knowledge. Especially in this room.
Okay, let's say this. I change your quote.
"The reason why D is 1.41…… is simply because we have defined D to be that constant. It is defined to be the ratio of the diagonal of square to its side and in our world it just happens to be 1.41…
Why D is 1.41? (I don't know if this number 1.41 is any use in our daily life as compare to 3.14...; 2.718...; 1.618...)
Because, as written in my previous post

This happens to be the case because of the way we have defined length in the Euclidean plane. The unit circle in the usual sense is the set of all $(x,y)$ such that $\sqrt{x^2+y^2} =1$. However, one could define a different way to measure length in the plane, and $\pi$ (along with the unit circle) would be different depending on how we define length.

Perhaps if we calculate the integral of x²+y²=1, we'll have the area. That's good enough because what we are looking for is not the area, but the circumference. But what the h*ll. We only have to divide the area by its diameter, then we'll get Pi.
Is that the easy way?
Is that the right way to find pi?
Now what is left is how to find the integral of $x^2+y^2 = 1$

 

[Mirero]

Wait, were you just asking for a way you could get a computer to numerically calculate $\pi$?

Why D is 1.41? (I don't know if this number 1.41 is any use in our daily life as compare to 3.14...; 2.718...; 1.618...)
Because, as written in my previous post
View attachment 87510

Well, that theorem technically only holds true because of the metric we have defined, and we could probably easily find a metric for which that isn't true.

Perhaps if we calculate the integral of x²+y²=1, we'll have the area. That's good enough because what we are looking for is not the area, but the circumference. But what the h*ll. We only have to divide the area by its diameter, then we'll get Pi.
Is that the easy way?
Is that the right way to find pi?
Now what is left is how to find the integral of $x^2+y^2 = 1$

Dividing the area by its diameter won't give you $\pi$, as $A=\pi r^2$

You can't integrate $x^2+y^2=1$ because it's not a function, as for any solution $(x,y)$ we also have a solution $(x,-y)$. What we can do, however, is take the equation for the upper half of the circle, $y=\sqrt{1-x^2}$ and integrate that from $-1$ to $1$. Multiplying that by 2 will give us the area of the entire circle, and since the radius is one, that area will represent $\pi$.

So in addition to the sequences given prior, one could also calculate $\pi$ from $\pi=2\int_{-1}^{1}\sqrt{1-x^2}dx$.

 

[HallsofIvy]

Dividing the area by its diameter won't give you $\pi$, as ##A=\pi r^2#.

I presume Stephanus intended to say "dividing the circumference by it diameter".

 

[aikismos]

Wait, were you just asking for a way you could get a computer to numerically calculate $\pi$?You can't integrate $x^2+y^2=1$ because it's not a function...

You can if you discard all the values in the third and fourth quadrant.

 

[jbriggs444]

You can if you discard all the values in the third and fourth quadrant.

That looks exactly like what Mirero went on to make explicit.

 

[Stephanus]

I presume Stephanus intended to say "dividing the circumference by it diameter".

I'm sorry HallsOfIvy. your statement is correct. "Dividing the circumference by its diameter", my statement is wrong. "Dividing area by its diameter", but that's what I mean.
I think it's area if you integrate x^2+y^2 = 1, but as Mirero points out, it's not a function.

 

[FactChecker]

[Edit] OTHERS HAVE GIVEN THIS ANSWER, BUT I CAN'T DELETE IT.
There are trig functions that are known to equal π. Their series expansions are used to approximate π to the accuracy desired, but they also prove that the value of π is 3.14159...

For instance, knowing that π/2 = arcsin( 1 ) means that you can use the Taylor series of arcsin to prove the value of π to any accuracy you want. I don't know if this example is practical, but there are many other similar ways. See https://en.wikipedia.org/wiki/Pi#Infinite_series