The Exact Value of PI. Well, not exactly ...

Zmurf · 2006-05-03 18:03:49

Ok this is my formula for finding PI. If anyone can work out a way to present infinity as a number (lol?) then hurray! It's pretty stupid but here it is, my formula for finding PI.
____________
PI = (180/θ)√(2(1 - COS θ))

The smaller θ is the more accurate PI will be found. So to find PI θ = 10^-∞. lol.

BUT when I was messing around on my calculator subbing in increasingly (or decreasingly) smaller numbers PI became for innacurate (in comparison to the first 20 decimal places of PI from google). But from my logic of the way i figured it out, a smaller number should yield more accurate representation of PI!

I did this. Break a circle into 4 RH Triangles. Find the hypotinuse (Spelling?) of all of them. You'll get a super innacurate measurement of PI. But it's split the circle into 4 90 degree segments. Using pythagorisis (Ok just overlook my spelling from here on out >.<) theorom c^2 = a^2 + b^2, the circumfrance^2 would be equal to (360/θ)*(b^2 + c^2) (θ being 90, to yield the 4 segments to multiply by th a^2 b^2 part of pythagorisis' theorum to get 4 times th elength of 4 congruentr triangles that made up my super rough circle) b and c would both be equal to the radius. But to make it more accurate I split the circle into more than 4 segments. segments of say 0.01 degrees each. but i can't use a^2 + b^2 to find the hypotinuse so I use extended pythag's theorum a^2 + b^2 - 2ab COS θ) since a = b i get 2r^2 - 2r^2Cos θ. whe nthe radius of the circle is 1/2 then the circumfrance is = to pi. which is why it is 180/θ not 360/θ so i wouldn't have to put the whole thing over 2. in pythag's theum the r^2 is consistent so it gets removed making it just r * √ which is how 360 gets halved. i lost the working out, now i forget how im getting places. you people are much better at this than me so i'm sure you'd understand, though my grammar drags, my grammar should be better since i'm in year 11 now ... anyways, yea thats pi .. still can't work out why a smaller number won't make it more accurate, anyone know?

yttrium88 · 2006-05-04 08:59:41

Good Work! The formula that you came up with is a good one. The only problem with it is the cos(θ) part. Since we do have caclulators and use them all the time, we kind of take for granted the fact that we can always have a perfectly accurate cosine function. The problem is, our calculators actually only give acurate trigonometric functions for the 10 digits that show on our calculators; or 11 digits, or 12, or 13...anyway, the calculator is not accurate for an infinite number of digits, so what this problem really comes down to is:

How can we get infinitly accurate trig functions, or more truthfully, functions as accurate as we want to make them?

Before proceding on, I think it would be useful to know how much math you've had, especially experience with: radians, trigonometric functions, inverse trigonometric functions, and sigma notation.

P.S. It is interesting to note that a few years ago, I myself pondered the same question about pi.

MathsIsFun · 2006-05-04 09:38:40

Good point - you would need to either use an expansion of cos θ (would have an infinite series of terms), or replace it entirely.

Archimedes followed a similar method using geometry - he used polygons with many sides, and then drew them inside the circle and outside so that he would get a lower and upper limit on Pi.

See if you can rework the formulas to get rid of cos θ. Note also that sin θ becomes very close to θ for small values - that may help you (see my page on Radians, near the bottom)

Zmurf · 2006-05-04 17:34:14

Unless theres another way find the side of a non-right angle triangle when given 2 sides and the angle between them, I don't know any way to get rid of cos. And a radian incorporates PI, wouldn't that make my formula infinitly long? Since 180 = PI Radians? Or is that only needed when converting radians and degrees.

I'm doing year 11 Extension 1 maths. I'm not sure what the equivilent is in america, but were covering interiro and exterior reatios of line intervals. We just finished 3d trigonometry (which was so easy, its just 2d trig with more than one triangle, I found it especially fun because everyone else found it difficult lol). I just started thinking, "I wonder how they figured out was PI was?" when I was supposed to be studying, so I spent the next hour devising this formula lol.

I'm going to make a console app in C++ to print the value of PI according to the formula for as many as possible, but I'll have to settle using 10^-999 until I can find infinity, I guess thats just for a later formula lol. (Anyone know how to grab the first decimal place of a number?)

By the way Admin, I love the quote in your sig, it's funny because it's true.

Last edited by Zmurf (2006-05-04 17:54:09)

MathsIsFun · 2006-05-04 19:47:44

How about Sin²θ + Cos²θ = 1,

So Cos θ = √(1-Sin²θ)

And for small θ, Sinθ ≈ θ, so Cos θ ≈ √(1-θ²)

Zmurf · 2006-05-04 23:34:10

So the new formula would be:

(180/θ)√(2(1 - √(1-θ²))) ?

MathsIsFun · 2006-05-05 00:52:27

Which approaches 180 as θ approaches 0, so that doesn't help. Ah well, worth the try

yttrium88 · 2006-05-05 04:35:48

One thing we can note is that the sine of 90 degrees is 1. Then, in radians, the sine of pi/2 is 1.

The arcsine (sin-¹) of 1 is 90 degrees, or in radians it is pi/2. So then we have:

pi=2*sin-¹ (1)

If we can find a way to calculate sin-¹ as accurately as we want, that will let us calculate pi as accurately as we want.

It turns out that there is a way to do this:

sin-¹(x)=x+1/2*(x^3)/3+(1*3)/(2*4)*(x^5)/5+(1*3*5)/(2*4*6)*(x^7)/7+...

However, this is a very sloooooooooooowwwww way to get pi. I know, because I put a program for it on my calc, and it took forever just to get the first 2 digits of pi correctly. There are lots of faster ways than this.

krassi_holmz · 2006-05-07 04:12:57

What about this:

Last edited by krassi_holmz (2006-05-07 04:21:41)

yttrium88 · 2006-05-07 06:27:42

I'm sorry, but could someone just show in quotes how to put Latex into this board. It should be simple, I know, but I can't get it to work. It's not just: "\[_code_\]".

Last edited by yttrium88 (2006-05-07 06:31:07)

Patrick · 2006-05-07 08:45:25

will give you..

gives

Last edited by Patrick (2006-05-07 08:46:12)

yttrium88 · 2006-05-07 09:44:53

Thanks Patrick!

Okay, so I was just wondering how fast the convergence on that series Krassi poster is.

I also figured that the sigma notation for that super slow converging arcsine series is:

The x term will just drop out if you are looking for the arsine of 1.

mikau · 2006-05-07 15:52:55

PI = (180/θ)√(2(1 - COS θ))

in that expression is θ supposed to be in degree's or radians? If its in degree's thats no good because the trig functions only work in radians, when you enter the value in degree's, the calculator multiplies the value by 2 pi /180 degree's. So technically you need pi before you can even use that formula.

But if your working in rads then nevermind.

krassi_holmz · 2006-05-07 17:17:57

What About this:

Last edited by krassi_holmz (2006-05-07 17:19:58)

krassi_holmz · 2006-05-07 17:28:51

This now coverages VERY VERY fast.
here's a proof:

Last edited by krassi_holmz (2006-05-07 17:29:43)

Zmurf · 2006-05-07 19:00:07

I came to this same conclusion over the weekend when I was reading more into radians. At first I thought that it would innacurate because radians were based around PI. The I realised that it wasn't based around the innacurate decimal PI but around an accurate constant PI. So the equation could be rearranged to generate an accurate representation of PI. I also found I get more accurate answers for questions using PI when i use this equation as PI is only accurate to 10 decimal places in my calculator, but I get many more ghosed decimals when I use this formula.

I'm currently working on a program to produce PI to many many decimals based on the forula, but I'm having trouble with with the Series for finding ArcSine.

Anyone experienced with C++?

MathsIsFun · 2006-05-07 22:51:56

Taylors Expansion for arcsin from Wikipedia:

yttrium88 · 2006-05-08 01:21:41

Unfortunately, this seemingly simple and straight forward expansion for arcsine is very, very, very slow. I tried it on my TI-86 and it took forever. It is simply not practical to compute pi by using 2*arcsin(1).

The image at the bottom of the page shows the Chudnovsky algorithm, which is the fastest way, according to MathWorld, to get lots of digits of pi. The Chudnovsky's are very smart. Their algorithm is complicated. It is based on harder math than arcsine.

Last edited by yttrium88 (2006-05-08 01:26:28)

Zmurf · 2006-05-08 18:38:16

These methods that involve creating more accurate representation of PI by increasing the number of time the SUM is repeated is impractical for calculating PI.

I've being looking at other methods, and there are algorithms that work by getting the n'th decimal of PI. Ie; Instead of increasing the accuracy each loop of the sum, it adds the next decimal place on.

Heres an example: http://fabrice.bellard.free.fr/pi/

I'm hoping to get the algorithm to work correctly in the computer program, then add functionality to check for redundancies and pattern. I'm not looking for the final repeating numbers and get the exact value of PI. I was thinking that perhaps there are patters. For example, any decimal place number which is a divisible of 'x' will always be 'y'.

Last edited by Zmurf (2006-05-08 18:38:47)

Zmurf · 2006-05-08 18:47:09

After reading up on it more I think I'm misunderstanding how it works ...

yttrium88 · 2006-05-09 01:54:28

If I understand you right, you want to try to find patterns in the decimals of pi. I would think the best way to do this is to start from the beginning (of pi). Granted, some of the power series for calculating pi converge slowly, and are therefor impractical. However, the Chudnovsky algorithm that I gave above converges very quickly, giving 14 decimal places of pi per term of the series.

An excellent site describing the methods for finding pi throughout history is the one given below.

http://www.cecm.sfu.ca/organics/papers/borwein/index.html

Last edited by yttrium88 (2006-05-09 01:55:16)

Zmurf · 2006-05-09 19:14:13

I was graphing:

To show the degree of error. I don't know why I couldn't visualise it. But the graph was the same as:

But with an asimtope of y cannot = PI. I don't know how to express that in the line equation, nor does my teacher no the name of the kind of line in that equation. We just call it the side-ways half parabola.

I though perhaps If we could mirror this line [vertically] and average the y values of the two lines it would gives us the exact value of PI. Because the average of the mirrored lines would then become the exact value the lines cannot be.

(BTW, Who designed the (math)(/math) bbc code? I've never seen it on another forum before.)

Last edited by Zmurf (2006-05-09 19:41:57)

Zmurf · 2006-05-09 19:39:10

Hmm, but I need to know PI in order to mirror the graph .... grrr.

yttrium88 · 2006-05-16 04:15:38

How were you able to graph that sum?

Zmurf · 2006-05-16 18:20:46

The x-axis becomes the value of n, and the y-axis becaomes the answer to the sum after the n'th iteration.

Math Is Fun Forum

#1 2006-05-03 18:03:49

The Exact Value of PI. Well, not exactly ...

#2 2006-05-04 08:59:41

Re: The Exact Value of PI. Well, not exactly ...

#3 2006-05-04 09:38:40

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#4 2006-05-04 17:34:14

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