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/sci/ - Science & Math

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3245799 No.3245799 [Reply] [Original]

So, here are some fraction approximations of Pi that are really accurate.

22/7≈π, accurate to within 1/24 of one percent.
201/64≈π, accurate to within 1/32 of one percent.
245/78≈π, accurate to within 1/55 of one percent.
267/85≈π, accurate to within 1/75 of one percent.
509/162≈π, accurate to within 1/82 of one percent.
465/148≈π, accurate to within 1/104 of one percent.
289/92≈π, accurate to within 1/108 of one percent.
421/134≈π, accurate to within 1/158 of one percent.
333/106≈π, accurate to within 1/377 of one percent.
377/120≈π, accurate to within 1/424 of one percent.
355/113≈π, accurate to within 1/11,776,662 of one percent.

And here are some interesting equations that get really accurate values for Pi, most of which I discovered myself by playing around on my calculator.

(π*2)^φ+φ^(π*2)≈π^(φ*2)–1/2, accurate to within 1/150 of a percent.
(π*2)^φ≈φ^(π*2)–1, accurate to within 1/282 of a percent.
(63/4)^(2/3)≈π*2, accurate to within 1/605 of a percent.
φ^2*6/5≈π, accurate to within 1/652 of a percent.
(π*2)^φ*(2^(5/47)+1)≈π^(φ*2), accurate to within 1/806 of a percent.
(π*2)/(φ*2^(4/167))≈LN(π*2)/LN(φ), accurate to within 1/1124 of a percent.
2^(2–3/φ)*1485/523≈π, accurate to within 1/1,601 of a percent.
(π*2)^φ*(2^(7/275)+1)≈(π*2)^2, accurate to within 1/2,740 of a percent.
(1947/100)^(13/21)≈π*2, accurate to within 1/4,590 of a percent.
φ^(π*10/3)–φ^(π*3)≈611/10, accurate to within 1/8,341 of a percent.
(17144/11)^(1/4)≈π*2, accurate to within 1/31,192,488 of a percent.

TLDR – yeah, I don't blame you. I've basicly just wasted the last four hours doing this. :/

>> No.3245810

Better men than you have wasted more impressive amounts of time tooling around with pi.

>> No.3245817

>>3245799
>φ^(π*10/3)–φ^(π*3)≈611/10, accurate to within 1/8,341 of a percent.
>using something ridiculously complex and including pi in it to calculate 2*pi with bad accuracy
>DERP

>> No.3245821

>>3245817
You clearly don't understand double-sided equations.

Go on Wolfram Alpha from time to time.

>> No.3245833
File: 127 KB, 1280x982, Screenshot-cf.rkt.png [View same] [iqdb] [saucenao] [google]
3245833

hai

>> No.3245837

can somebody remember that equation that gives 0.124999999999999999999 or something? I remember it is an integral with cos in it or something.

>> No.3245841

>>3245837
1/8

>> No.3245843

>>3245817
>>3245817
>>3245817
>>3245817
>>3245817
>>3245817
>>3245817
>>3245817

Gonna have to agree with you buddy.

>> No.3245846

Hmmm

How about

pi = 4arctan(1)

>> No.3245870

>>3245821
>You clearly don't understand double-sided equations.
>implying a mathematician has ever said "double-sided equation"
>implying you can have an equation, which by definition equates more than one thing, can have only one side
>implying a=b is a different equation than a-b=0
wow. i fucking hate you.

>> No.3245917

Bump.

>> No.3245932

>>3245917
Hey samefag, while your 9th grade yammerings about "go read wolfram alpha" are amusingly retarded to the 4 post-pubescent sci posters that actually managed to get one or more college degrees, i suggest you hit up some real books for the summer. maybe eventually progress to something where you can learn about floating point calculations in computers so you don't claim some dogshit is accurate to a greater degree than which it was actually calculated.

>> No.3245942
File: 7 KB, 590x251, wolframalpha-20110618105858149.gif [View same] [iqdb] [saucenao] [google]
3245942

>>3245870
Alright, you're clearly not understanding the concept, so I'll try to explain it.

Let's say we have φ, and an unknown variable A.
We know φ is equal to 5^(1/2)/2+1/2.
And let's say we have the equation φ^a–a^φ=1.

Now that equation might look redundant, because it seems to be using the variable A to define A. But when you think about it, it isn't actually redundant.

What you find is that you have two functions: f(a)=φ^a, and f(a)=a^φ. And what you will find is that, somewhere, there is a point where f(a^φ) is less than f(φ^a) by exactly 1. This can be approximated by the graph you see to the left of this post, which displays both functions, and the two points where they converge.

Now, actually figuring out what the values of A are for those two points involves a little arithmetic trickery that I haven't learned yet. But if you punch "φ^a–a^φ=1" into Wolfram Alpha, you'll find that it gives you the two solutions easily: a=0.3518309596134543…, or a=6.283407895627034….

You see? It is not stupid at all, and it is actually quite the coincidence that the equation you mentioned should approximate 2π so closely.

>> No.3245957
File: 3 KB, 210x230, 1301707155922.png [View same] [iqdb] [saucenao] [google]
3245957

>Pissing about with Pi
>Not trying to understand stuff that actually needs to be better understood and eveloped

>> No.3245971

Pi is what separates scientists from science fans.

Spoiler: OP is a science fan.

>> No.3245973
File: 8 KB, 590x259, wolframalpha-20110618110911576.gif [View same] [iqdb] [saucenao] [google]
3245973

>>3245942
Oops, that graph showed the functions f(a)=φ^a–a^φ and f(a)=1.

It should show the functions f(a)=a^φ and f(a)=φ^a–1. Here is the correct graph. (The solutions are still the same.)

>> No.3245980

>>3245957
>>3245971
>Pi is what separates scientists from science fans.
>Spoiler: OP is a science fan.

I don't see why a person can't be both. I happen to be in the summer after graduating from high school, so I haven't yet begun my college curriculum. But I'll have you know I'm majoring in physics and mathematics, and I'm planning to pursue a graduate degree in astrophysics at some point.

Just because I'm at a primitive stage in my education doesn't mean my mind or intentions are primitive.

>> No.3245985

>>3245942
>Now that equation might look redundant, because it seems to be using the variable A to define A.
No, it doesn't look redundant.
>Alright, you're clearly not understanding the concept,
I'm not even bullshitting you here: I have a BS and MS in engineering, and I'm presently a PhD student in computational mathematics. I'm guessing you are maybe 16.

>Now, actually figuring out what the values of A are for those two points involves a little arithmetic trickery that I haven't learned yet.
Right. You haven't learned it yet because you think "learning" involves typing out numbers into wolfram alpha. It's not.

>f(a)=φ^a, and f(a)=a^φ
ugh, for starters don't write like this, christ. basic math: f(a) should be the same thing as f(a). if f(a) /= f(a), our shit is fucked. you want to at least say f(a)=φ^a, and g(a)=a^φ. they're two different functions.

>a little arithmetic trickery
That equation (it's not "double-sided" you fuckin moron, don't make up math terms) is a *nonlinear* equation in one variable. It has three "roots", aka solutions for a. One method for root finding in an equation is Newton's method (or Newton-Raphson). There's a billion others.

seriously, get the fuck off sci for the summer or remain retarded for life.

>> No.3245996

hey faggot, find infinitely many rationals p/q which are within 1/q^3 of pi.

>> No.3246007

>>3245985
>I have a BS and MS in engineering, and I'm presently a PhD student in computational mathematics. I'm guessing you are maybe 16.

I'm eighteen. I clearly stated in the post you're responding to that I just graduated from highschool a couple weeks ago. Why are you getting all smug and superior on me? I clearly haven't gotten as far as you have in your education. It's not my fault for being younger and less experienced. Piss off if you're just going to be an asshole about it.

>ugh, for starters don't write like this, christ. basic math: f(a) should be the same thing as f(a)

Well sorry for making an honest mistake. g(a), then. God. Go be a pedant somewhere else.

>it's not "double-sided" you fuckin moron, don't make up math terms

I wasn't using it as a technical term. I was saying it was "double-sided" because it *is* double-sided. There are two separate functions, and the solution is where they converge. There are thus two sides to the problem.

>One method for root finding in an equation is Newton's method (or Newton-Raphson).

This is legitimately the only useful part of your entire post. Thank you for at least giving me something to go on. I like learning, and I will certainly look that up.

>> No.3246011

>>3246007
you should read about the irrationality measure of pi

>> No.3246015
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3246015

>>3246007
> doesn't know about newton raphson

>> No.3246026

>>3246015

Yes, I'm sure it makes you feel bigger and manlier to pick on someone for being younger and less educated than you.

>> No.3246028

If you want the best rational approximations to pi, this can be proven to be the convergents of the continued fraction for pi, where "best" is defined as "the rational with the least error for a given size of the denominator."

In fact, continued fraction convergents are best in this sense for all irrationals, not just pi.

Enjoy.

>> No.3246034

>>3246007
>Well sorry for making an honest mistake. g(a), then. God. Go be a pedant somewhere else.
Look sweetheart, if you want to go into physics/mathematics and an astrophysics phd (lol @ 18 year old with no math/sci knowledge that already knows they want to do a phd), it's best to not start off by bitching about pedants when it comes to math. formalism is pretty goddam important.

>I wasn't using it as a technical term. I was saying it was "double-sided" because it *is* double-sided. There are two separate functions, and the solution is where they converge. There are thus two sides to the problem.
This makes you sound dumb. There are not two separate functions. This is one function: >φ^a–a^φ=1
It's not two functions. It's one function with one unknown. Try this:
>f(a) = φ^a–a^φ-1
Now your problem is to find the roots of f, i.e. what values of a make f(a) = 0.

>Piss off if you're just going to be an asshole about it.
I wouldn't be if you hadn't tried to give me a lecture on your brilliant double-sided methods to plugging stuff into wolfram alpha and how i don't understand.

>> No.3246063

>>3246026
Welcome to 4chan. You should grow a thicker skin if you want to hang around here. Else this is not the place for you.

>> No.3246072

>>3246026
Aw look at the little baby

come under my wing, little one.

>> No.3246080

>>3246026
welcome to /sci/

>> No.3246092

>>3246072
you're a dumbass too.

>> No.3246096

>>3246034
>lol @ 18 year old with no math/sci knowledge that already knows they want to do a phd

Considering the horribly shitty school system I've been in so far, I would say that I am actually ahead of my classmates. Don't blame me; blame North Dakota.

>it's best to not start off by bitching about pedants when it comes to math. formalism is pretty goddam important.

Well hopefully I will one day know as much as you, and be able to make fun of freshmen for their inexperience.

>It's not two functions. It's one function with one unknown.

If you use the Newton-Raphson method, or one of the other "billions" of other methods one could use, sure. Then, the problem is solvable as one function. Not knowing the Newton-Raphson method, I resorted to using the graph and trace functions on my calculator, then closing the window to approximate the roots. Because of my ignorance, I had to use two functions. It was double-sided, because I didn't know how to unify it. I now concede that the equation is not inherently "double-sided".

>I wouldn't be if you hadn't tried to give me a lecture on your brilliant double-sided methods to plugging stuff into wolfram alpha and how i don't understand.

I apologize for assuming I knew more than you. I was mistaken; and it was a stupid assumption.

>> No.3246105

>>3245942
I still don't get it.
What is your definition of a "double-sided" function? And what is your purpose for them?

>> No.3246107

>>3246096
>be able to make fun of freshmen for their inexperience
I could care less about your inexperience. I know tons of people far far more experienced/intelligent/learned in many fields than I am. I was making fun of your arrogance.

>Considering the horribly shitty school system I've been in so far
I was in a shitty school system too. It's not an excuse. That's why the one piece of advice I have to give you is to pick up a real, dead-tree book of math. Stop dinking around on wolfram. I'm dead serious. Get off sci. Feedback and trollery from anonymous dipshits on your silly plug-and-chug isn't going to help you. I promise.

>I now concede that the equation is not inherently "double-sided".
no, you're still not getting it. all equations are "double sided" because "equate" means to compare two things as being the same. a+b is not an equation. a+b=0 is an equation. words mean things.

>> No.3246114
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3246114

Elen, see pic

>> No.3246117

>>3246107
> I could care less

should be

> I couldn't care less

inb4 this dumbass gets mad and says "thats how I say it hurr durr dont care if its wrong and makes me seem like a dumbshit" instead of just correcting himself.

>> No.3246127

>>3246107
>I was making fun of your arrogance.

Touché.

>I was in a shitty school system too. It's not an excuse. That's why the one piece of advice I have to give you is to pick up a real, dead-tree book of math. Stop dinking around on wolfram. I'm dead serious. Get off sci. Feedback and trollery from anonymous dipshits on your silly plug-and-chug isn't going to help you. I promise.

I'm not sure where you live. But here, try going to a public library and asking for a book on, say, celestial mechanics. The nearest Barnes And Noble, which is not in my county, does not have a single book in it, so far as I know, which even so much as mentions Johannes Kepler. I have never seen a copy of Newton's Principia or Euclid's Elements in my entire life. The very computer I'm sitting at I only got recently for a graduation gift, and I don't have any credit cards with which to order things online. My resources up to this point have been practically nil.

I appreciate what you're trying to tell me. And when I head off to Minneapolis, I'm sure there will be libraries and bookstores that have materials I can actually find useful.

>no, you're still not getting it. all equations are "double sided" because "equate" means to compare two things as being the same. a+b is not an equation. a+b=0 is an equation. words mean things.

I understand how equations work. And that's not how I was using the phrase "double-sided". I wasn't referring to the "sides" of an equation; I was referring to an equation which must be solved as two separate equations.

>> No.3246129

>>3246117
sorry, i should've written "couldn't."

good job nazi grammar fagging it up in sci though. we're all suitably impressed.

>> No.3246133
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3246133

>>3246127
>>3246127
ur owned

>> No.3246148

>>3246127
>I'm not sure where you live.
Originally from less than 500 miles away from wherever you are in North Dakota ;)

>But here, try going to a public library and asking for a book on, say, celestial mechanics.
Oh come on. I find it hard to believe that you have a public library that doesn't have at least some math/science texts that are above a high school level.

>I have never seen a copy of Newton's Principia or Euclid's Elements in my entire life.

Me neither. While i have great respect for them, why would I want to read a 2300 year old Greek text on material learned in middle school, or the original middle-English writing of the underpinnings of the physical laws of the world, when they've been rendered far more accessible in the centuries that followed? I have great respect for the works, but actually reading the originals is best left for historians.

>I appreciate what you're trying to tell me. And when I head off to Minneapolis, I'm sure there will be libraries and bookstores that have materials I can actually find useful.
If you really really really can't find a dead-tree edition of good books, there's a massive sticky of ebooks here.

>> No.3246154
File: 55 KB, 815x713, sshot_2011-06-18_19:10:49.png [View same] [iqdb] [saucenao] [google]
3246154

Over 9000 milliseconds in Mathematica

>> No.3246157

Oh the american high "school" system
>Graduated
>Doesn't know about newton raphson
>Creates this thread
>I would say that I am actually ahead of my classmates.
I forgive you.

>> No.3246161
File: 35 KB, 500x366, niggasaywhat128610146705757741.jpg [View same] [iqdb] [saucenao] [google]
3246161

>>3245799
...using pi in equations to get accurate values of..pi?

>> No.3246168

>>3246154
Same warning regarding comparison of floating point numbers. I don't know how mathematica does it, but I'm fairly sure you're just getting double precision operations there.

A common mistake is to compare two floats using the same precision they're stored in. You've lost a lot of precision in subtracting two very close numbers. You should really be using higher precision operations to compute relative error of two close numbers.

>> No.3246170

>>3246148
>Oh come on. I find it hard to believe that you have a public library that doesn't have at least some math/science texts that are above a high school level.

*shrug*
I do have a book on basic calculus. But truth be told, I've found it very difficult to wade through. I understand derivatives and anti-derivatives fairly well (I think). But I'm not going to jump ahead in my courses, when I could just as easily take a beginning course in calculus and have a teacher actually explain things. Maybe I'm just an audial (?) learner.

>While i have great respect for them, why would I want to read a 2300 year old Greek text on material learned in middle school, or the original middle-English writing of the underpinnings of the physical laws of the world, when they've been rendered far more accessible in the centuries that followed?

You get what I mean though. And anyway, I tend to learn better from lectures and course material, than from random books I find at the library that are geared for people of either a vastly higher or vastly lower level of education.

>> No.3246181
File: 47 KB, 1112x598, sshot_2011-06-18_19:17:31.png [View same] [iqdb] [saucenao] [google]
3246181

>>3246168
Seems accurate enough to me.

>> No.3246192

>>3246170
Oh and welcome to the tripfag initiation rite on /sci/. We've all been there.

>> No.3246195

I like you OP <3

>> No.3246295

>>3246181

you really don't seem to understand what he's saying, or how precision and numerical tools work at all

>> No.3246389

>>3246195
I don't!