# Proofs on the rationality of pi

#### lvlastermind

Im looking for some proofs on the rationality of pi. I also want to know what some people think about it.

Staff Emeritus
Gold Member

#### 1+1=1

does anyone know who actually discovered the "theory" known as pi? do you know how he got to it? i watched a tv program about it and i must say, it is rather interesting!

#### JonF

how is pi a theory? i thought it was a number.

#### abertram28

well, it is referenced by some as first showing up around 1650 bc in egypt.
this was though, that they didnt have irrational numbers, so they just estimated. they used 3 first and it wasnt accurate enough.

but it could be argued that the relationship between a circles diamater and circumference has existed since the invention of the circle, which predates humans, as there are a good many plants that use circles, therefore nobody actually invented it.

and since its not a theory but a naturally occuring relationship, the proof is in the pudding, or rather, its proved by definition... it exists, thats the proof.

youd think they might have actually told you something about it on that tv program

#### styler

pi

it is thought that pythagoras "knew" about pi..he certaily knew that not all numbers which were naturally occuring were rational....for "religious" reasons this information was surpressed.
there is a nice proof of the irrationality of pi the the great book by spivak titled simply "calculus"
but its is amazing that no matter how much math you learn you still find pi to turn up in strange places...it seems to be more than just a parameter that characterizes euclidean space

BTW the phrase is "the proof of the pudding is in the eating"
there was ever any proof in pudding all by itself.

#### HallsofIvy

Homework Helper
It was, of course, Archimedes who gave the first accurate approximations of pi.

#### fourier jr

I think the Chinese approximated Pi long before any western people. They said it was 355/113. (see the 1, 1, 3, 3, 5, 5? nice pattern) I don't think it can be dated though since the Chinese typically destroy everything from their past.

#### JonF

Archimedes gave the first accurate account of just about everything.

#### lvlastermind

fourier jr said:
I think the Chinese approximated Pi long before any western people. They said it was 355/113. (see the 1, 1, 3, 3, 5, 5? nice pattern) I don't think it can be dated though since the Chinese typically destroy everything from their past.

I have heard of this before also

#### 1+1=1

it is actually clever how archimedes found pi to = 3.14. and for his time, that is an accomplishment all in itself. not only did he find out what pi is = to, he also comtemplated the law of bouyancy? after getting into a tub of water and getting back out.

#### matt grime

Homework Helper
but pi isn't 3.14 is it 1+1=1?

#### abertram28

nobody has ever found pi to be equal to anything but pi, and if they used a numerical representation in any rational form they were just estimating. it wasnt even until like 17th century until it was found to a higher degree of accuracy.

rational approximations were used (22/7, 256/81, 25/8, 355/113...) for a long time, id say about 3500 years minimum. in fact, we still use rational approximations, but we generally have accepted the value of pi as an irrational number.

the paper definition of a ratio between the circumference and diameter has been dated to before 1650 BC, and although archimedies made the first decimal approximation the definition was already on paper. so the exact value had already been defined. in 1650, thus, archimedes could not have found the exact value of pi. rational estimations had also already been made for use in egypt. beginning with pi=3, then to 22/7. that was before archimedes, and 22/7 is as close to pi as 3.14.

there is also some evidence that babylonians used the 25/8 figure for the ratio, but had never defined the ratio to be its own value. that was like 2000 BC! so the first approximation was like almost 2 millenia before archimedes.

the origin of the use of the letter "pi" to represent this ratio didnt start until nearly two millenia after archimedes. so i dont see what he has to do with it, excluding his excellent geometric estimation that teacher still tell us to use if our calculators dont have a pi button, 22/7.

the proof that irrational numbers existed was what pythagoras objected to allowing to become public. and indeed, the student who proved it during class was killed some months after in a suspicious fishing accident. heres a little proof on irrationality
http://www.homeschoolmath.net/other_topics/proof_square_root_2_irrational.php

heres a good one
http://www.symynet.com/fb/pi_hoax.htm#

more use between 1700-1800 gained pi the rep that it has today, one of the most, if not the most, usefull trancendental number.

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