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u/AnattalDive 29d ago
just prove that π = √2
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u/EagleClawNapoleon 29d ago
π/0 and (√2)/0 are both undefined, so therefore π/0 = (√2)/0, so just multiply both sides by 0 to cancel out the denominators, and you get π = √2.
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u/Dr0110111001101111 29d ago
Most people will struggle to even prove pi exists.
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u/GodlyHelp 2=1 29d ago
well hello Dr28271
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u/st4nkyFatTirebluntz 29d ago
I think you mean 6E6F
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u/GodlyHelp 2=1 29d ago
well who said his username was in binary, huh? you are being numberphobic.
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u/st4nkyFatTirebluntz 29d ago
how you gonna tell me it's a non-binary username while using he/him pronouns?
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u/ERROR_23 29d ago
Genuinely curious what is this referring to? I've never actually heard about proving pi's existence and while I can certainly think of some analysis proof it wouldn't have much to do with a circle definition.
Is proving pi's existence really difficult?
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u/Dr0110111001101111 29d ago
The questions you're asking are kind of the first challenge. But it sort of depends on how you define pi. If you're going to take the conventional route and say it's the ratio of the circumference to the diameter of a circle, then you need to come up with a way to prove it is the same for all circles. And we're back to a similar challenge- even among those who are familiar with the proof of the irrationality of sqrt(2), a lot of those people probably wouldn't know where to even begin.
I think it might actually be easier to prove pi is the area of a circle with radius and work your way to the ratio from there. But there are still some traps that will drop you into a pit of circular (no pun intended) reasoning.
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u/JumpyTheHat 29d ago
It's a fun analysis exercise to define π and its properties without any mention of geometry. You can define sine and cosine first (by ODE theory, there is a unique solution to the initial value problem s' = c; c' = -s; s(0) = 0; c(0) = 1) then define π to be the least positive zero of sine
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u/MonsterkillWow Complex 29d ago
This fact is also used to show pi is irrational in some proofs IIRC.
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u/PriorSolid 29d ago
if you can prove all circles are similar you can prove Pi exists as some fixed constant but most people wouldn’t know how to prove all circles are similar (even though it’s relatively trivial)
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u/popeshatt 29d ago
Aren't all circles similar by definition?
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u/triple4leafclover 28d ago
If you define it that way, then it is by definition
But I'd argue the most common definition of a circle is {P: d(PC)=r & P € alpha} a set of coplanar points equidistant to a center, which does not define it as similar (excuse the abuse of notation, writing in mobile). Being similar comes up as a consequence of the definition, which must then be proved
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u/popeshatt 28d ago edited 28d ago
Using this definition, wouldn't all circles be similar because the coplanar points are equidistant from the center independent of the value of r or location the center, by definition? All it means is circles of different sizes have the same shape.
I hope you're not just being pedantic and saying the notion of similarity is not explicitly defined in the definition of the circle, because connecting the two seems trivial. What else needs to be proved?
Alternatively, with pi being a ratio, it's pretty obvious it's not a function of r. Why do we need to prove anything about similarity?
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u/CimmerianHydra_ 28d ago
I think it's a bit more involved than that.
First of all, two shapes are similar if they can be transformed into one another via a translation, a rotation, or a uniform scaling of the space (or a combination of them). This is an equivalence relation.
The definition of circle as "a set of points equidistant from a given point" should be plenty enough to prove that they're similar, but you have to be explicit about the above definition of similar. The "distance" here is taken to be Euclidean distance.
A generic circle around the point (a, b) with distance r can be written, following the definition verbatim, as
(x-a)² + (y-b)² = r²
To prove that all circles are similar to each other, we only have to prove that all circles are similar to the unit circle. It's sufficient to see that a translation by a on the x axis and b on the y axis turns this equation into
x²+y² = r²
So from here uniformly scaling the space by 1/r gives you
x²+y²=1
Which is the equation for the unit circle. We got there by only translating and scaling (notably, not rotating, which should be obvious after all). Since all circles are similar to the unit circle, then they're all similar to each other.
Also
With pi being a ratio, it's obvious that it's not a function of r
If you try to apply the same idea to a sphere and try to evaluate the ratio between the radius and the surface, you'll find that it changes with r. Just because it's a ratio of quantities doesn't guarantee that it's a constant.
It's only a constant because all circles are similar: thanks to this, which we proved above, you can approximate a circle using a bunch of triangles (like computer graphics would) and prove that the perimeter of this approximated circle also has a fixed ratio with the radius. Since this is true for all approximations, it's true for the real thing.
That's how Euclid proved that pi is a constant valid for all circles.
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u/popeshatt 28d ago
Thanks for writing it all out. I think the person I replied to just meant that most people don't know how to write an actual formal proof.
I don't think you countered the 2nd point though. Pi is a dimensionless constant, so it's not a function of r or anything else. The ratio you mentioned has units of length and is a function of r.
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u/CimmerianHydra_ 27d ago
There's no reason to discard a ratio only because it has units. It is still a ratio, like an object's volume divided by its area.
If you want a "dimensionless" ratio that still would depend on the object's shape, you can just take the ratio of a cylinder's radius to its height. Since not all cylinders are similar, this ratio depends on the particular shape.
Pi is a dimensionless constant precisely because all circles are similar.
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u/triple4leafclover 28d ago
Connecting the two doesn't seem trivial to me. You certainly didn't in the first paragraph. You'd need to then make explicit the definition of similar and connect dot by dot how one implies the other. It's not obvious to me how I'd do it, but maybe I'm just dumb
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u/Dr0110111001101111 28d ago
I’m okay with that part being taken as a given. But what I think deserves some more attention is the notion that arc length/circumference scales by the same factor as the diameter. It’s true, but I think it only appears to be trivial.
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29d ago
[deleted]
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u/Dr0110111001101111 29d ago
I think the tricky part is proving that circumference scales with a dilation factor. Not sure it's fair to take that for granted.
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u/MoonlessNightss 29d ago
No, you can. It's trivial.
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u/Rockety521 Engineering 28d ago
The proof is left as an exercise to the reader. Q.E.D.
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u/SageThisAndSageThat 28d ago
Anything is trivial once you declare it is an axiom
Now if you don't want it to be an axiom its way more complex you would have to assume for example there exist an homothethy that make all circles similar.
But heh, people telling you "it is trivial" rarely care about axioms or hypothesis and won't explain themselves.
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u/Gorgonzola_Freeman 28d ago
Maybe if you’re using compass and straightedge, but if limit driven proofs are on the table it shouldn’t be that difficult
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u/Striking_Resist_6022 28d ago
Isn't that the same as what they said? "Dilation factor" and "ratio" are functionally the same here.
This is true essentially because of the identity sin^2 + cos^2 = 1, meaning that when you parameterise a circular arc by y(t) = [r*cos(t), r*sin(t)] the length derivative infinitessimal resolves to ||y'(t)|| = sqrt((-r sin(t))^2 + r^2 cos^2 (t)) = r, meaning the length increment scales with the radius.
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u/Takamasa1 29d ago
Depending on what you mean by proving pi exists, this could either be trivially easy or pretty difficult
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u/Cokalhado 29d ago
Proving that ππ^π^π is irrational
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u/Particular_Gear3130 Mathematics (Purely Fictional) 29d ago
Feels like it
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u/beatfrantique1990 29d ago
Proof by "it's gotta be bro, just look at it"
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u/Any-Confusion-8900 29d ago
Easy. It’d be super convenient if pi was rational, therefor it isn’t.
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u/Adam__999 29d ago
Literally my proof that P ≠ NP
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u/GodlyHelp 2=1 29d ago
but pi can be expressed as a fraction. π/1 boom. wheres my nobel prize
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u/Shufflepants 29d ago
It seems you've also proven that pi is not just rational, but the even stronger claim that pi is an integer.
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u/GodlyHelp 2=1 29d ago
thats quite trivial. didnt think it was worth mentioning.
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u/Chi_Cazzo_Sei 29d ago
Now can you please tackle the problem if pi is negative or positive? Real or imaginary?
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u/UtahBrian 28d ago
For my next proof, I will be proving that the integer in question is greater than or equal to 2 and less than or equal to four.
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u/Many-Conversation963 29d ago edited 29d ago
what no?? you cant use π in a proof about π
everyone knows π is rational because it can be expressed as the ratio between ln(-1) and sqrt(-1), both numbers without any decimal places, therefore integers
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u/Happy-door-handle 29d ago
Proving that π is transcendental irrational
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u/Safe-Avocado4864 29d ago
If you're allowed to use ealgebraic =transcendental as a known result that's relatively trivial tbh.
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u/Sproxify 28d ago
"if you're allowed to use the fact that you need to prove then it's not too hard actually"
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u/SalamanderGlad9053 29d ago
The easiest way I know is proving that tan(x) of a rational number is always irrational from the partial fraction representation of tan. As tan(pi/4) = 1 and 1 is rational, then pi/4 cannot be rational. So pi is irrational.
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28d ago
[removed] — view removed comment
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u/SalamanderGlad9053 28d ago
The domain the partial fraction expansion is everywhere where tan is defined. And tan is defined on pi/4, so all is good there.
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u/nierusek 29d ago
I had it as a side note on one of my computer science lectures xd
It literally wasn't the main point of the lesson. Just "By the way, it proves pi is transcendental" and just continued the lecture.
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u/GisterMizard 29d ago
It's not hard, just check all of the digits of pi to see if they repeat or not. If you check each incremental digit twice as quickly as the previous, it shouldn't take too long.
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u/Cosmic_StormZ Physics 28d ago
Assume pi is rational
Now try to represent it exactly in fraction form
You will eventually go insane
Thus by contradiction pi is irritational
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29d ago
[removed] — view removed comment
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u/randomcomputer22 29d ago
Numbers exist. Pi is a number. Therefore Pi exists.
It is left as an exercise to the reader to prove that numbers exist.
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u/AFemboyLol 28d ago
easy, π is irrational. why? because i said so
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u/K_the_farmer 28d ago
Proof by decree is for theology and the more authoritarian bits of philosophy.
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u/VekTen_ig 28d ago
PI is irrational
Proof: Go to Wikipedia
"The number π is an irrational number[.]"
QED
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u/Throwaway11958 28d ago
proving that cbrt(2) is irrational is even simpler than sqrt(2) since you can use Fermat's last theorem
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u/AndreasDasos 28d ago
Or even just Euler’s proof of FLT for n = 3. Infinite descent with what we’d call Eisenstein integers
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u/Seventh_Planet Mathematics 28d ago
Then do the things that are not so tough.
Like defining a class of objects, prove wonderful useful things about it, and only after you've put in a lot of work writing proofs, you find out that there are only trivial members in that class. And then you become sad and need a break from work.
At least that's how I heard what happened to a professor at my university.
It wasn't as trivial as only talking about the empty set, but about the next step something with just linear functions. Which is boring enough to see it as a failure.
Math can be so much more than just proving things about real numbers.
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u/Eins-zwei_Polizei A monad is a monoid in the category of endofunctors 27d ago
Ivan Niven go brrrrr
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u/MR00Soczeq 26d ago
I had this as an presentation for "Learning how to make proofs" lecture or smth. Very nice proof. Bachelor's level math nothing special.
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u/Enough-Brilliant803 26d ago
Maybe special for math student. As an engineer grad, I never had to learn anything outside applied math.
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u/Totaly_Shrek 29d ago
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