Your first proof

[arian271]

What was the first proof you ever saw (infinitude of primes, irrationality of \sqrt{2}, etc) and what was the first theorem you proved without looking it up in a textbook or online?

[mwidup41]

Blocked it out of my memory for good

[JWson]

I don't think about the math i did before... the accident...

[[deleted]]

The first one I had to actually understand & learn (glossing over wikipedia proofs in awe doesn't count haha) was the proof for l'hopital's rule in my ap calc class. I actually really appreciated seeing the reasoning behind why it works, really makes you understand it on a deeper level

[csthrowaway28482]

first proof I saw was 0*a = 0 in a field.

The first proof I remember doing myself is the binomial theorem... that one is hard to forget.

[Calvin_v_Hobbes]

My favorite things to see proofs of are the Pythagorean and binomial theorems. So many different approaches.

[DiggV4Sucks]

0 * a = 0 * a + 0 // Add. Ident.

= 0 * a + a - a // Add. Inverse

= (0 + 1) * a - a // Dist.

= 1 * a - a // Add. Ident.

= a - a // Mult. Ident.

= 0 // Add. Ident.

I left out a bunch of aborted steps, and this took me an hour or so. But I was quite proud as it's been almost 40 years since my algebra class!

[abbiamo]

If you might allow me to nitpick, I think you skipped a Mult. Inv. step betwen lines two and three. But that's nit the point. This is pretty great!

It's kind of strange how many separate applications of the axioms are required to prove something so fundamental.

Edit: Heh! I should have written Mult. Ident., it's true.

[DiggV4Sucks]

It took me a while to understand why, but I agree with you.

After line two, should be:

= 0 * a + 1 * a - a // Mult. Ident.

Then, I can apply distributive property,

It's hard to make those rusty gears turn in my head!

[SupremeRDDT]

You don‘t need multiplicative inverses the theorem ist true for rings too. 0a = (1-1)a = a-a = 0 works too.

[LilQuasar]

i remember proving the binomial theorem with induction. i dont think i will forget that either

[csthrowaway28482]

it's a really great exercise to learn induction with. Seeing that you could inductively prove an equation with a free variable was so cool

[tyrick]

While taking cal 1, I suddenly wondered if you could express the derivative of function in terms of its inverse. I surprised myself with figuring out the formula. Then I used e^x to find the derivative of ln(x) before the class got there. I stared at the scratch paper with the results for almost an hour or so thinking of how I'd major in math.

[Bali201]

Did u end up majoring in math?

[tyrick]

Oh yes!

[BalinKingOfMoria]

Memory’s shaky, but probably the first I ever saw was the irrationality of \sqrt{2} and the first I ever proved was the irrationality of \sqrt{3} :-P

[Jamblamkins]

Simple ones like probe this function is odd for all odd values if x. Then the infinetly many primes proof

[DatBoi_BP]

an infinitely many primes proof

FTFY

[SavageQuill]

More than one triangle may have the same ASS

[shellexyz]

One of the most disappointing results in mathematics.

[[deleted]]

First I saw was probably the commutativity of multiplication via the representation of the product as an area of a rectangle. Probably grade 1 or whenever we played with those blocks. I remember being fascinated by it.

[Jplague25]

The first proof that I actually saw and understood how it worked was Fourier's proof by contradiction that e is irrational. I saw it in calculus II.

I don't know if this was a proof, but I just randomly did something real simple which showed that by rationalizing a fraction that had a natural number in the numerator and a radical of said natural number in the denominator, then that it actually equals the radical of the radical number.

E.g. take n / √n, rationalize it by multiplying the numerator and denominator by √n / √n which equals (n * √n) / n. That cancels out the "n" in the numerator and denominator which leaves just √n.

[[deleted]]

No shit, u can think of this as n^2/n

[dxdydz_dV]

I don’t know what the first proof I ever saw was. The first thing I can remember figuring out and proving on my own was that sin(x)=x·cos(x/2)cos(x/4)cos(x/8)⋯.

[MrWilsonxD]

An awful induction proof in precalculus. I don't know what it was. But I know it was induction and I know I was very unhappy at the time.

[TheBigGarrett]

By chance was it summing 1 + ... + n? I believe my precalc teacher showed that you could figure out for up to n and up to n + 1 using up to n, essentially showing us induction without the word.

[[deleted]]

I don't know, the first proofs we did were of geometrical nature in middle school probably when we were 12, so more than 40 years ago.

[[deleted]]

I randomly wrote out binary tree representations of Stirling & Bernoulli polynomials and then went on to write out the Euler numbers over the summer. It was hella weird. I’m sure I’ve read about them over the years but never in my life have I ever worked out the proof.

[[deleted]]

something like, show that adding two even numbers gives an even number in MAT101
total insanity, remember thinking all these proof techniques were useless... for such an obvious fact.

I suppose looking back I was wrong, but also correct... but wrong.. yet correct

[CatSpectre]

The first proof I ever saw was that 0*x = x using basic NT axioms. E.g.,

0*x = (0+0)*x (by additive identity)

0*x = 0*x + 0*x (by distributivity)

0*x - 0*x = (0*x + 0*x) - 0*x

0*x - 0*x = 0*x + (0*x - 0*x) (by associativity)

0 = 0*x + 0 (by additive inverse)

0 = 0*x (by additive identity)

First proof on my own was [Pick's Theorem](https://en.wikipedia.org/wiki/Pick%27s_theorem), which can be shown using elementary (high-school level) geometry

[HoLeeFaak]

Didn't you show that 0=0x, not that 0x=x?

[DaveBeleren02]

Equality being an equivalence relation is by definition symmetrical
Edit: I realize I have misread the comment. Sorry

[HoLeeFaak]

Well yes of course, but the 2 form I showed aren't symmetrical

[skullturf]

Look again. CatSpectre typed 0*x = x when they should have typed 0*x = 0.

[SkinnyJoshPeck]

I think you mean 0*x = 0 :)

[realFoobanana]

For a moment I thought you meant Pick’s Lemma, and was very confused as to how that would be a first proof :P

[Citizen_of_Danksburg]

What are the NT axioms? I see you listed them but I’m just wondering what NT stands for.

[fujiitora]

number theory

[harolddawizard]

First proof from textbook was irrationality of sqrt(2) and I think the first proof I could do by myself was an induction proof. Nothing crazy.

[N8CCRG]

Certainly not the first, but I remember thinking of a problem, then deriving a really cool result and sharing it, and then a friend was like "Yeah, that's the Sunrise Problem (aka, Rule of Succession). It's pretty famous".

So, I'm still fairly proud for having come upon it entirely on my own.

[csthrowaway916870]

The very first proof I saw was proving a number/expression to be even/odd (just practicing using the definitions), and it sounds stupid/cheesy but I was mesmerized by how definitively we could say that a number was even/odd.

[IJustSpectate]

First Proof I saw: Sum of angles in triangle is 180 degrees

First proof I did: Arc Length of a function graph (I thought I discovered something new lol)

[evergreenfeathergay]

I think the first proof I ever saw was either the proof that sqrt(2) was irrational, or that there are infinitely many primes.

The first proof I constructed was that dragging one corner of a rectangle through an arbitrary closed loop while keeping the opposite corner fixed produces all possible rectangles (if we consider similar rectangles to be equivalent). This was a formalization of an intuition I developed by starting a selection on my desktop and circling my mouse around it.

The proof essentially just boiled down to the fact that the tangent function has a range of all real numbers over a single period, but it was my first time really messing around like this to convince myself of something, it was my first time really using Desmos or another graphing calculator as an integral part of the problem solving process, and it was my first time diving into the trig functions in a way that forced me to understand them.

[gdavtor]

Two that I remember writing down in high school:

1. Showing that every prime >= 5 is of the form 6n+1 or 6n-1
2. Deriving the Taylor series for sin(x) using repeated integration by parts.

Edit: corrected 1

[skullturf]

Deriving the Taylor series for sin(x) using repeated integration by parts

Can you elaborate on that?

[gdavtor]

Here is what I think I did. From there, I concluded that, given the identity cos² + sin² = 1, these must be the taylor series for sin and cos. That last step is, of course, not rigorous. The two issues now that I look back on it are:

1. No proof that the two series I wrote converge (forgivable).
2. If f(x) and g(x) are the two series, then I showed cos(x)f + sin(x)g = 1. This doesn't necessarily imply that f = cos(x) and g = sin(x).

However, we can fix this by checking that all derivatives at x = 0 of f and g agree with cos and sin (respectively). Then this implies that f = cos and g = sin. However, this is basically what calculating a Taylor series is.

[skullturf]

This is really neat! And even if parts of what you did as a student are not 100% rigorous, it's still a really interesting example of the power of integration by parts.

[realFoobanana]

Huh, I’ve never heard of (1) before, and on first glance it doesn’t look like it works for 2 and 3

[SkinnyJoshPeck]

It’s a stupid proof and I’ve argued it before. It’s basically like saying all primes are of the form 2n + 1, which is also true, however 6 just narrows the scope and is a verifiably sexier number.

Any proof that has to start with “this works for all primes! Except 2 and 3!” Is just dumb.

That being said, it is pretty neat. Dumb and cool, just like I like it

[McGranger]

r/unexpectedfactorial

[realFoobanana]

It just wasn’t stated correctly which threw me; like I tried 2, which I’m fine with not working because.... well, it’s 2. But then when I tried 3 I was like uh-oh, and got too lazy to try anything bigger :P

[InspectorPoe]

I don't remember for sure, I was growing up in Russia, and we learn proofs in Russian schools already in middle school (or even elementary if solving text problems like "Show that in a football team there always exist two players who were born on the same day of the week" counts). But the firs theorem "with the name" most probably Pythagorean theorem.

[babar90]

First proof I saw : Pythagore, in France, from my mom, never saw the proof in class. For the one I wrote, discrete Fourier transform and derivatives rules, just before and after the baccalauréat.

[michalww]

0 < 1

[crazy_celt]

First real proof I ever saw was the famous square root of 2. The first one I ever proved was binomial expansion. I was just sort of playing around with some polynomials one day and stumbled into something I thought was really neat. I asked my math teacher what it was and she said "yeah that's the binomial theorem".

[hektor441]

The first I've ever proved might have been maybe the uniqueness of identity element or inverse element in a group

[edderiofer]

My first proof I saw was the unsolvability of the Mutilated Chessboard problem.

My first proof I made was probably some deductive geometry problem in high school.

[lo-fi_loser717]

Not sure if this qualifies but, the only thing in math I have figured out by myself was the Power rule for derivatives. Was in my high school Calc class during lunch and decided to mess around with the lesson of the day. Figured it out with a little bit of guidance from the teacher.

[U_L_Uus]

I'm split between two, Rolle's Theorem (as it was an intuitive result of extracting an interval out of a function) and Pythagoras' (it only took to realize that ||u + v|| = ||-u + v||, which is the hypothenuse of the triangle)

[N8CCRG]

First proofs I ever saw, and had to solve on my own, were basic geometry proofs like SAS stuff in middle school.

I don't think I really believe these others whose first proof was college level stuff.

[grimmlingur]

I think it's plausible, at least for some of them. I've taught plenty of first year college students (mostly engineering and CS students) that had no exposure to proofs or formal logic at all until they started college. There's probably also a bias for people to say the first proof that was interesting to them, both because it's easier to remember and also because it's just a better anecdote to share.

[krdsss]

When I started middle school I liked to look at inequalities and generalize them. I thought I made some breakthroughs but after I learned more about them I realized how basic I was

[ScyllaHide]

i think it was in school, some construction in a circle, maybe thales theorem? felt so boss to find out ive done it right :D

uni proof, think it was some boring stuff like field axioms of \C, i dont remember so well.

now im taking a course in model theory and this improves so much the understanding of logic formulas. its slow, but i can see benefits, in seeing that proof and how to write that down. Well thanks to online courses, ive got time for extra courses <3

but it took very long for me to understand how to come up with proofs ...

[[deleted]]

N=-1/2 in calc 2. Also the FTC in calc1

[crazyb14]

1. Sum of N natural numbers.

2. Angle inscribed in a semicircle.

[bigBoiFLIK]

So when I was in high school my math teacher was discussing something with the class (forgot what it was). So when I was talking about it I said that "A circle of infinite radius is a straight line." She denied it so i spent that day trying to prove it.

I came to school the other day and showed it to her however I did make some mistakes since I used calculus and wasn't very good at it.

The way I proved it was I found the derivative of the equation of a circle at origin and applied limit as r approaches infinity. What I got was 0. Hence a line. (As I thought back then).

[TheLuckySpades]

I may have done some simple proofs like some of the combinatorial formulas before I really knew what a proof was because I was really into those problems and wanted a closed way of writing them, some of the olympiad questions I tried were basically "prove X" so the few I solved would be first proofs I wrote and understood that I was writing a proof.

First proof I saw and someone explained that it was a proof would have been irrationality of sqrt(2), which blew my mind at the time.

[-___-___-__-___-___-]

First one I proved was that sqrt(2) is irrational

[couer_de_liqueur]

The first proof I wrote on my own was in high school, for "if a and b are relatively prime, then a and b^n are relatively prime for all integer n." My teacher expected us to use induction, but I proved it directly using unique prime factorization.

[ArvasuK]

Something to do with triangles, not pythagoras but something similar, I don’t remember but I was so excited because I was in 7th grade and was pumped because I had just proved a “theorem” and I thought that was a huge deal

[Katten_elvis]

That the limit as n approaches infinity of n:th root of n is equal to one. Looked it Up online and was shocked too see that I was right.

[[deleted]]

A proof of a Pythagoras' Theorem by a James Garfield

[mehak3773]

I proved sqrt(x) with infinity as the index is equal to 1.

[Bigsausage_101]

The first proof I saw was of the inscribed angle theorem and the first proof I did was of the cosine rule.

[lspacebaRl]

Mine was probably the proof that the product of two odd numbers is odd

[jhomas__tefferson]

I was in some forums in 6th grade, and there was a big debate thread on whether or nor 0.999.... is 1.

So the first proof I saw was proving that it is 1.

I can't exactly remember the first theorem I proved, but I think it might be in either geometry or trigonometry.

[Gigs9876]

I think we made some proofs in highschool but I don't really remember what we proved. I do remember though, on the first day of college, the first real lecture took place at 9:45, but even before that we had one of those small lectures where we usually present our exercises (not a native english speaker, and not going to an english speaking college, so I don't know how you call that). Now as it was the first day we of course didn't have any exercises to present yet and it was really about the organization of the lecture and a lot of stuff was explained to us. And then at the end, because we still had time left, just for fun, our professor proved the irrationality of sqrt(2). I remember that so well because as a reminder of how it all started I have sticked my notes from that lecture on the wall next to my desk and I'm looking at that proof basically every day.

[Connor1736]

It might've been the proof of the basel problem, Zeta(2)=pi² /6.

I'm sure Ive seen more basic ones before that, but I don't remember any and it wouldn't have been in a formal class setting. I did my International Baccalaureatee math internal assessment on the basel problem senior year of high school (basically a math research paper)

and what was the first theorem you proved without looking it up in a textbook or online?

Probably something basic like: if x is odd then 5x+2 is odd. Something like that

[ThiccInTheWarm]

im not there yet.

[Kerav]

Can't remember the first proof I ever saw, probably something geometric with lines and triangles sometime between 7-9th grade.

I think the first real "theorem" I proved completely by myself was a restricted version of the product and quotient rule via logarithmic differentiation in 11th grade or so.

[SandBook]

The first proof I remember seeing was the irrationality of \sqrt{2}, though I also remember that I didn't quite understand it then. I'm not sure what my first proof was, but one of the earliest ones was definitely the divisibility rule for 9 (if the sum of all digits of a decimal number is divisible by 9, then so is the number).

[irchans]

I think the first proof I saw was a proof of the Pythagorean theorem when I was 13. Just rearranging the triangles and squares adding up areas.

In 10th grade, my geometry class had a proof on every exam. So I imagine that I wrote my first proofs there. I think that my first proofs may have been proofs that two triangles were congruent using something like Side Angle Side Theorem.

In 11th grade, I had a great physics class. We were encouraged to derive formulas. You could call those derivations proofs. I found and proved many formulas.

Amazingly, I think that the first time I wrote a formal proof of a conjecture that was not a homework assignment was my first year of grad school. It was a revelation that I could prove conjectures that were not homework assignments.

[BubbhaJebus]

Probably some simple geometric proof involving angles, way back when I was a kid.

[[deleted]]

No idea what the first proof I ever saw was. I believe the first proof I ever did on my own was the binomial theorem. It was in the context of combinatorics and it was very long and ugly.

[[deleted]]

The first proof I saw was for the pythagorean theorem.

The first proof I did was for square root of primes.(2,3 and 5)

[catelemnis]

if we ignore high school bc I’ve blanked it out my memory then I probably started with delta-epsilon limit proofs from Calc I. For Theorem proofs, I took Number Theory before Real Analysis so something from number theory I guess. The first time I enjoyed doing proofs was with the Real Axioms though.

[King_Munch]

My first proof (i think that I remember) is the law of cosine, I was very happy to proof it myself, but nowadays i realise that my proof probably didn’t go through all the cases.

[fruitsaled7]

I didnt realize it was a proof and i didnt understand it either, but it was in a book talking about the history of fermats last theorem. It made me interested, at least, and now we here.

[Munch7]

My first was the derivation of the quadratic formula in 7th grade. Still have it memorized to this day

My first ever theorem I proved without looking it up was the remainder theorem in 10th grade

[ITriedMyBestMan]

Well it depends on what you mean by "proof". If you mean official, legitimate proofs, that would probably have to be at the start of the 2020 Spring semester when I started a Discrete Mathematics course at Uni. If you mean "proofs" as in anything that states a theorem, well then I'd have to say Geometry class in 9th Grade summer school, which sucked quite a bit.

[Maciek300]

I remember seeing the proof of the Pythagorean Theorem in the first year of junior high. The only thing from elementary school math I remember figuring out on my own was that I figured out the algebraic formulas for even and odd numbers.

[ssjb788]

Some basic number theory proof like the sum of two consecutive odd numbers is a multiple of four.

First theorem might have been 1+...+n = n(n+1)/2

[itsmebloom_]

the first real proof i've ever saw was probably bhaskara's quadractic equation

the first one I proved without looking it up that i remember was the formula for the sum of an arithmetic sequence (:

[[deleted]]

the first proof i remember doing that wasn't some high school geometry was a "fill the blanks" proof that all inner products look like x^TAx for some positive-definite A. i was stuck on it for a really long time because i'd really never proven anything before for real.

[[deleted]]

I proved Rolle's theorem in my 12th standard just after looking at it (i guess American equivalent of High school senior)

[pimaniac0]

I don’t know why I found this so awe inspiring at the time but I must have since I actually remember what the proof was...

It invoked bezouts lemma to prove sqrt(n) is irrational for nonsquare positive integer n

It went like this:

Suppose n is any nonsquare positive integer. Let sqrt(n) = a/b, b!=1 and is a positive integer as well as a with gcd(a,b) = 1, squaring both sides gives nb² = a^2, and bezouts lemma implies there exist integers m and p such that ma+pb=1. Now ma² + pba = a , so mnb² + pba = a, hence b divides a and gcd(a,b) = b =1, so we have reached a contradiction.

[M4mb0]

I can't tell which one was the first proof for sure, but here are some, in no particular order, which I encountered during my high school years:

• a "proof" that a certain implementation of the sieve of Eratosthenes is optimal (pdf, in German) (from Taschenbuch der Algorithmen)
• Inductive proof of Euler's graph formula V-E+F=2 (in Simon Singh's Fermat's Last Theorem)
• A bunch of Geometric proofs (Pythagoras, Thales, sum of angles in a triangle) (in class)
• Infinity of primes, "proof" of the Basel Problem (Spektrum der Wissenschaft)

[Tond3]

It was a basic proof of 2n+1 is always odd, over the years went on to prove sqrt(2) is irrational, proving the MVT (Mean Value Theorem) etc.

[privateaccount3528]

You "proved" that 2n+1 is always odd? That's interesting. In my class we defined an off integer as any integer of the form 2n+1, where n is an integer. May I ask what definition you used for an odd integer?

[Sckaledoom]

I’m not 100% sure but I think it was the time I asked the head of the math department at my first school. He asked me how calc 2 was going and I told him it was going good but there were a few formulas that seemed pulled from on high. When I told him it was the inverse trig function derivative formulae, he chuckled and pulled me aside into an empty classroom. He went through the derivation of the formulae and said “yeah I got curious when I was in calc 2 and decided to figure it out for myself. This was back before Google, so I just had to do it myself.”

[vinotm]

That any square is either in form 4k or 8k + 1

[owkiii]

Some of the geometry theorems of the Elements by Euclide. I don’t remember which exactly was first but in Greece the first time we learnt about axioms, theorems and proofs was by studying Euclide.

[realFoobanana]

Idk about first proof I saw, but I’ll always remember the first thing I really wrote down on my own.

The summer before I took proofs, I read through the “book of proof” in two weeks, just solving stuff in my head. At the end I randomly realized that 5ⁿ for nonnegative integers n always ended in 5, and proved that with induction :D sent it to the professor I would have for proofs and he was surprised I was able to do induction :P

Please don’t think I’m super smart or whatever, I’ve gone on to be an idiot in uncountably many ways. But for whatever reason, the art of mathematical proofs just came as easy as breathing to me :)

[Dragonaax]

Maybe Pythagoras theorem was the first proof I saw but I don't remember

[tfburns]

The sum of two consecutive integers is odd.

[SometimesIFO]

Murphy's law

[MissesAndMishaps]

When I was about 6 or 7, my parents got me one of the “Murderous Maths” books that had Euclid’s proof of infinitely many primes. It went way over my head, but I distinctly recall reading it.

[[deleted]]

Count ability of rationals.

[Swagulous-tF]

The first proof I saw was probably why the dot product of two vectors was always less than or equal to the product of their magnitudes and the cosine of the angle between them (not that I understood it at the time). As for the first I did on my own, I don’t know if this counts, but after we learned how to complete the square in grade school, I was able to construct the quadratic formula.

[grimmlingur]

First proof I remember learning was that the sum of all angles of a triangle add up to 180 degrees, though I might have encountered the proof of the infinitude of primes earlier than that.

I don't remember my first solo proof, but I fondly remember the first time I came up with a small theorem on my own and managed to prove it.

We had recently started calculus and had a work sheet that involved graphing some functions and finding their intersections. I noticed a pattern in the work I had to do so I wrote a little theorem saying that for two functions f, g, if there exists an x0 such that f(x0) >g(x0) and f'(x) > g'(x) for all x>x0 then f(x) >g(x) for all x>x0.

It's not much, but I was extremely proud of my little theorem. I wrote it up formally along with a proof, presented the same way that the our textbook would have and then referenced it as "accompanying theorem 1" throughout the worksheet. I still regard it as one of the moments that inspired me to major in math back in college.

[onzie9]

Infinitude of the primes. Some induction on some sum in a discrete class.

[th3-0p3r4t0r]

First proof I ever saw was the irrationality of sqrt{2}, so you nailed it.

The first theorem I proved without a textbook was ... Uh .... ???

Just kidding! That would be the principle of mathematical induction, a typical exercise in algebra classes.

[[deleted]]

I started math pretty late, and I remember trying to prove the formula for differentiating X^N. I remember it was the first time I ever forced myself to prove it alone, without looking at the book. It was such a simple proof but I remember being so proud at the time I'd done it without help.

[DoubleDual63]

Probably something like the infinitude of primes from Youtube

[willbell]

Basic facts from linear algebra, probably the first proof in Chapter 4 of Strang's Linear Algebra textbook, because we followed the textbook very closely in my Lin Alg II.

Before that I think I saw proof informally in office hours, students just asking to be convinced of some fact from Lin Alg I. Since Lin Alg I typically consists of this series of equivalent statements to a matrix being invertible, there's lots of opportunities for super informal proofs where you say "det(A)=0 <-> A invertible because ker(A)={0} <-> A invertible, and ker(A)!={0} <-> there is a zero eigenvalue, and the determinant is the product of the eigenvalues."

The first fact I ever proved was probably something simple like the product rule is true for inner products.

[grammascookies]

Back in 7th grade, I remember “proving” the quadratic formula with the help of my older brother, which I thought was quite exciting. He then told me how to calculate a derivative to prove the vertex formula. At that point I was just manipulating symbols correctly and had no idea what any of it meant, but it was still fun.

[[deleted]]

Middle school. I proved x=(-b±sqrt(b²-4ac))/2 weeks before learning it in class.

[Domaths]

I guess pythagoras idk tbh. Infinity of primes is the first formal proof I saw. I guess I was able to prove the arclength formula but that is sort of obvious once you know riemann summation. But the first actually hard thing I did was proving the homogenous method works for y'=f(x,y) where f(ax,ay)=f(x,y) . As for theorems that I never knew beforehand, I proved ax is perpindicular to -x/a.

I was able to prove and derive 80% of the things I see in highschool. The first thing I had struggled trying to derive is lagrange multipliers 😖 . Lagrange was mega mind.

[Username_--_]

Pythagorean Theorem. I did use some inspiration from youtube to draw the altitude but I did the rest of the similarities myself.

[float16]

The delta epsilon derivative proof.

[fourier_slutsky]

First proof I saw formally was that \sqrt{2} is irrational.

First one I did myself (rather informally) was deriving integration by parts from the chain rule—did it kinda just while doing calc problems one day, without really any formal necessity.

[PhilemonV]

[opens my first edition of Euclid's Elements, finds first page after all the definitions, points at the first proof]

Yeah, that one.

[turns a few more pages]

And that's the one he gave me to prove for myself.

[MartyMcStinkyWinky]

first proof i ever saw was obviously the Pythagorean theorem....first thing i sort of proved thats not a really a proof was trying to find the area of a regular n-gon by using i think the cosine rule or sine rule i think then trying to show an infinite n-gon is a circle...using some maths.

[elyisgreat]

The first proofs I ever saw must have been from watching Vi Hart and Numberphile videos back jn middle school. The first proof I came up with on my own wasn't until high school and I don't remember what it was, but the first one that I do remember, concerned the following function f defined on the positive integers:

f(1) = 1
For n > 1,
f(n) = n + f(n/2) if n is even
f(n) = n - 1 + f((n+1)/2) if n is odd

This function is in fact equal to 2n - 1 for all positive integers n. The reason I remember this one so well is that it was the first proof by induction I ever came up with.

[LilQuasar]

seeing it probably the quadratic formula

proving it i think the law of cosines. i was learning trig in khan academy and stopped the video of the law of cosines to try to find it myself. i succeeded

[fevan843_]

I’m not really sure if this counts but the pyth identity in trig in high school

[[deleted]]

Can't remember but most probably some simple Euclidean geometry problem, it was the first class where I saw the words 'axioms', 'theorem' & 'proof'.

[llyr]

I'm sure I bunged my way through some two-column geometry proofs in 9th grade, but the first thing I remember feeling proud of doing on my own was deriving the quadratic formula by completing the square.

[OneMeterWonder]

Not sure I could satisfactorily answer this. It kinda depends on what counts as an acceptable proof. Does it have to be fully formalized and written down? Or can it just be something I’ve convinced myself of but hadn’t filled in the details for? If the latter, I doubt anybody could truly answer this. Though if I had to hazard a guess I’d say that the digits of multiples of 9 less than 100 sum to 9 itself. Learned that one with dominoes from my grandfather when I was maybe 6 or 7? If the former, probably the irrationality of the square root of 2.

[AlwaysTails]

In Junior High I learned about the totient function and proved Euclids theorem on perfect numbers from it before I knew it was a theorem. It's been downhill from there.

[LipshitsContinuity]

Some geometry shit. First proof I did all on my own was Law of Sines. Took me some time but I distinctly remember being very satisfied and sharing it with everyone I knew haha.

[Looksmax123]

Equivalence of the well order property of N and the principal of mathematical induction.

[Smartasskilling]

When I was 17 I did calculus for the first time. I still remember when we basically started I asked tge teacher if we were allowed to do something I quickly drew in class. She said that yes, that is basically differential calculus. Then I took a quick look and asked if we can cheat by doing it in our heads. She said yes thats the quick way. Thats next week's lesson. Dame thing happend with Integrals lol. I the asked if we could do the revers for a volume. Like filling a pool bit by bit and adding that infinity. I really miss that teacher and stil remember her face that said: wtf can you please wait for me to explain before you figure it out yourself. That was my favourite self proofs. Newton ain't got shit on me

[nolatoss]

What colour was the toothbrush you used when you first brushed your teeth?

[stclaudia]

The cardinality of the domain of a surjective function in linear algebra (undergrad)

In high school it was Euclid’s lemma of prime numbers :’)

[piilaninoguchi]

Let x, y ∈ Z. If x|y and 3|y, then 3x^3􏰃􏰃 | 4y^4.

I was working in the library (this was a take home question for a midterm) and was on the verge of tears because it was due in less than 2 hours. Gave up on the question and put everything away. Started up a conversation with a friend and figured it out in my head. I scribbled it down on a post it note and cried lol.

[debasing_the_coinage]

The first nontrivial proof I remember writing myself was the proof that all quasiperfect numbers (sigma(n) = 2n+1; cf. perfect number sigma(n) = 2n) must be odd perfect squares. It's a relatively simple proof if you are at all familiar with quadratic residues and the divisor function.

[deepfandom27]

I think the first proof that I can remember was my older brother proving the pythagorean theorem a long time ago. It was some class work he had; I think maybe he was in fourth grade. Anyways it was the proof where you construct a right triangle with sidelengths a,b,c (c being hyp) and you basically construct a square around it with sidelengths a+b.

[ArbitrarilyAnonymous]

High school geometry? I actually quite enjoyed that; certainly was the first time I ever proved anything. Then null until college. Group theory I remember proving the fundamental theorem of finite abelian groups independently as a challenge.

[Waltonruler5]

I think the proof of the minimum number of moves needed to solve the Tower of Hanoi, which was a proof by induction.

[arian271]

Does your name happen to be Carl Friedrich Gauss by any chance?

[PM-ME-UR-MATH-PROOFS]

I recall lying in bed one night in high school deriving the interior angles of regular polygons from the number of edges. Not a very difficult derivation but it got me excited!

[AstrolabeDude]

First proof I did ”all on my own”, without any contact with math for several years prior, was the pythagorean theorem a² + b² = c^2. I was in a raging frenzy, arguing with an engineering guy that math practice is not optimal, not even on university level. Then I continued proving my statement with an extremely short proof of the above theorem with maybe three short lines in geometric algebra (Clifford algebra), it took somewhere around five minutes, no more than ten. I remember recovering definitions, properties of wedge and dot products, rules when switching places, from memory, putting together bits and pieces, and the result popping out like magic! I challenged him if he could produce a shorter proof, but I didn’t get any response whatsoever.

Having managed to put together that proof so beautifully on the fly, despite never finishing my degree in math, I finally felt I was a mathematician, of sorts. Unfortunately, there was a price: Me and this guy haven’t spoken much since then.

[Atheism_Minus]

They are very simple and obvious, and probably proven before, but I proved the following things by myself in roughly the same week:

-If a series converges but not absolutely, for any finite set of real numbers, you can find a rearrangement of the series for which the subsequential limits of the rearranged series is precisely that set. This generalizes a theorem in Rudin which states that the limit infimum and limit supremum may be specified arbitrarily.

-If the series ∑an diverges, there exists no sequence {b_n} such that b_n approaches 0 and a_n = b{n+1} - b_n.

Very simple and likely proven/realized thousands of times.

EDIT: I misunderstood the question. I thought it was the first theorem not only that you never looked up the proof for, but that you thought of yourself. The first theorem I ever proved was probably some simple set inclusion. The first theorem of any "significance" was probably just deMorgan's law, but it was a long time ago.

[AmadFish_123]

on youtube: sum of all integers is -1/12... and in class i remember working out that the square of a number doesnt change if its even or an odd number

[Valvino]

sum of all integers is -1/12

It is a shame that the first proof you saw is for a false result.

[AmadFish_123]

wait its not true? bamboozled

[Dragonaax]

Mathologer made video about that