A Dozen Proofs: Sum of Integers Formula (visual proofs)

Рет қаралды 54,303

Күн бұрын

In this video, we explore the famous formula for the sum of the first n positive integers. In particular, we present twelve proofs of the sum formula using induction, area-based techniques, combinatorial techniques, physical techniques, and by using a couple of deep theorems. All of the proofs except the first are visually inspired or have a visual component. #SoME2 #manim #visualproof
Comment with your favorite of these twelve or let me know if you have a different favorite proof of this fact!
This video is my submission to the "Summer of Math Exposition 2" contest. The key takeaway is that we can gain exposure to many areas of mathematics by "thinking deeply of simple things" as suggested by mathematician Arnold Ross.
0:00 Introduction : Think Deeply About Simple Things
1:08 Proof by induction
2:49 Classic visual proof and "reverse and add"
4:45 Triangle area proof
5:30 Fundamental theorem of calculus proof
7:04 Trapezoid area proof
7:55 Double counting proof
9:15 Bijective proof
10:40 Linear recurrence proof
12:10 Pick's theorem proof
14:11 Euler's formula proof
16:10 Water flow diagram proof
17:49 Center of mass proof
19:19 Concluding remarks
20:38 Citations
#sumformula #sumintegers #integers #mathvideo #math #mtbos #animation #theorem #pww #proofwithoutwords #proof #iteachmath #mathematics #3b1bsome2 #combinatorialproof #combinatorics #integralcalculus #area #areas #bijection #trapezoid #triangle #physics #moments #weight #centerofmass #waterflow #recurrences #linearrecurrence #gauss #doublecount
This video is based on an expository paper written by Tom Edgar and Enrique Treviño compiling 35 proofs of the fact that 1+2+3+...+n = n(n+1)/2; the paper is linked here: bit.ly/2UPRzep
That paper includes many references, but here are a few more relevant sources for proofs from this video:
Ian Richards, Proof without Words: sum of integers, Mathematics Magazine (March 1984 page 104): www.maa.org/sites/default/fil....
Joe DeMaio and Joey Tyson, Proof without words: A graph theoretic summation of the
first n integers, The College Mathematics Journal 38 (2007), no. 4, 296. (www.jstor.org/stable/27646507)
Jaime Gaspar, Proof without words: using trapezoids to compute triangular numbers,
Math. Mag. 91 (2018), no. 3, 206-207. (www.jstor.org/stable/48665541)
Tom Edgar, Proof without words: matchstick triangles, College Math. J. 47 (2016),
no. 3, 207. (doi.org/10.4169/college.math....)
Tom Edgar, Proof without words: a recursion for triangular numbers and more, Math. Mag.
90 (2017), no. 2, 124-125. (doi.org/10.4169/math.mag.90.2...)
Loren C. Larson, A discrete look at 1 + 2 ++ n, The College Mathematics Journal 16 (1985), no. 5, 369-382 (doi.org/10.2307/2686996)
David Treeby, A moment’s thought: centers of mass and combinatorial identities, Math.
Mag. 90 (2017), no. 1, 19-25. (doi.org/10.4169/math.mag.90.1.19)
Another visual proof of this fact using eight triangular arrays can be found in this video here: • Video
If you enjoyed this video, please like and subscribe. Also feel free to leave a comment noting your favorite of the 12 proofs!
To learn more about animating with manim, check out:
manim.community
__________________________________________________________________
Music in this video:
Ambiment - The Ambient by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. creativecommons.org/licenses/...
Source: incompetech.com/music/royalty-...
Artist: incompetech.com/

Пікірлер: 208

@MathVisualProofs Жыл бұрын

Tell me which of the 12 is your favorite! Or maybe you have a favorite I didn't include?

@michaelmam1490 Жыл бұрын

I like the triangle area proof. It's so elegant

@MathVisualProofs Жыл бұрын

@@michaelmam1490 triangular numbers from triangular area-makes so much sense! Thanks!

@AssemblyWizard Жыл бұрын

Great video, I think a basic proof similar to others shown is missing: linking 1 with n, 2 with n-1, etc, each with a half-circle arc, giving a diagram that looks like a rainbow or a Menorah, then the sum of each pair is n+1 and you have n/2 pairs (need to split into even/odd cases). This is very similar to the proof where you duplicated the sum and flipped it, but all of the proofs are similar so I think this also counts as different

@MathVisualProofs Жыл бұрын

@@AssemblyWizard for sure. That’s a nice one. Thanks for sharing!

@peterbrough2461 Жыл бұрын

I like Gauss': (first plus last) times (half the number of terms) Works for any arithmetic sequence. Works also for arithmetic sequences missing terms in a symmetric way like say, the sum of numbers in a square on a calendar.

@peytonritchie3986 Жыл бұрын

You have a great reading voice, I could easily see you having success in voice acting/audiobooks

@MathVisualProofs Жыл бұрын

Thanks! Probably not in my future, but good to have options :)

@Mr_Happy_Face Жыл бұрын

This is amazing. I’ve always wanted to find a KZfaq video that proves the same thing in many many different ways. I think in the future you could do a similar video on the connection between binomial expansion, Pascal’s triangle, and the number of ways to choose from a collection.

@MathVisualProofs Жыл бұрын

Thanks! Definite a good suggestion for the future - I will put it on the list :)

@johnchessant3012 Жыл бұрын

If f(x) is the generating function for a sequence {a_n}, then f(x)/(1-x) is the generating function for the partial sums of a_n. Using this fact, we see that the generating function for 1, 3, 6, 10, ..., is 1/(1-x)^3. So the sum 1+2+...+n is the x^(n-1) coefficient in the series expansion of 1/(1-x)^3, which we can evaluate as 1/(n-1)! d^(n-1)/dx^(n-1) [1/(1-x)^3] = 1/(n-1)! 3*4*...*n*(n+1) / (1-0)^(n+2) = n(n+1)/2.

@MathVisualProofs Жыл бұрын

Excellent one for sure! I wasn’t sure how to do visualization of gen funs, so I didn’t include it. But I love it! Thanks!

@joel-wg4bp Жыл бұрын

This channel is hidden gem!

@MathVisualProofs Жыл бұрын

Thanks for thinking so! Appreciate it :)

@tylerboulware6510 Жыл бұрын

I agree. I hope we can help make it less hidden!

@Mutual_Information Жыл бұрын

I haven’t seen all your videos-you have many-but I’ve seen a lot.. and this is my favorite one. It offers a ton in a short amount of time. Excellent

@MathVisualProofs Жыл бұрын

Thanks! This was fun to make :)

@monsieur910 Жыл бұрын

The center of mass proof is amazing! All of them are, but that one is the best one

@MathVisualProofs Жыл бұрын

Thanks! I definitely like that proof.

@behackl Жыл бұрын

Great job with this collection, kudos! 👏

@MathVisualProofs Жыл бұрын

Thanks!

@buffalocat1096 Жыл бұрын

Here's one that came to me after I couldn't stop thinking about this video: Consider the function f(x) = 1 + x + ... + x^n = (x^(n+1) - 1) / (x - 1). Evaluating f'(1) in the first expression gives the sum of the first n positive integers. Starting from the right expression and computing lim x->1 f'(x), we can use l'Hopital's rule to get the desired formula! edit: I see I wasn't the first to come up with this!!

@MathVisualProofs Жыл бұрын

Yes! This one is excellent (love the geo series formula). It’s in the linked paper but I couldn’t come up with a great visualization for it. Thanks!

@mathflipped Жыл бұрын

Great job, Tom! And good luck with this impressive #SoME2 submission. I'm still working on mine. It will be an opener for the Visual Group Theory series.

@MathVisualProofs Жыл бұрын

Thanks! Oh that's great! I look forward to seeing what you do for that. I'll keep my eyes out for it.

@edkhil Жыл бұрын

I came here from KZfaq shorts and I was wondering if you had a public repository with all the manim scripts. That would be very helpful!

@MathVisualProofs Жыл бұрын

I have only put up some of the code: github.com/Tom-Edgar/MVPS . I have been learning a lot over the past almost two years, but much of the code is quite messy. Those ones are some of the best (though none are documented).

@edkhil Жыл бұрын

@@MathVisualProofs Thank you so much!!!

@williamweatherall8333 10 ай бұрын

very cool. I started to get lost during the combinometric proofs, but the water one got me back. Maybe one day I'll return to this and see it more deeply.

@MathVisualProofs 10 ай бұрын

The combinatorial one is a challenge for sure. Thanks for checking it out!

@markzuckerbread1865 Жыл бұрын

subscribed before i even started watching :D great video :D

@MathVisualProofs Жыл бұрын

Thanks! Glad you enjoyed it :)

@mostly_mental Жыл бұрын

This is really well done. Lots of beautiful connections (some I'd never seen before), all explained clearly. I'm always a fan of recurrence relations (and the hidden generating functions), but I think Pick's Theorem is my favorite.

@MathVisualProofs Жыл бұрын

Thank you for the comment! And check the link in the description for even more proofs :) I think I agree with you about Pick's theorem. Especially because it ties together area and counting proofs :)

@patrickgambill9326 Жыл бұрын

This is amazing! I subbed

@MathVisualProofs Жыл бұрын

Thanks!

@RafaelCouto Жыл бұрын

amazing content! keep on it

@MathVisualProofs Жыл бұрын

Thank you! I’ll see what I can do :)

@anyoung1818 Жыл бұрын

This video is great thank you! And your thumb’s nail is beautiful!

@MathVisualProofs Жыл бұрын

Thanks for watching!

@neizod Жыл бұрын

Reminds me of Philip Ording's 99 Variations on a Proof. Great job!

@MathVisualProofs Жыл бұрын

An excellent book! Thanks :)

@chhaganarammali4573 Жыл бұрын

I am really impressed by this video, just beautiful....

@MathVisualProofs Жыл бұрын

Thank you! It was fun to make!

@keinKlarname Жыл бұрын

9:45 This bijection has had an aha-effect on me! So easy - and so brilliant.

@MathVisualProofs Жыл бұрын

Glad you liked it!

@AllThingsPhysicsYouTube Жыл бұрын

Very nice. And nice to see you're getting some traction!

@MathVisualProofs Жыл бұрын

Thanks! This has been the best traction yet... still a long way to go :)

@imperatoreTomas Жыл бұрын

I love this

@MathVisualProofs Жыл бұрын

Thanks for checking it out!

@estee1209 Жыл бұрын

Thanks!

@MathVisualProofs Жыл бұрын

Woah! Thank you. I appreciate it :)

@samueldeandrade8535 6 ай бұрын

This guy right here is the real MVP of Mathematics on KZfaq. 3blue1brown? Who is that?

@MathVisualProofs 6 ай бұрын

Hah! Thanks. I still use his software and my videos are typically only short. Not long explainers. I have a long way to go. Still appreciate the sentiment :)

@TheDitronik Жыл бұрын

awesome keep it up Thanks

@MathVisualProofs Жыл бұрын

Thanks for checking it out!

@tutorchristabel Жыл бұрын

amazing lesson

@MathVisualProofs Жыл бұрын

Thanks! 😃 I'm glad you liked it

@Axel_Arno Жыл бұрын

Hey man, wonderful video here ! Animations are gorgeous. Do you have a github in order to see the Manim code you used to animate this video ?

@MathVisualProofs Жыл бұрын

The code for this video is not up there yet. I have a small repo with code from some videos: github.com/Tom-Edgar/MVPS

@hamarana Жыл бұрын

all I know about math is that 2+2=5. the rest of it I don´t know much.. great videos it gives insights to make us start loving what the universe of numbers is all about!

@egohicsum Жыл бұрын

great video

@MathVisualProofs Жыл бұрын

Thank you!

@pengin6035 Жыл бұрын

Very nice video! I appreciate the work that went into this. I also didn't know some of the proofs, I really liked the one using Euler's formula! However, we used that the number of faces inside of the graph is n². This is equivalent to the summation formula we want to prove so it feels like we hide something at that step, right? Here is one proof I just thought of: Consider the expression f(x)=1+x+x²+...+xⁿ If we calculate f'(1), we can either use the summation rule to obtain the sum 1+2+...+n. On the other hand we can first use the geometric formula for this expression which gives f(x)=(xⁿ⁺¹ - 1)/(x-1) (for x≠1) Now we use the quotient rule we obtain f'(x)=((n+1)xⁿ *(x-1) - (xⁿ⁺¹ -1))/(x-1)² We can't just plug in x=1 because this is undefined. But we can use L'Hôpital twice to obtain f'(1)=((n+1)(n(n+1)-n(n-1)) - (n(n+1)))/2=n(n+1)/2

@MathVisualProofs Жыл бұрын

Thanks! I love that proof too! ( it is in the linked article but I couldn’t figure out how best to visualize it for this). As to the Euler’s proof. I think you can get the number of faces by a scaling argument too but yes it is also equivalent to the sum formula so maybe a bit of a stretch. Still fun to see that theorem in this context :)

@maloukemallouke9735 Жыл бұрын

i love your chanel

@MathVisualProofs Жыл бұрын

Thanks! Appreciate the feedback :)

@dysfunc121 Жыл бұрын

Well I'm convinced.

@MathVisualProofs Жыл бұрын

😀

@faresalahd 9 ай бұрын

فيديو مذهل، أقدّر هذا المجهود الكبير وأحييك على إتقانك أتساءل إن كان هناك برهان باستخدام علم المثلثات

@MathVisualProofs 9 ай бұрын

Good question! I would guess yes :)

@s90210h Жыл бұрын

I was thinking of a few before I watched and only had a few. The water flow is my favourite! Such a nice one which I didn't know about and totally for me as I had the center of mass in my head all video long. I wondered: Is there a way the Pythagorean theorem is right around the corner in this, seen as there as it triangle galore all over?

@MathVisualProofs Жыл бұрын

Great thought. I don’t know one that uses PT explicitly but perhaps it’s there. You’d want a n by square root of n triangle. Can you fit twice the sum of integers in there nicely?

@ProfeJulianMacias Жыл бұрын

Good Topic

@MathVisualProofs Жыл бұрын

Thanks!

@artsmith1347 Жыл бұрын

Before shading the area below the line, y= x + 0.5, at 05:50, it would have been interesting to see that line drawn on the grid at 05:00. It was not immediately obvious to me that lifting *_both_* ends of the sloped line at 05:00 by 0.5 does slice the top squares in such a way that the top portions of those squares can be rotated and shifted to fill the valleys below the raised line.

@MathVisualProofs Жыл бұрын

That’s a good one! Wish I had thought of it.

@msinkusmeowmeow1442 Жыл бұрын

Good video! What manim version did you use?

@MathVisualProofs Жыл бұрын

Thanks! This is done with manimgl. I haven’t moved to ce yet.

@andychen7016 Жыл бұрын

A few years ago, I used the formula “(n(n+1)) / 2” to make another formula, where instead of counting up by “1”, you count up by “x”. (n(n+x)) / (2x)

@050138 Жыл бұрын

Lol don't think this is correct.... If you know what an Arithmetic Progression means 😁

@tonaxysam Жыл бұрын

This is really cool. I was blown up by the bijective proof. Very engaging indeed!

@MathVisualProofs Жыл бұрын

Thanks! Yes. That bijective proof is totally amazing - the kind of thing I wished I had thought of myself :)

@jamesking2439 Жыл бұрын

My favorite was the bijective proof.

@MathVisualProofs Жыл бұрын

Suck a cool one, right?

@LuisHernandez-ip7gx Жыл бұрын

Interesante las diversas interpretaciones, a lo que yo considero la inducción, especialmente la de n(n+1)(0.5)

@MathVisualProofs Жыл бұрын

Thanks for checking it out!

@myrus5722 Жыл бұрын

Awesome stuff! I love how esoteric and field-expanding some of the proofs got. My favorite proof is this: > Let’s assume we already know 1 + 3 + 5 + … + 2n - 1 = n^2 > Let’s add 1 to each term in the sum on the left. There are n such terms, so to keep equality, we should add n 1’s on the right: 2 + 4 + 6 + … + 2n = n^2 + n > This looks like our target result. We just have to divide the left side by 2, and what does this give on the right? (n^2 + n)/2 Thoughts: Pretty simple and cute. I always love manipulating equations in cool ways, so this has a soft spot for me ever since 7th grade. Try thinking of a good visual animation of this too… it’s not anything mind blowing, but it’s still fun.

@MathVisualProofs Жыл бұрын

Excellent! Nice one. Thanks!

@MonkOrMan Жыл бұрын

I think u mean "n" not "1" but I love this!!

@myrus5722 Жыл бұрын

@@MonkOrMan Fixed! Thanks for seeing it

@050138 Жыл бұрын

@@myrus5722 where did you fix.... It's still '1' has to be 'n'

@myrus5722 Жыл бұрын

@@050138 Oh I had it wrong in two places maybe? Should be completely fixed now

@3bdo3id Жыл бұрын

great to know! thanks

@MathVisualProofs Жыл бұрын

👍

@blacklistnr1 Жыл бұрын

@1:14 Ah yes.. the classic "use divination to find the answer then prove it's true". It's like a swimming school throwing somebody in the ocean and if they swim it's proof that they learnt to swim. (at some point... also don't be distracted by the dead bodies on the ocean floor)

@MathVisualProofs Жыл бұрын

Haha. It’s unfortunate that that appears to be the most common way people encounter the formula…

@zihaoooi787 8 ай бұрын

I don’t need sleep. I NEED ANSWERS

@GraysonGranda Жыл бұрын

Now, I admit I'm not familiar with even the term linear recurrence, let alone whatever field it's drawing from, but I feel like I'm missing a LOT of steps in that "proof." I'm certain these formulas come from somewhere, but I can't immediately tell where.

@MathVisualProofs Жыл бұрын

Yes. That one requires some knowledge of the theory of linear recurrences. I don’t think this video was the place. Here is a video showing how to deal with two term linear recurrences : kzfaq.info/get/bejne/j8Bzg8J-zb_VgZ8.html. But it also only gives the idea and doesn’t delve into repeated roots too much. The idea in this video is to inspire you to learn about that area of mathematics now :)

@gitterrost-4 Жыл бұрын

At 14:52 how did you determine the number of faces? Don't we first need to know (or prove) that the sum of the first n odd integers is equal to n*n?

@MathVisualProofs Жыл бұрын

One way to get it is to use the formula. Another way is to use a scaling argument.

@afifudinlisgianto1640 Жыл бұрын

Try to proof with telescopic, multiply with 2/2, first term multiply with (2-0)/2, second term (3-1)/2, third term with (4-2)/2 until n^th term with ((n+1)-(n-1))/2 , we get telescopic form

@Osniel02 Жыл бұрын

Where did the y = x + 1/2 came from in the FTC proof?

@MathVisualProofs Жыл бұрын

Used it because it works! That's the trickiest part of that proof: knowing which curve to study. But if you differentiate (x^2+x)/2 (the eventual sum), then you get x+1/2 as needed.

@ayushsharma8804 5 ай бұрын

How do you know so many diverse things, do they teach these topics in ug math?

@MathVisualProofs 5 ай бұрын

I’ve been around a long time and have thought a lot about math. You pick things up over time.

@ZachAbueg Жыл бұрын

hey! i'm confused why, in the FTC proof, the function y = x + 1/2 was used

@MathVisualProofs Жыл бұрын

Sort of because it works out! It’s interesting that the area under that curve is n^2/2 +n/2. That’s kind of expected because the derivative of x^2/2+x/2 is x+1/2.

@filipsperl Жыл бұрын

@@MathVisualProofs nvm, I get it now. The integral is calculated in two ways, both of which are equal. The integrand is really chosen just because it works

@aliledra4887 Жыл бұрын

Can u give us a link to ebook of sequences and series , that give visual explaination and ulstration

@MathVisualProofs Жыл бұрын

I don’t know of such an ebook. Roger Nelsens three Proofs without words books will have most of the visualizations. I am working my way through them : kzfaq.info/sun/PLZh9gzIvXQUsgw8W5TUVDtF0q4jEJ3iaw

@theweebrt 6 ай бұрын

1:05. The triangle area...

@bluestrawberry679 Жыл бұрын

14:50 I'm confused about how you got the result that that triangle has n^2 faces, without already having a proof of the sum of integers After all, the faces are made up of the upward facing triangles, and the downward facing triangles, meaning that the number of faces is just T_n + T_(n-1) Which is equal to T_(n-1)+n+T_(n-1)=2T_(n-1)+n This means that knowing that that Triangle has n^2 faces seems equivalent to knowing the sum of integer formula already, making the proof a circular Argument in a way

@MathVisualProofs Жыл бұрын

Is a good point but I think you can get the argument by scaling.

@minhducphamnguyen7819 4 ай бұрын

I've read numerical proof for the sum of n integers but the proof never really stick in my head. The moment you arrange the coin in a triangle however it clicks for me that it has something to do with the area of the triangle.

@MathVisualProofs 4 ай бұрын

😀👍

@RSLT Жыл бұрын

Very Inserting!

@MathVisualProofs Жыл бұрын

Thanks!

@lukeinvictus69 Жыл бұрын

Can someone explain how the equation at 11:30 arises? I understand where the coefficients come from but not what x is supposed to represent.

@MathVisualProofs Жыл бұрын

Here is a video I did that gives some idea about where that comes from : kzfaq.info/get/bejne/j8Bzg8J-zb_VgZ8.html. But that is only two terms and doesn’t really explain why we handle the repeated roots the way we do. I suggest searching for “solving linear recurrences” to find some more in depth notes.

@lukeinvictus69 Жыл бұрын

@@MathVisualProofs Thanks for the prompt reply! Enjoyed the video

@MathVisualProofs Жыл бұрын

@@lukeinvictus69 thanks!

@MonkeySimius Жыл бұрын

When I was in precalc I got bored as I couldn't really understand what the professor was saying. So I ended up inventing that formula using what we had just learned about differential equations. Although mine was (N^2+N)/2 Most useful equation ever when playing a lot of card/board games. Now I could computer arbitrarily large sums of ascending numbers in my head instantly and I looked like a genius when I could very quickly add up scores in my head.

@MathVisualProofs Жыл бұрын

Very cool! One of my top fave formulas.

@gytoser801 Жыл бұрын

How about thinking simple about deep things

@thegreatsibro9569 Ай бұрын

I'm a year late, but I just found another proof: The sequence of sums of the first n positive integers is 1, 3, 6, 10, 15, 21, 28, 36, etc. For our purposes, we're going to start the sequence with a 0 so it looks like 0, 1, 3, 6, 10 etc. and considering the 0 to be the term we get for n=0. When we take this sequence's difference (meaning we do f(n + 1) - f(n) for each term), we get 1, 2, 3, 4, 5, 6, 7, 8, 9, ... because the next number in the sequence is always the last one plus the next greatest positive integer. When we take the difference again, we end up with a row of all 1's. What we are doing here is basically calculus with sequences. Now that we have our original sequence with an added 0 at the start, the first difference, and the second difference, we can use the Gregory-Newton interpolation formula to come up with a formula for our sequence. This is where we take the n=0 terms of our sequence and its differences, multiply each by the binomial coefficient n choose k where k is which difference each term represents (k=0 for our original sequence), and add them all together to get a formula for our original sequence. For the original sequence we get 0, for the first difference we get 1 x n, and for the second difference we get 1 x n(n - 1)/2. Now we just need a bit of algebraic manipulation to finish the proof. Distributing the second difference term gives us (n^2 - n)/2. Let's also manipulate the first difference term to give us 2n/2. Now we can combine both fractions into (n^2 - n + 2n)/2 which can then be condensed into (n^2 + n)/2 aka n(n + 1)/2. And that's the proof!

@leesweets4110 Жыл бұрын

You can also do it the way mathematicians have always done it when the problem is too hard to solve. Call it an axiom.

@MathVisualProofs Жыл бұрын

Hah! Yes. I suppose that’s one way.

@oncedidactic Жыл бұрын

Reminds me a Christmas song 😋

@MathVisualProofs Жыл бұрын

Should have phrased it that way maybe 😀

@oncedidactic Жыл бұрын

@@MathVisualProofs haha I liked it as a surprise how long it kept going, really amazing seeing how everything is connected! and I have explored around some of these approaches before! Gives me Mathologer vibes how it ties things together ^_^

@MathVisualProofs Жыл бұрын

@@oncedidactic his videos are the best.

@floppy8568 Жыл бұрын

(n²+n)/2

@MathVisualProofs Жыл бұрын

But here's twelve different ways to get it!

@drizer4real Жыл бұрын

Langland’s dream …

@christopherellis2663 Жыл бұрын

The late Twentieth Century

@ojas3464 Жыл бұрын

👍

@MathVisualProofs Жыл бұрын

😀

@emanuellandeholm5657 Жыл бұрын

Had to do a double take there. YTs algorithm threw an ad with grifter and con deluxe Elon Musk at me before the video.

@MathVisualProofs Жыл бұрын

Weird. I don’t understand the algorithm at all.

@adki231 Жыл бұрын

In my opinion this sum equals to -1/12

@MathVisualProofs Жыл бұрын

Hah! Only if you never stop ;)

@hexagon5610 Жыл бұрын

One small tip: Please consider also international viewers and don't use sizes like gallons. But otherwise, a very interesting video!

@MathVisualProofs Жыл бұрын

Yes of course :) Luckily it is really just a problem about rates, so you can assume liters per minute. Thanks!

@Jkauppa Жыл бұрын

generate all sets of pyhtagoran triangle integer solutions (up to certain number like 100k), all the possible variations, integer combinations

@MathVisualProofs Жыл бұрын

I am not sure I quite understand this one... can explain?

@Jkauppa Жыл бұрын

@@MathVisualProofs meh, its just an algorithm description suggestion

@Jkauppa Жыл бұрын

@@MathVisualProofs what you use it for is another thing

@Jkauppa Жыл бұрын

@@MathVisualProofs maybe find the 3d box shape that has only integers as the sides and the connecting lines

@Inspirator_AG112 Жыл бұрын

Unrelated, but what is the theorem that 2 + 2 = 4, 2 × 2 = 4, 2 ^ 2 = 4, etc. called?

@MathVisualProofs Жыл бұрын

I’m not sure I know if that’s a theorem. It’s just a coincidence?

@Inspirator_AG112 Жыл бұрын

@@MathVisualProofs: What is it called though?

@Inspirator_AG112 Жыл бұрын

@@MathVisualProofs: There are very few posts online about it.

@rafiihsanalfathin9479 Жыл бұрын

Its not a theorem, why would it?

@Inspirator_AG112 Жыл бұрын

@@rafiihsanalfathin9479 : What is the 'rule' called then?

@sj00100 Жыл бұрын

Here is a proof using telescoping sum: a(n) = n or n[ a(n) -a(n-1) ] Let S(n) = a(1) + a(2) +...+a(n). a(1)= 1[ a(1) - a(0) ] a(2)= 2[ a(2) - a(1) ] a(3)= 3[ a(3) - a(2) ] ..... .................... ..... .................... a(n) = n[ a(n) -a(n-1)] Summing all equations: S(n) = -S(n-1) + n^2 S(n) + S(n-1) + n = n^2 + n. (Adding n on both side) S(n) = (n^2 + n)/2.

@analopes5983 Жыл бұрын

-1÷12

@MathVisualProofs Жыл бұрын

only if you let n go to infinity ;) So n(n+1)/2 - > -1/12 as n goes to infinity I guess...

@grahamfinlayson-fife73 Жыл бұрын

-1/12

@BLVGamingY Жыл бұрын

no. I don't use induction. aren't so much of these the same proofs?

@MathVisualProofs Жыл бұрын

I think once you know what’s going on you eventually see them as the same proof - the interesting bit is how they can be reframed in different places and are then adjacent to or include so many ideas and techniques.

@tejarex Жыл бұрын

The fundamental idea is that if you put n*n + n markers on a table, you always have the same number of markers if you just shuffle them around and regroup them however and never add or delete any. Ditto for an area cut into squares, with some squares cut along a diagonal as needed.

@piwi2005 Жыл бұрын

First, no one learns this formula by induction. Second, Gauss' proof is 2*S=(1+n)+(2+n--1)+(3+n-2)+....+ (n+1)=n*(n+1)

@MathVisualProofs Жыл бұрын

Many many textbooks use this as a classic first proof by induction. Second, the rectangular array visualization (which was known to the Greeks) is a literal translation of “Gauss’s trick” to a visual proof

@TreeLuvBurdpu Жыл бұрын

Induction CAN NOT be our last resort because ALL DEDUCTION depends on prior induction.

@MathVisualProofs Жыл бұрын

I think the visual proof of stacks of squares should be first. Induction is too dry and requires that you already know a formula.

@TreeLuvBurdpu Жыл бұрын

@@MathVisualProofs I see what you're saying. It just sounds a little like you're saying that your proof doesn't depend on induction. That first there is proof and then induction depends on the proof, which is not how thought works.

@schweinmachtbree1013 Жыл бұрын

You're confusing philosophical induction and mathematical induction

@MathVisualProofs Жыл бұрын

@@schweinmachtbree1013 thanks!

@TreeLuvBurdpu Жыл бұрын

@@schweinmachtbree1013 mathematical induction also depends on philosophical induction

@line8748 Жыл бұрын

Induction is not a proof btw

@MathVisualProofs Жыл бұрын

Why not?

@line8748 Жыл бұрын

@@MathVisualProofs Because no matter how many times you find something to be "right", you'll never be sure that next time you do it it's going to be "right" again. That's why everything in physics is a "right" theory until proven wrong. In math instead, we define what is "right" and what isn't (e.g. 1+1=2). With induction you can't prove anything, because for example no matter how many times you see a stone fall on the ground, there's no way to determine for sure what would happen if you did the "experiment" once more. Physics is all about sensible guesses, which we assume to be laws of nature until proven wrong. Math is all about truths we define ourselves.