Root 2 is Irrational from Isosceles Triangle (visual proof)

In this short, we use a famous argument by Tom Apostol to prove that the square root of two is irrational by infinite descent using a right isosceles triangle. We also go a bit further and show how this proof hints at the number theoretic construction of the convergents of the square root of two, which are the best rational approximations of root 2.
If you like this video, consider subscribing to the channel or consider buying me a coffee: www.buymeacoff.... Thanks!
For an alternate visual proof of this fact, see this video: • Visual irrationality p...
This animation is based on an argument due to Tom Apostol from issue 9 of the 2000 American Mathematical Monthly: doi.org/10.108...
To learn more about the convergents argument and the relationship between this proof and convergents, see this wonderful article by Doron Zeilberger:
sites.math.rut...
and here you can learn more about convergents:
en.wikipedia.o...
#irrationalnumbers #realnumbers​ #manim​ #math​ #mtbos​ ​ #animation​ #theorem​​ #visualproof​ #proof​ #iteachmath #mathematics #irrational #triangle #righttriangle #isoscelestriangle #proofbycontradiction #root2 #algebra #infinitedescent #numbertheory #convergents
To learn more about animating with manim, check out:
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[@ed.puckett]

There is a lot here to digest. Thank you!

[@MathVisualProofs]

👍

[@skvttlez1263]

Infinite descension of integers is impossible because 1 is the smallest positive integer.

[@error_6o6]

That’s literally the point of the proof

[@bmx666bmx666]

Wow, this is the simplest way to find the approximate value of sqrt(2). So easy to remember - start with 1 / 1, double the bottom and sum with the top = save on top 3, sum top and bottom - save at the bottom 2, got 3 / 2, etc ... .

[@MathVisualProofs]

😎

[@terdragontra8900]

I am fascinated that I’ve never seen this particular proof before, wow, pretty cool. In fact, a/b = sqrt(n) implies (nb - a)/(a - b) = sqrt(n), but these new integers are smaller than the originals only for n < 4… but at least it works for 3!

[@benjaminbailey3907]

Very nice. It's very interesting that you didn't need any divisibilty arguments.

[@guessundheit6494]

Just like Fibonacci numbers, there's and easy calculation for subsequent numbers. The difference is, the initial value for the numerator must start at 1, and the denominator start at 0. Numerators: 1 { 1x2+1 = 3 > 3x2+1 = 7 > 7x2+3 = 17 > 17x2+7 = 41 > 41x2+17 = 99 > etc. Denominators: 0 { 1x2+0 = 2 > 2x2+1 = 5 > 5x2+2 = 12 > 12x2+5 = 29 > 29x2+12 = 70 > etc.

[@sankojuprithvi]

Very genius and informative and spectacular video on mathematics and illustrations in this video very magnificient a I deeply congratulate the content maker of this video he is a spectacular mathematician

[@MathVisualProofs]

😎

[@alanthayer8797]

VISUALS Visuals visuals is Key! Thanks as usual fa da VISUALS

[@MathVisualProofs]

:)

[@SeanSkyhawk]

This also has an interesting implication: there is no number such that taking its square and adding that square to itself results in a square of a different integer.

[@BenWard29]

Except for 1, right?

[@majorproblem8796]

@@BenWard291+1=2 sqrt(2) is not an integer

[@landsgevaer]

What is interesting about that implication?

[@BenWard29]

@@majorproblem8796 Yeah you're right- I'm an idiot. I misread your comment.

[@cheeseeater4434]

@@BenWard29 no problem with that mate, owning up to your mistakes already puts you ahead of 90% of the online community

[@張洪鈞]

Simple is the best, thank you for proving that.

[@michaelgonzalez9058]

The isocoslis is 3°~3•1=1/2 which equals 0

[@cupatelj52]

But Terrence Howard told me that it is Rational... 🤣🤣🤣

[@Larsbutb4d]

what?

[@Larsbutb4d]

oh waut nvm

[@cupatelj52]

​@@Larsbutb4d This is a joke. There is a Hollywood actor, Terrence Howard, who recently on podcasts and other platforms, has been making ridiculous scientific claims, including one that the square root of 2 is a rational number. :)

[@samueldeandrade8535]

​@@cupatelj52 why tf are you invoking Terrence Howard ... Ffs, why humans need so much to demean other humans, using them as objects of jokes? You know what, yeah, he said 1×1=2 and √2 is rational. But ... I guess that considering a*b := 2ab those statements may be true? So ... not a big deal at all? He would just be changing the product ...

[@cupatelj52]

​@@samueldeandrade8535As if he was speaking about a topic that isn't as strict as mathematics. Of course, he should be seen as a joke for making such childish claims.

[@pelayomedina2174]

Congratulations, you just got a succesion of rational numbers that converges to an irrational number This is as if I say: The square root of 2 is 1.4, 1.41, 1.414, 1.4142,... Also is impressive because a and b can be any number so if you follow that rules you are going to approach square root of 2 Oh it is explained later :)

[@wompastompa3692]

7 × (2, 2, 3, 5, 9)

[@charliestein9350]

Very clever! I don't think I ever heard an argument like this before, but it is very sound. If I can repeat a process infinitely many times, and reduce the size of some variable in each iteration, the variable must eventually reach zero. Nice!

[@tarekmesto]

Great videos. what software do you use?

[@MathVisualProofs]

I use manimgl. That’s the python library developed by 3blue1brown

[@tarekmesto]

@@MathVisualProofs great thank you

[@PC_Simo]

I’m surprised that Mathologer hasn’t covered this particular proof, in his video of shrink proofs, or ”Visual Irrationals”. 😮

[@ignacioelia759]

1:15 we are assuming that those tangents are equal (which they are, no argument there), but I missed the proof. kzbin.info/www/bejne/qGalep2sa897ibM explains why the tangents from the same external point are of the same length, and since the arc is less than semicircular, we know there will be such a point, which in this case it is the rightmost vertex of the smaller triangle with bottom angles 45deg. 2:08 "infinite list of decreasing positive integer ratios". Yeah, but I missed the explanation there too. The assumption that those lengths are integers comes from the assumption that a and b are integers, so then the expressions "a - b" and "2b - a" will also be integers (positive integers, by construction) because the operations applied here will leave us in the same number set.

[@eonasjohn]

Thank you for the video.

[@MathVisualProofs]

Thanks for watching!

[@TupperWallace]

Neatly done!

[@MathVisualProofs]

Thanks!

[@DanielPoupko]

That is a very cool proof

[@MathVisualProofs]

👍😀

[@mauisstepsis5524]

How do I decide initial values (a0, b0) for initial values to approx sqrt(2)?

[@MathVisualProofs]

1/1 is the best approximation with denominator 1. Also the simplest solution to 2b^2-a^2=1. I should have mentioned it. :)

[@Woah9394]

A sequence that has a much better covergence is an=(an+2÷an)÷2 The 3rd term is already equal to 577÷408

[@MathVisualProofs]

I didn't mean (or say) that this sequence is the best convergent sequence to root 2. I said the convergents are the best rational approximations to root 2. So each number in this collection of rationals will always be the closes to root 2 compared to any other rational with same or smaller denominator.

[@landsgevaer]

Yet you get the same terms from the other sequence, just skipping intermediate steps (I think you double the index).

[@Woah9394]

@@MathVisualProofs ik I just wanted to add

[@orisphera]

I prefer considering the smallest such triangle

[@Smartas599]

Beautiful

[@MathVisualProofs]

Thank you

[@Aman_iitbh]

is that seq relates with pell equation with approximates root n

[@MathVisualProofs]

Yes. For sure !

[@CristianBaeza-rh7zq]

Haha, amazing! Simple and nice ❤❤❤

[@gonzus1966]

This is Pell's equation in disguise, right?

[@MathVisualProofs]

Yes. This is related to Pell’s equation for sure :)

[@tazepatates4805]

I didn't understand anything but this is fascinating

[@Evan-Delauney-Meyer]

Why do you say " where a and b are positive integers". I know it makes sense because you are talking about lengths but its not the definition of a rational number.(They can be negative)

[@letswait30days]

sqrt2 is clearly positive

[@sagarsaha935]

Sir I said that this channel has a conic section playlist ?

[@MathVisualProofs]

I don’t have many about conic sections.

[@sagarsaha935]

@@MathVisualProofs Sir it is very good to make video on conic section I am currently having a lot of trouble understanding conic sections....... please sir give some thought to this

[@brrrrug565]

amazing

[@MathVisualProofs]

Thanks!

[@IIScarletKingII]

nice

[@MathVisualProofs]

Thanks!

[@avijitpramanik5425]

Visualising is the best way to understand math Do you agree 👍

[@tired5925]

This video does not explain why it is a problem, if i have a triangle and take half of it then half of what is left then it will go forever, why is it a problem and a contradiction?. And even if it is, that just shows ur method creates a loop and doesnt arrive at an answer.

[@MathVisualProofs]

I note explicitly that shrinking the triangles would produce an infinite list of decreasing positive integers. This contradicts the well-ordering principle.

[@pythontron8710]

Proof by infinite descent

[@qwertek8413]

why is everyone saying positive integers instead of natural numbers

[@MathVisualProofs]

if you say positive integers, you can avoid the arguments about whether or not 0 is a natural number :)

[@qwertek8413]

@@MathVisualProofs yea, that was one of my guesses, thanks :)

[@aznstride4325]

Sorry, I think I believe Toward Herrence who says that it is rational

[@MisterSnail1234]

Easy: 0/0 = √2

[@johnkardier6327]

This "non-standard" proof might be the oldest one.

[@MathVisualProofs]

The citation is from 2000. But I guess maybe this idea was known before though I haven’t seen it written down before the Apostol article.

[@rivenoak]

@@MathVisualProofs afaik the "usual" numeral proof was made by Euclid ? we know he was busy and tinkered with pythagorean triangles back in ancient times, so perhaps the visual proof was known to him as well ? math lessons ~300 bc in alexandria must have been interesting times :D