A Proof That The Square Root of Two Is Irrational
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- čas přidán 6. 05. 2024
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/ tweetsauce
/ electricpants
Vsauce PO Box: PO Box 33168
L.A. CA 90033
**CREDITS**
Hosted by
Micorn Stevens
Edited by
Hannah Canetti
( / hannaynaycanaynay )
Curiosity Box VFX by
Eric Langlay
( / ericlanglay )
Music by Jake Chudnow and from AudioNetwork
( / jakechudnow )
(www.audionetwork.com)
**VSAUCE LINKS**
Vsauce1: / vsauce1
Vsauce2: / vsauce2
Vsauce3: / vsauce3 - Zábava
Michael: Or is it
2 million people: *Literaly creaming themselves*
Fits with your profile picture
OH GOD YES MICHAEL
Easily amused
That face you make when people overuse the word _"literally"_
ew
Rational
Ratio
I have never realized this and now am freaking out.
you made me think of other words, and rations, and now im dying
What about irratio?
An irrational number divided by an irrational number.
π/e
Never realized that either... I’m French, irrational numbers are called irrationnel, but ratios are called fraction......
you're not alone on this
Proof by contradition is like kryptonite for infinity-related questions, isn't it
I mean, proof by induction is more so suited for infinity questions
@@unconscious5630 what’s proof by induction? I just started discrete math and I don’t think I’ve learned that one yet
@@joeman1063 It's basically the domino effect as a method of proof: If you want to show that a statement that somehow depends on a natural number n is true for all natural numbers n, it's sufficient to first show that it is true for n=1 and then show that your statement being true for some n implies it being true for n+1. In other words: you show that the first domino piece falls, and then you show that if some domino falls, the next one will fall as well. Therefore all domino pieces fall and your statement is proven for all n. One typical example for proof by induction is the proof of the closed formula for summing the first n natural numbers provided by little Gauss. Generally a lot of statements involving finite sums lend themselves to proof by induction, for example if you want to show that laws like distributivity hold in general for a finite numbers of addends.
Contradiction and Induction method are both same logically. Here your n is a/b , so it holds for any a/b and also a/b + 1 . So by induction this contradiction holds for any a and b ratio.
Interesting thing is that,..can we prove existence of God with this method???
I was taught this proof in school yet was somehow still entertained enough to watch the whole video. There is my proof that Michael is an incredible host.
Same
same
same
The proof is like Star-Wars-once-you-see-The-Lord-Of-The-Rings. It's something that looked like "greatness" until you saw what *true* (Academy-Award-Winning) greatness actually looks like, and then you can never go back. Let's do it here now.
(1) The Fundamental Theorem Of Arithmetic applies to fractions, too, not just to integers. Every positive rational number has a *unique* factoring into a product of *integer* powers of primes.
(2) If q is rational, n is a positive integer, and qⁿ is an integer, then the prime factors of qⁿ all have *non-negative* multiples of n as their exponents. Therefore, the prime factors of q all have non-negative exponents. Therefore q is integer.
(3) Therefore, no n-root of any integer z > 0 is rational, unless it's an integer and z = qⁿ is the n-power of an integer.
(4) For square roots, that means √z is irrational, unless z = 1, 4, 9, 16, ... for cube roots ∛z is irrational, unless z = 1, 8, 27, 64, ... (or z = -1, -8, -27, -64, ... if you expand this to negative values) and so on.
Alternative title: Michael having fun doing his hobby.
Michael is alternate Albert Einstein
This stuff is all just so lovely! I’m happy he does this stuff
I had no idea you could prove irrationality. Proofs are a mathematician’s witchcraft.
He likes it , but I ...
or does he?
16:05 the music synchs perfectly with the sound of the pencil drawing on the paper
this is extremely satisfying
This was planned prove me wrong
"synchs"
Damn right X"D
16:06
Omg thats so cute
Literally had this as a proof by contradiction question in a maths test and wanted to curl up in a ball
i really love it when these math videos give me a slight shift in perspective about something that is incredibly common/natural to me-, like, reducing fractions
i know in my head that 4/6 is equal to 2/3, easy, and i know that 4/6 can be reduced into 2/3 without changing its value. if you were to ask me _why_ the value doesn't change, my answer would be something like "because the ratio in relation to the numbers stayed consistent, the proportions of 4 to 6 are equal to the proportions of 2 to 3."
but here-, michael just gives the simple answer of "since both 4 and 6 can be divided by 2, we can divide this fraction by the fraction 2/2. and since 2/2 is equal to one, this is equivalent to dividing by one, which will not change the value of our number."
like-, "huh, that *_is_* true-, it is just like dividing by 1, never thought of it like that"
it makes absolute sense and shows a different way to describe the same thing-, it's so cool to me.
Your approach is essentially the geometric proof. Suppose you have two similar right triangles, one with legs of 4 and 6, the other with legs of 2 and 3. For them to be similar, the angles must match, which means the tangent of the angles must match. Since tangent is opposite over adjacent, which is 4/6 for one and 2/3 for the other, those fractions must be the same value.
Michael would be an incredible teacher. Not even just for Maths, he seems to absolutely love learning and teaching
Downside is I dont think teachers get to choose what subject to teach, youtube gives him more freedom and we all get access
@@marialawal7449 they do lol why would they teach a subject they're bad at
Is he not already a teacher?
@@ichthyostegaxd3727 They're ✨forced✨, didn't you know?
@@rexon112 In which country?
The way you write "8" gives me nightmares.
edit 2 years later: You're free to draw the 8 any way you want, it's on me to not have the nightmares ^^ Also... please buy my indie game? I'm poor. Super Hiking League DX. Thank you.
Me too
@@mariomario-ih6mn Coincidentally, my real name is also Mario. So is my father's.
@@NesrocksGamingVideos My name is not Mario it's Alexander I named my channel after Super Mario bros
Ah cool :)
its mechanical drawing
Me in Xth class waiting for the statement " To the contrary, lets assume that √2 is rational. "😂😂
and to end with "this contradiction has arisen because of our wrong assumption that √2 is rational " 😂😂
fr 😂 😂
Indians joined the chat...
@@madhurmishracss im afghan
This was the first video I've seen on this channel, and I thought that was phenomenal! My favorite math class in college was number theory. I love this kind of stuff. Subscribed.
Doctor: What would you like to name your child?
Michael: I think i'll call it "c".
Nate Sullivan
I c what you did there
@@bukucinho i wanna B as cool as you
Filip but U and I can’t
Tor McLean at least you R continuing it like us
ichbin luis Y
As a former Indian 10th class student, I can confirm that our 1st chapter REAL NUMBERS is all about this video. All concepts very well explained.
yea me too but in our ch it i profed by saying that the result is co prime i was quite intrested to know how actually we condisedered the a=2c part .... never knew it was about even no
true
yeah bro
I remember watching this a couple years ago and not understanding a single thing. It feels good watching it today and understanding everything
"Negative... thirteen **exhales in disappointment** is odd"
-13 = -12 + 1 ¿
*Disappointed*
@@koushioni YOU MEAN NEGATIVE 14!!!
@@augdawg6170 -13 = 2 x -7 + 1 ¿
*Disappointed*
Fixed
Saw this as it was happening lol
@@koushioni that would be -11
Everybody gangsta til Michael hits us with that "OR IS IT?"
Thor Christensen xdxdxd
What im i doin here 😢
Yes michael, numbers are lines used by evolved hairy apes to count things, question answered now end the video- “But are we evolved apes? And are numbers really just meaningless lines? First we need to define ape, but what does define mean? First we need to define define so we know how to define apes.”
@@generichomosapien4666 but we need to know what is define before defining define. So we end up in a loop.
I dont know how to end this, I'm confused, choose one of the two below
1. But what is a loop? *VSauce music plays*
2. OR DO WE?
or are they?
That's why I love maths. I think it's really interesting how you play with numbers to solve/prove things.
Okay, this is the best CZcams Video I've ever seen. It's so simple, yet so beautiful and inspiring and I think this is, what makes it so good
*watches this instead of studying for math test*
same unfortunately, normal algebra is way more fun than linear algebra
Same, pre-calc test tomorrow
good luck, I bombed mine, hopefully you don't suffer the same fate
I just watched it after calculas II EXAM, this looks silly but reasonable🤔
I think you're mistaken... u don't need to study if you're watching VSauce
The ancient Pythagoreans would have literally killed you for this act of heresy. They really did not like even the idea of irrational numbers. They drowned the man who suggested that pi might not be some divine ratio like 22/7.
Robinson Kaspar oh yeah like wtaf was wrong with people in maths and sciences back then?
@@nabranestwistypuzzler7019 the pythagoreans were more of a cult that thought rational math was the explanation to everything than what would be considered a modern style group dedicated to math or science.
Gareth Baus Oh wow
@@garethbaus5471 a math cult, awesome
the guy that made this proof was pithagorean. he was killed/banished
I know it may seem weird but I have to really thank you bur specifically this video, this honestly helped me getting actually better at math because the first time I watched it it reminded me the deep connection between every mathematical concept and how important are the basics concepts.
Today I'm litterally one of the best people in math in my class and I think that this video totally did it part in this, so thank you.
This is for me the best video to introduce maths to literally everyone.
Let's prove that the square root of 2 is irrational...
BUT FIRST! What is a number?
...and how much does a number weigh?
And where is it in time
Can I eat it?
It's odd to have even numbers..
... and how does it smell in space ?
When he wrote the decimal number 0.8 and said YIKES i felt that
My linear algebra professor literally talks about matrices like they are sentients beings. Math teachers are kinda weird
yyyyyyOWza
0:13
Then proceeds to dissociate from space time
I wish I was on that kinda high
Damn these videos are so compelling to watch. Michael's enthusiasm and simple explanations are captivating.
I can't believe I watched the whole video. This was surprisingly very entertaining. I wish my math teachers had been like him
How come I’ve never been told “rational” just means “able to be expressed as a ratio”?
*I invite you to join the 200+ commenters in advising that I didn't pay attention in school or that ratio is in the word. Please, showcase your originality for us all to see.
Sameee
You haven't been told that? I pity your school.
@@danieldavid3766 There's many of us who haven't been told that... esp. if you're not in an english speaking country and rational is a word but ratio isn't
@@lucasroach7835 Just add a word meaning ratio to your language. There, problem solved.
@@That_Awesome_Guy1 who are you so wise in the ways of science
2:32 when u are not sure of ur answer on a math test
14:25 when you are not sure of an answer in an open book math quiz
So good😂
i had a math test in my school today
If √2 can't be the exponential of 2 can't be either.
Exactly.
I also love how the thing continues all the way to 2:51
always fun to rewatch these videos years later now i know more about mathematics and statistics and so on
Thanks for saying zero is an even number. This makes me happy
'Unless our teachers have been lying to us our entire lives'
"YoU wOn'T aLwAyS hAvE A cAlCuLaToR wItH yOu"
Here most of the job related exams don't allow calculator imoooo 😭😭😭😭
@@mohiburrahman8331 Practice maketh perfect
Oh yeah one of my teachers told me that so i carried a calculator in my pocket for five solid months straight. The teacher was not amused.
You carry a library containing more books than the one in Alexandria, and a calculator and a supercomputer in your pocket. Try going back in time to tell your teachers that😅
Oh wait, we don't carry a time machine in our pockets, YET
Well to be fair they didn’t lie, they just didn’t know
He protecc
He attacc
But most importantly,
He bacc
... on the wrong channel
Second comment
@@jtjs2890 why u gotta announce that
VN :)))
Though Kevin and Jake are still active
Great video! This gave me flashbacks to when i was first learning how to do a basic proof. I just know younger me would have just kept going back and forth proving the various variables were even and completely missing what i was even supposed to be doing.
Memories; Your video was fun to visit. It brought back my intro to my mathematics studies. Your presentation was well done. Thank You.
Did anybody notice at 00:13 he distorted himself when he said yikes lol
LOL
*"y i ḱ e̳͠ s҉ s̤̩̜̬̝͍͞ s̴̨̢̠̳̹̖͎̺ s҉̶̹̜̟̖̬̘̬̦͓̪̺̙ͅ ş̸̴҉̭͓̰̫͎͚ s̵̞̬̳͍̥͙̪͔̎͒̓ͧ͠ͅ s̢̞̥͓̗̩̠̙̩͇̥̰͕̬̟̃̐ͪ͛͗̚͠"*
yup
I did not lmao
I though it's just my eyes so I didn't replay that part until now. lmao
I wish someone would look at me the way Michael looks at the number 8.
Oooh look at the beauty
You’ll find them ☺️
Sorry, I cannot keep looking at you.
You are just too bright 😎
Lmfao
Some men love money.
Some love the smell of napalm in the morning.
But the truly scary ones love reduced fractions.
My favorite moment is when Michael finishes the proof. We all knew where this was going. Michael certainly knew where this was going. But that moment facial catharsis after drawing those final two underlines at 16:39 is the pure feeling of discover, understanding, and enlightenment.
I mean Aristotle knew where this was going, and for reminder he didn't even know that the Earth was spinning around the Sun
*Y O W Z A S*
What's that?
@@NodenGaming ???
christian Saldaña 11:22
Oooooohhhhhh...okay.
Яуза ))) A river in Moscow!
having ptsd flashbacks of doing proofs for exams
Haii
Music man make more music.
No, i was watching like 6 minutes of the video and then realized i was thinking about other things while listening michael speaking something faint in the background. But i was still watching the video. It feels amazing but its so sad that i zone off and moom fail
I'm currently doing that
Yep, I felt like I was back in college math classes. I remember doing the even/odd proofs.
You learn this in undergraduate linear algebra. It was very useful for me to have seen this video years before we covered this topic, as it really made it easy to get into.
Probably one of the best math teacher I have ever seen. And arguably the best on CZcams.Just Superb
Irrational? Like Michael’s decision to not upload on the main channel ever?
Ahaha
He spends months on making the main channels vids so they are as detailed as possible but on dong he can make quick vids also he definetly makes less money making those than other things he could be doing.
He has said why before
even if he uploaded a video of a compilation of his "hi vsauce, michael here", i would watch it
This is too damn true... The extro said Vsause tho so idk
He draws "8" like I'm back in primary school
Yaaa thought the same.. xD
That irritated me so much 😄
@@williammay451 Or did you?
@@steelforge8577 I mean what is writing
(vsause music starts)
Beautiful! Understood more than ever in classes, in school!
This is literally 8th-9th grade exact maths, he is teaching in such a way that its part of our dopamine dose. Amazing
The way he subtly integrates easy to understand definitions for complex and simple math at the same time is incredible and would make it very easy for children to learn if teachers did math like this.
sad life
the difference is teachers teach multiple children all at once. Here michael is almost talking to you one on one and i dont think teachers get paied enough to have a full conversation w/ all their kids about the subject.
@@skrimish7145 except he's not asking you questions or anything that is impossible to do with multiple people, the teacher could write all this on a board (or paint if they're online), and then ask questions after to verify.
@@skrimish7145 a youtube video is quite literally the polar opposite of a one on one conversation. This video could be a conversation with everyone in the world at the same time.
@@henry497 I mean, A teacher cannot check all student at the same time.
Binomial Multiplication: *appears*
Me: So we meet again...
how did u americans not learn that (x+y)² = x²+2xy+y²
@@squibble311 No, I am not American, and yes, I did learn that.
Meanwhile i know that (x+y)^5=x^5+(x^4)(5y)+10(x^3)(y^2)+10(x^2)(y^3)+(5x)(y^4)+y^5
sini harshan we did learn it. It’s not hard
@@jacksonbrewer2380 idk about western education (i heard it is slow) but asians learn basic algebra at 5th or 6th grade
man i needed that proof for my sophomore exams but did not expect vsauce to have a video on it
This is pretty cool. I'm currently in an introductory abstract math course and I'm learning almost exactly these things.
I have seen this proof maybe 50 times now, but no one gets as hyped as Michael. I love it! "OH MY GOODNESS GRACIOUS! IT'S EVEN!!!!"
2:31 he says the thing
What even is that pfp
ROLL THE 🎶
it scares me
I unironically want to kiss Cirno's ocks on your profile picture
Or did he?
I loved your video. I hadn’t felt so much emotion since abstract algebra. Keep up the great work, man.🎉🎉
This has and will forever be my favorite proof. Simple, elegant, and easy.
Maybe it's irrational because of a traumatic experience
@@owltaku5757 xD
It saw 7 8 9
What if the number 2 identifies as a 3?
Me: Doesn’t like math
Also me: watches man do math for 17 minutes
Can relate
I mean, Michael somehow teaches way better than my math teacher. Don't know about you guys.
@@williammay451 same
It's not that you don't like math, it's that you have been taught to fear math by the school system and bad teachers. Math by itself can be fun when it's not connected to pressure.
I'm your 666th like, you're now the devil
Its been more than 10 years since I left my school, and this video made me understand why multiply becomes divide on the other side of the equation and vice versa. I was never told why just taught what.
Learning more info from CZcams in 17 minutes than my school can provide in an entire lesson, math edition
15:40 "Oh my goodness gracious!" Lol. Getting so excited for an even number
It's like he can't even
@@aleksandarvlasev4030 LOL
+Alexander Vlasev u made my day 😂
Michael: writes 8 as two circles instead of a single interlocked line
Everyone, while going through content aware scale: *ＹＩＫＥＳ*
8 as two circles are quite.. perfect to look at. You can replay the beginning to see that. Granted.. it can take a bit longer to move your wrist and draw the other circle than to do it in a continuous line :P
@@user-fs3ch6cc3j yeah, everytime i write an 8 there will always be a point coming out of it, that trigger me everytime, and i do it with a single line.
yikes
I write my 8's as 2 circles.
Just because it's not the most common way of doing so doesn't mean there's anything inherently wrong with it.
What does it mean to be going through content aware scale?
You explained this better in 17 minutes than my math teacher in 45 minutes did
He is indeed one the of the best maths teacher I found while scorilling.
God if every math teacher was like this everyone would have their Master's
yeah like in 40 years
@@mikkihintikka7273 lol
Yep , the video was just him explaining basic algebra most of the asians already learnt in 6th or 7th grade
Idk about western education tho
@@padmanabanraghavendran3570 yep and they learn it in tenth of the time
@@padmanabanraghavendran3570 western education is optional beyond algebra and some geometry. Highschool usually has you take algebra 1 and 2, geometry, and trigonometry or pre calculus. After that, in college, a lot of degrees only require like... Calc 1 which is pretty easy, you learn derivatives, optimization, integrating, implicit derivatives, and stuff like that mostly.
In some majors you have to take more, like I am a CS major and i have to take calc I, II, III, and then either calc IV or Matrix Algebra. If i take two extra math courses which i can easily do over summer or something, I will actually be able to get my minor in math. All engineering majors are about the same in that regard
Hey Vsauce, Michael here. Vsauce is my main channel
Or is it? 🎵🎵🎵🎵🎵
Hahaha I love this comment
This is the video I would like to make myself. But, it could never be as good as this, so now I’m not going to bother!
One minor issue. The statement "a^2 is even implies a is even" doesn't follow from "a even implies a^2 even". It follows from the contrapositive of the statement "a odd implies a^2 odd".
In this case it seems trivial but the phrasing comes off as "A implies B is true because B implies A is true", which in general is NOT logically true. Converses are not equivalent statements, but contrapositives are.
Excellent point
Can we rename this channel to Vsauce4 - Mainly DONG and sometimes Original Vsauce
I love this kind of stuff and its actually the main reason i am subbed. I miss the original Vsauce videos and he hasn't uploaded anything but mindfeild for so long. This kindof stuff should go of Vsauce 1
Or just new vsauce 1 then rename vsauce 1 to vsauce premium
This is Vsauce4 - Ponto em Comum (Made in Brazil):
czcams.com/video/0z2XzdqAMVQ/video.html
It is an amazing documentary about vaccines.
"Olá Criaturas da Internet, aqui quem fala é Davi Calazans."
SAP: Hey Vsauce, Davi Calazans here.
i wish my math teachers were as engaging as you, god damn
Math teachers are.. but they have to teach you both how to do, and why to do.... and a lot more than just one thing. Most of class would not be able to follow this anyway.
it's hard to get your students' eyes to zoom in on your face at comedic moments
Ryan G-P underrated comment
0:12
@@Tor.G why wouldn't they. While I agree there's a plan the theachers have to follow this is basic math, most youngsters out there would understand pretty quickly
I had no idea what this man was talking about but somehow i was entertained enough to watch the entire video
This video is so comprehensible and deeply inspiring, thank you for such a wonderful video.
Michael: * says something *
me : yes yes very good * stares intensely at t-shirt *
Hello reyan
close but no cigar ;-;
anyway hi ¯\_(ツ)_/¯
Did I ask for this?
Absolutely not
Did I need this?
Absolutely not
Did I love this?
Absolutely
Not
Did i need to tell this: Absolutely not
Am i telling this: Absolutely
I am your 111th like
i needed this
@Modar Bilal01 but first let us see, what is need...
I needed this
Better energy than my college prof. Perfect example of proof by contradiction
When he proved it at the end, I was so excited that I pumped my fist and said, "Yes! We did it!"
Wait, Michael writes 8 as one circle on top of another.
Let him live in peace with his circle 8.
this reminds me of a joke i once read:
two zeroes are trekking though the desert when they spot an eight under a palm; one zero says to the other: i can't understand how they can do it in this terrible heat.
THERE ARE DOZENS OF US
Normies
Y O O O O W Z E R S
When you wake up on Monday and remember that you didn't do ur homework
0:13
haha
underrated
I didn’t notice how he wiggled XD
Dan Slater woah I just noticed that
Oww....
That was actually pretty beautiful
Now if only my math instructors in school could have explained things as thoroughly and with such enthusiasm, instead of just glossing over everything.
I too enjoy binomial multiplication.
as if theres any other kind
it's all about polynomial multiplication
Think about remarkable products...
Ah a fellow intellectual, I see.
I do not. Especially when it involves exponets and 2 variables.
14:07 "Oh my goodness gracious"
15:40 "Oh my goodness gracious"
Look at that. LOOK AT THAT
look at what we have now!
Thanks
I'm on a physics degree but somehow Vsauce michael is the first person to tell me the rational ratio name thing.
This is my favorite video put out by Michael and I have no idea why.
Dong status: fully expanded.
Channel name “Dong”: fully retracted.
D!NG*
Mandela Effect at it's finest
Add 852 ??
@@remlo5482 r/woosh
Micheal: Do you know what an even and an odd number are?
Me: I do!
Micheal: ...or do you?
Me: 😦😦😦
Alex Baldwin how did you spell it wrong twice
@@sickomode2761
Micheal? That's how autocorrect tells me to spell it. :(
Alex Baldwin Can you turn off autocorrect?
Spelled 'Michael'
It's Mikál
I just finished my chapter real numbers and learn about proving Irrational and CZcams recommendations this to me
I find it really beautiful that the sqrt(2) is the ratio of 2 even numbers. Numbers so even that their ratio is irrational.
Its really fitting for 2, the most quintessentially even number, that at its root is numbers so even our minds cannot comprehend it, in a way.
sqrt(2) cannot be expressed as the ratio of two even numbers
sorry to burst your bubble man
root 2 cant be expressed this way
it was a proof by contradiction
root 2 is assumed to be rational and set as a fraction in its lowest form
on solving the fraction we figure out it can still be dividing lower
this contradicts the statement that its in its lowest form
the contradiction basically tells ya that root2 being a fraction aint a thing and stuffs gonna break if you try to make it so
Sorry if I'm dumb but I don't really get it. For me the proof Michael shows at 5:55 only proves that an even number squared always produces an even number, NOT that the square root of any even number is also an even number. For example the square root of 6 isn't an even number, since by definition an even number has to be an integer (not a fraction). And this kinda ruins the whole proof at the end in my mind. Pls help me out. Edit: Ok I finally get it, for those that are still confused I hope this clears it out (thanks to FlamingJM for making me understand it fully):
The proofs at 5:55 indeed don't prove that the square root of any even number is also an even number, like I said. They just prove that all even integers squared produce even numbers, and all odd integers squared produce odd numbers. This means that IF the square root of an even number is an integer, then it must itself also be an even integer, (it can't be an odd integer since all odd integers squared produce odd numbers).
In the final proof at the end, we know that a and b are integers (because numerators and denominators in a fraction are always integers after all). Therefore since by squaring either of them you get an even number, they themselves must also be even numbers.
All of this means that if the square root of an even number is not an integer, then it must be an irrational number. Apparently this applies to odd numbers too. So the square roots of not just 2, but all other non-square numbers, are also irrational numbers. I only learned this now. Hope this helps. And as always, thanks for watching!
the square root of 6 is also an irrational number, just like the square root of two, so you've basically just discovered the same contradiction that Michael uses to prove that the square root of two is irrational
I thought we all knew that most even numbers have irrational square roots 🤷🏾♂️
Remember that he defined a and b to be integers. This means that a² and b² are squares of integers. While it is true that sqrt(a) is not necessarily an even integer if a is an even integer, we know that a is an integer, and we're squaring it to get a². So if a² is an even integer, we know that a (the integer that was squared to get a²) is also an even integer.
He doesn't prove that the square root of an even number is even, but rather the square root of c^2 (where c represents an even number) is even, which is common sense: squaring and square root-ing cancel reach other out and oh my God you have over a million subs
Another way to think about it is that, since he proved that all odd numbers squared are odd, and a² is even, then a is definitely not odd (he already defined a to be an integer) thus a is even.
This is kind of a fusion of numberfile and vsauce
And for some reason it is on DONG
4 years later and all I can say is I miss this channel and the good old VSauce...
I used to think the square root of 2 was just another number, but it turns out it's the sneakiest number of them all!
Michael looking at 8 : *ahegao faces*
Michael looking at 0.8: *incoherent screaming*
Me: I already know this proof, it’s pretty basic, I’ll just move on
...
*clicks anyway*
Yup. I had to learn it when graduated from high school 3 years ago. Still nice to refresh your brain with some lovely smiling 'or is it' Michael.
I had to learn it in middle school...
@@can.of.beans101 It seems I did, good thing you remembered to write one for me. Thanks champ :)
@@stephendonovan9084 that comeback😳
OR IS IT BASIC?
Learned this in my discrete math class my last semester. Had to prove a lot of things like this such as there are infinitely many primes. That one is similar to this problem
And this proof works for the square root of any non perfect square numbers
the way Michael writes the number 8 is mildly infuriating
Ethanagor S-D that’s how I write 8s
@@SuperTurtle0 youre a bad person
Alexander Geppert not my fault I’ve always written them that way
@@SuperTurtle0 style, eh?
But it looks much better
Let’s just take a moment to appreciate how straight all of his lines are.
Those things were straighter than I am!
But that’s not saying much.
@@oldmanjeffrey ayo
@@oldmanjeffrey 😏😏 okay daddy
most brilliant video of all on vsauce channel
There's an oversight in this proof. The fact that an even number squared is also even does not mean that the root of an even number is necessarily squared. This needs further proof, if we are to remain rigorous.
Yes, however, in this proof, a and b are defined to be integers in the initial assumption
“Let’s prove that the square root of 2 is irrational”
“Yay!”
“But first, we have to reteach all of the math you learned in 1st grade”
“Yay?”
In 1st grade, they teach you how to do arithmetic. In this video, he teaches you why the arithmetic works. I guarantee you your teacher was not proving the arithmetic rules in 1st grade.
Wait you did binomial multiplication in first grade
isn’t numbers just lines made by some evolved hairy apes used to count things so the mammals look cool?
yang ye also it's used for some other stuff somehow idk tho