A Proof That The Square Root of Two Is Irrational 

Michael: Or is it 2 million people: *Literaly creaming themselves*

16:05 the music synchs perfectly with the sound of the pencil drawing on the paper this is extremely satisfying

Proof by contradition is like kryptonite for infinity-related questions, isn't it

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.

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.

Rational Ratio I have never realized this and now am freaking out.

Literally had this as a proof by contradiction question in a maths test and wanted to curl up in a ball

Michael... Can you please pay Terrence Howard a visit.

Alternative title: Michael having fun doing his hobby.

having ptsd flashbacks of doing proofs for exams

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.

Who else is here after watching the Terrence Howard video?

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.

Michael would be an incredible teacher. Not even just for Maths, he seems to absolutely love learning and teaching

Terrence needs this in his life

This is a beautiful introduction about how mathematics is done. Exploring to find the proof.

"Negative... thirteen **exhales in disappointment** is odd"

Doctor: What would you like to name your child? Michael: I think i'll call it "c".

someone send this video to terrence howard and joe rogan

Terrance howard enters the chat:

Irrational? Like Michael’s decision to not upload on the main channel ever?

*watches this instead of studying for math test*

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.

Someone should send this video over to Terrance Howard I think he needs it.

'Unless our teachers have been lying to us our entire lives' "YoU wOn'T aLwAyS hAvE A cAlCuLaToR wItH yOu"

Let's prove that the square root of 2 is irrational... BUT FIRST! What is a number?

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.

I remember watching this a couple years ago and not understanding a single thing. It feels good watching it today and understanding everything

Everybody gangsta til Michael hits us with that "OR IS IT?"

Me in Xth class waiting for the statement " To the contrary, lets assume that √2 is rational. "😂😂

Okay, this is the best RU-vid Video I've ever seen. It's so simple, yet so beautiful and inspiring and I think this is, what makes it so good

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.

When he wrote the decimal number 0.8 and said YIKES i felt that

I have never seen anyone write an 8 like that, but it is much neater looking that my sideways infinity 8.

learned this one in college, crazy what Euclid goes on to prove using irrational numbers, the icosahedron construction proof at the end is mindblowing

Did anybody notice at 00:13 he distorted himself when he said yikes lol

He protecc He attacc But most importantly, He bacc

Damn these videos are so compelling to watch. Michael's enthusiasm and simple explanations are captivating.

This has and will forever be my favorite proof. Simple, elegant, and easy.

*Y O W Z A S*

He draws "8" like I'm back in primary school

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 pretty cool. I'm currently in an introductory abstract math course and I'm learning almost exactly these things.

2:31 he says the thing

I wish someone would look at me the way Michael looks at the number 8.

Why does vsauce write his 8 like that?

That's why I love maths. I think it's really interesting how you play with numbers to solve/prove things.

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!

2:32 when u are not sure of ur answer on a math test

I can't believe I watched the whole video. This was surprisingly very entertaining. I wish my math teachers had been like him

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.

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.

Me: Doesn’t like math Also me: watches man do math for 17 minutes

Memories; Your video was fun to visit. It brought back my intro to my mathematics studies. Your presentation was well done. Thank You.

Probably one of the best math teacher I have ever seen. And arguably the best on RU-vid.Just Superb

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!!!!"

God if every math teacher was like this everyone would have their Master's

I watched this a year ago, i didn't understand and today in math class, we studied rational numbers and when i saw this video, i immediately understood what was in the video. It feels so awesome to finally see something you see in vsauce like 4 years before you study in school

always fun to rewatch these videos years later now i know more about mathematics and statistics and so on

15:40 "Oh my goodness gracious!" Lol. Getting so excited for an even number

i wish my math teachers were as engaging as you, god damn

This video is so comprehensible and deeply inspiring, thank you for such a wonderful video.

I loved your video. I hadn’t felt so much emotion since abstract algebra. Keep up the great work, man.🎉🎉

Did I ask for this? Absolutely not Did I need this? Absolutely not Did I love this? Absolutely

Michael: * says something * me : yes yes very good * stares intensely at t-shirt *

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.

This is literally 8th-9th grade exact maths, he is teaching in such a way that its part of our dopamine dose. Amazing

Michael: writes 8 as two circles instead of a single interlocked line Everyone, while going through content aware scale: *ＹＩＫＥＳ*

Maybe it's irrational because of a traumatic experience

Never seen someone so excited about the definition of even and odd 😂

Vectors, scalars and quantities are like putting a ruler against the universe and expecting a graph.

Binomial Multiplication: *appears* Me: So we meet again...

This is kind of a fusion of numberfile and vsauce And for some reason it is on DONG

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!

Thanks for saying zero is an even number. This makes me happy

the way Michael writes the number 8 is mildly infuriating

“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”

He is indeed one the of the best maths teacher I found while scorilling.

That was actually pretty beautiful

Dong status: fully expanded.

Wait, Michael writes 8 as one circle on top of another.

I feel like math really gets Michael going. I could totally feel what I interpret to be his normal everyday cadence and voice slip out a few times in this vid, as opposed to his more calculated, mysterious, and scripted youtube voice. Perhaps this has been seen in other videos before and I've missed it, but this felt very humanized from the perspective of someone showing you math from a kitchen table sort of vibe.

"Or is it?" always gets me fr

14:07 "Oh my goodness gracious" 15:40 "Oh my goodness gracious"

Michael looking at 8 : *ahegao faces* Michael looking at 0.8: *incoherent screaming*

Very neatly done.

This is my favorite video put out by Michael and I have no idea why.

Micheal: Do you know what an even and an odd number are? Me: I do! Micheal: ...or do you? Me: 😦😦😦

" 0:11 but what about a number like 0.8? Y I K E S " I don't know why but that made me laugh a lot

most brilliant video of all on vsauce channel

When I come across the term "rational numbers", I never imagined "ratio" comes to mind. I simply thought they are numbers that "makes sense". Very educational video indeed. Especially the concept of proofing via contradiction. Please make a video to proof that 1+1=2

Let’s just take a moment to appreciate how straight all of his lines are.

I too enjoy binomial multiplication.

Vsauce saying "yikes" with the distortion filter just made me giggle

And this proof works for the square root of any non perfect square numbers

When you wake up on Monday and remember that you didn't do ur homework 0:13

Can we rename this channel to Vsauce4 - Mainly DONG and sometimes Original Vsauce

Some men love money. Some love the smell of napalm in the morning. But the truly scary ones love reduced fractions.

This helped me a lot

Vsauce: "atleast it isn't irrational" Me: "oh thank god" Vsauce: "or is it...?" Me: OH MY GOOOOD NOOOO