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

Alternative title: Michael having fun doing his hobby.

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.

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

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.

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.

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

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

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?

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

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 send this video to terrence howard and joe rogan

having ptsd flashbacks of doing proofs for exams

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.

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

Terrance howard enters the chat:

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

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

Terrence needs this in his life

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

*watches this instead of studying for math test*

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

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

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

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

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

Why does vsauce write his 8 like that?

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.

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

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.

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

Absolute breath of fresh air after listening to 3hours of Terrence Howard rambling on jre

*Y O W Z A S*

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

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

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

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

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

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

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

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

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!!!!"

7:31 I love how he underlines the numbers once for the odd numbers, and twice total for the even numbers. Almost poetic.

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

He protecc He attacc But most importantly, He bacc

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

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

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!

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.

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

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.

I'm pretty sure this is the video that started Terrance Howard's suspicion

2:31 he says the thing

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

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.

/Terrence Howard has entered the chat

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

Maybe it's irrational because of a traumatic experience

I Prove it like this Let √2 be rational i.e √2 can be expressed as a/b (were a and b are co-prime integers) √2 = a/b by squaring on both sides 2 = a^2/b^2 transposing b^2 to l.h.s 2.b^2 = a^2 therefore a^2 is divisible by 2 (if p^2 is divisible by a then p is also divisible by a) then a is also divisible by 2 now let a = 2c sub it in 2.b^2 = a^2 then we have 2.b^2 = (2c)^2 2.b^2 = 4.c^2 b^2 = (4.c^2)/2 b^2 = 2.c^2 therefore b^2 is divisible by 2 (if p^2 is divisible by a then p is also divisible by a) then b is also divisible by 2 now a and b have a common factor 2 this contradicts with the fact that a and b are co-prime integers therefore our assumption is wrong √2 is irrational Hence Proved.

I love knowing the proof for multiplying fractions. This was so cool to watch.

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

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

the solution of square root 2 in nearest fraction is 1871/1323 or 1970/1393 or 3124/2209 or 3363/2378 or 5234/3701 or 7105/5024 or 8119/5741 or 9611/6796 the solution of square root 2 is 25/18 + 1/42 + 1/660 = 19601/13860 = 1.4142136 1.4142136 = (2)^(1/2) the solution of square root 2 is 11/8 + 3/83 + 6/1955 = 1835819/1298120 = 1.4142136 1.4142136 = (2)^(1/2)

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

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

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

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 think we would all be a lot better at math if we learned why things were true rather than that they were true, really gives you an understanding for the mechanisms on the inside. great video

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

Hey Vsauce, Michael here. Vsauce is my main channel Or is it? 🎵🎵🎵🎵🎵

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

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

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

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

1x1=2

Someone send this to Terrence Howard

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

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!

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

Tell Terrence Howard.

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

“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”

Learning more info from KZfaq in 17 minutes than my school can provide in an entire lesson, math edition

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

Dong status: fully expanded.

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

Someone has to get Michael to draw his 8s another way, it’s driving me nuts

I love how the comments here are so calm and reasonable, meanwhile in vsauce it's pure chaos

the way Michael writes the number 8 is mildly infuriating

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

What is irrational is the way Michael draws the number 8

At about 16:00 he says "since an even number squared is even, then if b^2 is even then b must be even". That is using the implication in the wrong way.

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

When he proved it at the end, I was so excited that I pumped my fist and said, "Yes! We did it!"

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