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.
this was my first hw assignment in math class in college. solving it made me taste the tip of the iceberg of math. a start of a life long love affair with the subject.
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.
@@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.
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.
@@nexor7809 Seriously? This is so elementary that this is going to be used in your everyday life, the use of the concept of fraction is so ubiquitous to explain quantities in comparison to other quantities as example. If you don't understand that fractions can be reduced, you are going to be easily fooled by people intentionally misleading you.
Wait, isn't this how algebra is just taught by default everywhere? I mean I have been thought this approach many times by different teachers. I don't mean the specific example of reducing fractions, but more in general about how we do algebra. It's always some form of adding/taking away a number from bith sides of an equation which is okay to do, because if you do the same operation on both sides they cancel out so you are adding 0. Same with multiplication/division except you are multiplying with 1 so it stays okay to do. I mean we were always shown why an operation, approach ir trick is okay to do, why it works
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
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.
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
Terrence is where he is because he refuses to listen to people. He doesn't care. He's your typical slightly-smarter-than-average person who wishes he was the smartest person in the world, but the only way for that to be true is for everyone smarter than him to be wrong. His goal isn't to find useful truths.
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.
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.
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
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.
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.
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?
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
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 never knew there were 2 different 8's until this video, a sideways infinity 8, and a snowman 8. His snowman 8 is much neater than my sideways infinity 8 so I may have to try adopting the snowman.
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
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.
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
@@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.
@@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
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
This is Vsauce4 - Ponto em Comum (Made in Brazil): ua-cam.com/video/0z2XzdqAMVQ/v-deo.html It is an amazing documentary about vaccines. "Olá Criaturas da Internet, aqui quem fala é Davi Calazans." SAP: Hey Vsauce, Davi Calazans here.
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.
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.
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.
@@Zephy9273 Asians trying not to brag about their school systems (obviously making 6 year olds repeat the multiplication table 1000 times a day is a good way to teach, great work Asia)
@@bighillraft i am indian and literally every video of a hard problem i see some IIT dude is like "smh i did this problem when i was 11 give something harder", it annoys us too
that proof is the brainchild of Euklid and thus ~2300 years old. means the stuff is ancient and fortunately basic math. you can argue still if you should be able to think about the proof yourself or if the knowledge of "squareroot 2 is irrational" is enough on its own .
I am 37 and I just understood what rational and irrational means. As a kid I always thought that rational numbers are which we can comprehend or wich numbers sort of make sense and irrationals are some kind of alien numbers. Jesus... RATIOnal... it makes so much sense! Thank you for this eyeopening lesson!
I think that it would be really cool if you guys made a video talking about a particular denominator. Im 16 and a few years ago, I discovered something during a test. Its a beautiful pattern. If you take any number and dived it by 11, its the numerator times 9 in decimal form. For example: 1/11 = .090909 etc. 2/11 = .18181818 etc. Or even 7/11 = .63636363 etc. As if that pattern wasnt cool enough, it works the other way around. Numerator over 9 equals numerator times 11 in decimal form repeating. For example: 1/9 = .11111111 etc. Or 4/9 = 4444444 etc. I love this and would like you to explore it further. This is my favorite thing about math and its something that I haven't seen anyone talk about. There has to be a reason for this. For me, figure it out lol
First step is to understand how long division works. Take 1/11 for example. 1. 11 goes into 1 0 times, so the first digit is 0 2. Next, move over one decimal place and multiply the numerator by 10 to get 10/11. Still, 11 goes into that 0 times so we have 0.0 3. Next, you get 100/11. 11 goes into 100 9 times leaving a remainder of 1 and our number is 0.09 4. next you move over and you get 10/11. That's familiar right? We just repeat the step 2 and go on from there leading to an infinite loop of 09 so the result is 0.090909090909... If we did exactly the same thing except instead of moving one decimal over each time, we moved two decimals over, it would look like this 1. 11 goes into 1 0 times so we get 0 2. 11 goes into 100 9 times so we get 0.09 with a remainder of 1 3. repeat step 2 ad infinitum If we make the numerator 2, we can do the same thing again except lets just skip step 1 and assume the first digit is 0 1. 11 goes into 200 18 times with a remainder of 2 2. repeat again and again to get 0.181818181818... We can now generalize it by seeing that since with 1, if we multiply the whole operation by a number "x", you would end up with x*100/11 inevitably and since 11 goes into 100 9 times, it'll go into x*100 x*9 times and the remainder will always be a since the initial remainder is 1 and 1*x is just x. Now, why does this happen with 9? lets do a similar thing with 9 100/9 is 11 with a remainder 1. The remainder is once again the same as the original number, so it will repeat continuously and the same method applies to being able to multiply it so we have x/9 = 0.xxxxxxx... given that x is a single digit number. You also didn't mention another number that has a similar pattern which is with 3. 100/3 = 33 with remainder 1 thus x/3 is 0. x*3 repeating for any number between 0 and 3. So, what is the relationship between 9 and 11 or 33 and 3? They are all factors of 99 because 100-99=1 which is the remainder we need to get this relationship. In fact, you can get this relationship by taking any factors of any number that is equal to 10^y-1. For example, 999 has factors 37 and 27. Lets test it by doing the long division of 1/27 1000/27 gives us 37 and a remainder of 1 so the result is 0.037037037037037... So any number x/27 would be x*37 repeating every 3 digits therefore, 11/27 = 0.407407407407... because 11*37=407 Advanced notes: If we were to try to generalize one of these to any integer numerator, not just ones below the denominator, we could get a formula that still works. Lets do 11 for example, it would look like x/11 = Σ x*9*10^(-2n) so you could have a number like 15/11 which you could figure out by 9*15=135 so the result would be 1.35 + 0.0135 + 0.000135 + ... resulting in 1.363636363636363636... What happens if you do this for something like 11/11. 11/11=1 right? So it breaks the rule right? Actually no. 11/11 using this method means 11*9=99, therefore, 11/11=0.99999...=1 If you are skeptical about this, there are plenty of interesting videos explaining why 0.999... = 1.
I never realized why this proves anything, until now: if you can keep dividing a and b forever by 2 and still get an integer, then they must be infinite.
@@mahimagrawal825 We know that a is an integer, we made it so. We know that a^2 is an even integer. It follows that a is an even integer. Yiu are WRONG.
A minor correction to my original comment: a and b could both be zero, but it's not defined in elementary algebra. I think it's obvious, but I'm a poindexter in these things😁
@@mahimagrawal825 I'm sorry but the former statement is correct. It does not imply however that the "root of an even number is even". The square of a number is literally the opposite of a square root of a number. [(x^2 is even) ⇒ (x is even)] does not imply [(x is even) ⇒ (sqrt{x} is even)] Note: the '⇒' sign means "implies" The first statement was already proven in the video. The statement "if x is odd then x^2 is odd" proves this statement because the contrapositive of a proposition is always equivalent in truth value of the original. and if x is not odd then it should pretty much be even, granted that x is assumed to be integer which is the case in the video. Your second statement does not hold absolute. EDIT: at 15:58 vsauce essentially said that if b is even implies that b^2 is even then it means b is even.... An error in his part....
For a beginner that wants to learn mathematics, you have helped me a lot to understand different theorems and how proofs work :) simple and well defined.
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.
I am 37. None of this has any bearing on my life whatsoever. I will almost certainly never need this knowledge, however I cannot resist a video of Michael explaining stuff to me. So, 17 minutes of my life gone and now I know it. I'll comment again if it ever comes up.
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
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?
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
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
this was my first hw assignment in math class in college. solving it made me taste the tip of the iceberg of math. a start of a life long love affair with the subject.
666 likes hahahaha
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
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?
@Jul W In which country?
@Jul W this is the dumbest thing I've seen in 2 years
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 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.
so why would i ever need that and why was it taught in schools again?
@@nexor7809 Seriously? This is so elementary that this is going to be used in your everyday life, the use of the concept of fraction is so ubiquitous to explain quantities in comparison to other quantities as example. If you don't understand that fractions can be reduced, you are going to be easily fooled by people intentionally misleading you.
@@manghariz2211 no
Wait, isn't this how algebra is just taught by default everywhere? I mean I have been thought this approach many times by different teachers. I don't mean the specific example of reducing fractions, but more in general about how we do algebra. It's always some form of adding/taking away a number from bith sides of an equation which is okay to do, because if you do the same operation on both sides they cancel out so you are adding 0. Same with multiplication/division except you are multiplying with 1 so it stays okay to do. I mean we were always shown why an operation, approach ir trick is okay to do, why it works
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.
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
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.
Michael... Can you please pay Terrence Howard a visit.
Terrence is where he is because he refuses to listen to people. He doesn't care. He's your typical slightly-smarter-than-average person who wishes he was the smartest person in the world, but the only way for that to be true is for everyone smarter than him to be wrong. His goal isn't to find useful truths.
And today my friends we learned that... Binomial multiplication is Michael's favorite
Hs Sm wait until he figures out Pascal’s triangle in polynomial multiplication
(a+b)^2=a^2+2ab+b^2 ... ?
but he said theyre his fav in a vsauce video
What I see: (a+b)^2 = a^2 + b^2
What others see : (a+b)^2=a^2+2ab+b^2
Sardorbek Omonkulov
If you say that (a+b)^2=a^2+b^2, then 2^2 is (1+1)^2, so 2^2 is 2
"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
*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
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...
bro how was ur exam?
literally scrolled down to the comments to find this lmaoo
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
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
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.
@Lucas Roach 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
@@omeraydindev congratulations
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
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.
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?
*Y O W Z A S*
What's that?
@@NodenGaming ???
christian Saldaña 11:22
Oooooohhhhhh...okay.
Яуза ))) A river in Moscow!
learned this one in college, crazy what Euclid goes on to prove using irrational numbers, the icosahedron construction proof at the end is mindblowing
'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
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
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
Who else is here after watching the Terrence Howard video?
We are being watched and listened to. 😂
Me 😂
Me 😂
Pretty much
Hahahaha that’s what made me watch this
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
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)
I never knew there were 2 different 8's until this video, a sideways infinity 8, and a snowman 8. His snowman 8 is much neater than my sideways infinity 8 so I may have to try adopting the snowman.
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 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
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
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
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!!!!"
This is a beautiful introduction about how mathematics is done. Exploring to find the proof.
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
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
@@春樹-e1b 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?
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?
Thanks for saying zero is an even number. This makes me happy
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???
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?
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
That's why I love maths. I think it's really interesting how you play with numbers to solve/prove things.
This is kind of a fusion of numberfile and vsauce
And for some reason it is on DONG
Hey Vsauce, Michael here. Vsauce is my main channel
Or is it? 🎵🎵🎵🎵🎵
Hahaha I love this comment
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 😂
Terrence needs this in his life
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):
ua-cam.com/video/0z2XzdqAMVQ/v-deo.html
It is an amazing documentary about vaccines.
"Olá Criaturas da Internet, aqui quem fala é Davi Calazans."
SAP: Hey Vsauce, Davi Calazans here.
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
Damn these videos are so compelling to watch. Michael's enthusiasm and simple explanations are captivating.
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
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
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?
Someone should send this video over to Terrance Howard I think he needs it.
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....
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
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.
Dong status: fully expanded.
Channel name “Dong”: fully retracted.
D!NG*
Mandela Effect at it's finest
Add 852 ??
@@remlo5482 r/woosh
“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
-68 , we were on the verge of greatness
We were this close
Nah
-138÷2=-69
Nice
It sounded like Dr Seuss
This being a star wars quote I assume verge of 66?
Yes
Terrance Howard watched this video and is claiming he broke science and solved the universe
Michael: * says something *
me : yes yes very good * stares intensely at t-shirt *
Hello reyan
close but no cigar ;-;
anyway hi ¯\_(ツ)_/¯
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
Michael looking at 8 : *ahegao faces*
Michael looking at 0.8: *incoherent screaming*
7:31 I love how he underlines the numbers once for the odd numbers, and twice total for the even numbers. Almost poetic.
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 find the lack of brown paper confusing
👍👍🏼👍
go up
@@anishmaharjank please don't tell me I was the only one to get it
lots of ppl gets it, lots more dont
Sorry, but us numberphile fans prefer brown paper, but srsly, who gives a damn
So veritasium is doing vsauce and vsauce is doing numberphile !!
Epic threesome.
Which means math is the ultimate king :)
( ͡° ͜ʖ ͡°)
This is a reality that I'm okay with.
The circle will be complete when numberphile does 3B1B, 3B1B does Mathologer, and Mathologer does Veritasium.
Why does vsauce write his 8 like that?
Don’t question the sauce.
Literally had this as a proof by contradiction question in a maths test and wanted to curl up in a ball
Vsauce: "atleast it isn't irrational"
Me: "oh thank god"
Vsauce: "or is it...?"
Me: OH MY GOOOOD NOOOO
Lol this was in our maths course class 10 of CBSE in India
I’m surprised rest of the world isn’t taught this in school
@@Zephy9273 Who said it isn't?
It is taught in the UK.
@@Zephy9273 Asians trying not to brag about their school systems (obviously making 6 year olds repeat the multiplication table 1000 times a day is a good way to teach, great work Asia)
@Samridhi Pucha kisine ( who asked?)
@@bighillraft i am indian and literally every video of a hard problem i see some IIT dude is like "smh i did this problem when i was 11 give something harder", it annoys us too
*Y O W S E R S*
*Y O W S E T T E*
YIKES
Dippin'Dots ok now this is e p i c
*Z* *O* *I* *N* *K* *S*
@@dippin4dots oof
meanwhile 10th std CBSE students memorizing this sum for their syllabus: look what they need to mimic a fraction of our power
I was looking for this one😂
FREAKING THANK YOU I HAVE NEVER ENJOYED LISTENING TO A MATH LESSON BEFORE AND I'VE NEVER UNDERSTOOD IT SO CLEARLY
I love how you explain it so simple. Like you are explaining this to a very beginner.
This should be way of explanation for maths or a science related topics.
that proof is the brainchild of Euklid and thus ~2300 years old. means the stuff is ancient and fortunately basic math.
you can argue still if you should be able to think about the proof yourself or if the knowledge of "squareroot 2 is irrational" is enough on its own .
@@oldmanjeffrey why?
@@nicomatf idk, i don't remember replying that, he's obviously very experienced. it's deleted now so i guess it doesn't matter
@@oldmanjeffrey oh ok, have a good day man
When you have a math presentation with that requires a specific word count
2:13
Lol turn it in .com doesn’t transcribe videos yet ...
I'm pretty sure this is the video that started Terrance Howard's suspicion
*Or is it????*
I was waiting for the sweet Vsauce music
*_(weird music play)_*
√2 : "Michael should stop posting because his content isn't as good anymore"
*The square root of 2 is irrational*
COMMENTING SO I CAN SHOW I APPRECIATE THIS DAD JOKE MORE THAN THOSE OTHER 177 PASSERS-BY
SECRET MYSTERIOUS incredible
It's Bk This took me the square root of 2 seconds to get the joke. Very good.
It took me longer to get than it should have.
interesting but what is the existence of irrational numbers? what does it mean to the physical world.
BiNOmiAl MultiPliCatiOn
Vsauce saying "yikes" with the distortion filter just made me giggle
i was lost from the moment he said : "hey vsause micheal here" wait.....
he didnt say it in almost a damn year now !
Michael really out here teaching people proofs, and making them enjoy it. What a good service.
You know you’re actually doing high school math when a Vsauce video is easy to understand
DeadMehmed high school mathS is piss easy though try doing A-level maths bro
DeadMehmed ikr, what I was thinking, I actually understand this xD
Cammy Dough I’m fifteen and the most advanced thing I’ve covered is functions
Oh boy, as someone in calculus, I'm almost jealous of you
@@cammydough3184 r/gatekeeping
I am 37 and I just understood what rational and irrational means. As a kid I always thought that rational numbers are which we can comprehend or wich numbers sort of make sense and irrationals are some kind of alien numbers. Jesus... RATIOnal... it makes so much sense! Thank you for this eyeopening lesson!
" 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
0.8 is 4/5
I think that it would be really cool if you guys made a video talking about a particular denominator. Im 16 and a few years ago, I discovered something during a test. Its a beautiful pattern. If you take any number and dived it by 11, its the numerator times 9 in decimal form. For example: 1/11 = .090909 etc. 2/11 = .18181818 etc. Or even 7/11 = .63636363 etc. As if that pattern wasnt cool enough, it works the other way around. Numerator over 9 equals numerator times 11 in decimal form repeating. For example: 1/9 = .11111111 etc. Or 4/9 = 4444444 etc. I love this and would like you to explore it further. This is my favorite thing about math and its something that I haven't seen anyone talk about. There has to be a reason for this. For me, figure it out lol
mathforum.org/library/drmath/view/63848.html
First step is to understand how long division works.
Take 1/11 for example.
1. 11 goes into 1 0 times, so the first digit is 0
2. Next, move over one decimal place and multiply the numerator by 10 to get 10/11. Still, 11 goes into that 0 times so we have 0.0
3. Next, you get 100/11. 11 goes into 100 9 times leaving a remainder of 1 and our number is 0.09
4. next you move over and you get 10/11. That's familiar right? We just repeat the step 2 and go on from there leading to an infinite loop of 09 so the result is 0.090909090909...
If we did exactly the same thing except instead of moving one decimal over each time, we moved two decimals over, it would look like this
1. 11 goes into 1 0 times so we get 0
2. 11 goes into 100 9 times so we get 0.09 with a remainder of 1
3. repeat step 2 ad infinitum
If we make the numerator 2, we can do the same thing again except lets just skip step 1 and assume the first digit is 0
1. 11 goes into 200 18 times with a remainder of 2
2. repeat again and again to get 0.181818181818...
We can now generalize it by seeing that since with 1, if we multiply the whole operation by a number "x", you would end up with x*100/11 inevitably and since 11 goes into 100 9 times, it'll go into x*100 x*9 times and the remainder will always be a since the initial remainder is 1 and 1*x is just x.
Now, why does this happen with 9? lets do a similar thing with 9
100/9 is 11 with a remainder 1.
The remainder is once again the same as the original number, so it will repeat continuously and the same method applies to being able to multiply it so we have x/9 = 0.xxxxxxx... given that x is a single digit number.
You also didn't mention another number that has a similar pattern which is with 3. 100/3 = 33 with remainder 1 thus x/3 is 0. x*3 repeating for any number between 0 and 3.
So, what is the relationship between 9 and 11 or 33 and 3? They are all factors of 99 because 100-99=1 which is the remainder we need to get this relationship.
In fact, you can get this relationship by taking any factors of any number that is equal to 10^y-1. For example, 999 has factors 37 and 27. Lets test it by doing the long division of 1/27
1000/27 gives us 37 and a remainder of 1
so the result is 0.037037037037037...
So any number x/27 would be x*37 repeating every 3 digits therefore, 11/27 = 0.407407407407... because 11*37=407
Advanced notes:
If we were to try to generalize one of these to any integer numerator, not just ones below the denominator, we could get a formula that still works. Lets do 11 for example, it would look like x/11 = Σ x*9*10^(-2n) so you could have a number like 15/11 which you could figure out by 9*15=135 so the result would be 1.35 + 0.0135 + 0.000135 + ... resulting in 1.363636363636363636...
What happens if you do this for something like 11/11. 11/11=1 right? So it breaks the rule right? Actually no.
11/11 using this method means 11*9=99, therefore, 11/11=0.99999...=1 If you are skeptical about this, there are plenty of interesting videos explaining why 0.999... = 1.
@@bobbbbEE it was nice of you to explain this. 😊😊
Let me know if that helps, if a part is confusing, or if I just did something incorrectly ;)
Try making a table for y=9^x
And take a look at the first few digits of each number
0:07, I've never heard someone having an erection over the number 8
I'm not sure erections are even audible.
6:07 as well
7:05
FinBoyXD my erections are always audible. Sort of a “fwooosh__ ftinggg” sort of sound. It’s a bit annoying at movie night... lol
Don't forget 11:22, that's a big one
That’s a freakishly long pencil
Y O W Z E R S
Hybridjunkie no, Y O W Z A
Hybridjunkie i read this as he said it lmao
Yikes!
Hybridjunkie Z O Y I N K S
oh my goodness gracious
When u get the two channels confused:
- puts vsauce outro on D!NG
👍
Nice spot 😂
I never realized why this proves anything, until now: if you can keep dividing a and b forever by 2 and still get an integer, then they must be infinite.
It has been 5 minutes since the video ended and you're the reason why I understood the last bit more intuitively. Thanks!
You can just think of it as that it contradicts the original statement that a/b is in its simplest form, therefore root 2 must be rational
@@mahimagrawal825 We know that a is an integer, we made it so. We know that a^2 is an even integer. It follows that a is an even integer. Yiu are WRONG.
A minor correction to my original comment: a and b could both be zero, but it's not defined in elementary algebra. I think it's obvious, but I'm a poindexter in these things😁
@@mahimagrawal825 I'm sorry but the former statement is correct. It does not imply however that the "root of an even number is even". The square of a number is literally the opposite of a square root of a number.
[(x^2 is even) ⇒ (x is even)] does not imply [(x is even) ⇒ (sqrt{x} is even)] Note: the '⇒' sign means "implies"
The first statement was already proven in the video. The statement "if x is odd then x^2 is odd" proves this statement because the contrapositive of a proposition is always equivalent in truth value of the original. and if x is not odd then it should pretty much be even, granted that x is assumed to be integer which is the case in the video. Your second statement does not hold absolute.
EDIT: at 15:58 vsauce essentially said that if b is even implies that b^2 is even then it means b is even.... An error in his part....
Terrance howard enters the chat:
It's hard out here for a pimp, Mayne.
Terrance howard is too dumb and/or delusional to comprehend this video
@@bakedpotato420❤
For a beginner that wants to learn mathematics, you have helped me a lot to understand different theorems and how proofs work :) simple and well defined.
Anyone got bothered he didnt simplify 8/10 to 4/5?
I did
YES
I think he didn’t do that to keep things simple for younger audiences, because he explains it 10 minutes later and it might’ve been confusing
Man, that hurts😓😅
merp merp i got brothered that you didnt simplify 4/5 to 2/2.5
When he said "or is it?"
i thought: "finally the music!"
Then i remembered where was i.
"Damn..."
What
We were on the verge of greatness, we were this close
*I was. 🤓
TornByTheSeams music happens on the vsauce channel which he also runs whenever he said or is it
Why can’t he also have the music here?
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
Me trying to explain to my mom that everyone else in the class failed the test.
ua-cam.com/video/TVqewTxRtyM/v-deo.html
I am 37. None of this has any bearing on my life whatsoever. I will almost certainly never need this knowledge, however I cannot resist a video of Michael explaining stuff to me. So, 17 minutes of my life gone and now I know it.
I'll comment again if it ever comes up.
Rage Against My Hairline interesting..
I like to think that just by following along we use our brains in a way we usually don’t, and there’s value in that
adding a comment so i get a notification if it actually comes up again.
Same
did u love it tho