# A Proof That The Square Root of Two Is Irrational

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JDYoonaddict2 hours agoWhy do you like to make your life complicated? If you want to prove that a/b*c/d = ac/bd, or a/b * a/b = a^2 / b^2, then all you have to do is to replace the letters with random numbers in a way that if you divide the numbers of the fractions, u get an integer (or not, I just want to give a very simple example)... for example assume that a=4, b=2, c=9, d=3... then its 4/2 * 9/3 = 36/6 = 6... and we all know that 4/2 = 2 and 9/3 = 3 so 2*3 = 6.. if this works with these numbers, then it works with any numbers you have... Here you are! I proved that a/b*c/d = ac/bd and so I proved that a/b*a/b = a^2 * b^2... I know, all mathematicians try to prove everything the hard way, but its not bad to simplify things sometimes if possible...

Lowkii16 hours agoIt triggers me how you write the number 8

Pretty Girl Strks16 hours agoHeres a tip Micheal:

Post these types of videos on your Vsauce channel and move your new videos to this channel then all of your followers are happy!!! :) LIKE SO HE SEES THIS.

Eristofan Dedaj17 hours agoCan michael be my teacher pls pls pls

Keith Thedford II18 hours agoOutro = Lit

Jeremiah Gerald Luna22 hours agoAll numbers are rational by removing the decimal point and having them denominated by 10^(how many actual decimal places the number has) :v

Grace HsuDay agoLove this episode

SpicyMelonDay agoSo if we could learn this without the universe by using our own mind. Then doesn’t this mean that our mind is the universe, or a part of the universe. So in turn our consciousness, our thinking, was created by the universe.

TimmacTRDay ago^{+1}Mathemagics

TimmacTRDay ago^{+2}1/3 = 0.3333333... in 10 based system

but 0.4 in 12 based system

slapensee2Day ago(√-1) Love You!!

Michael WhiteDay agoMichael, I'm offended that you'd dare to draw your 8 as two separate Os...

Justin LittleDay ago1 isn't prime, so 14/15 don't have co-primes?

Daniele untouchableDay agoOmg

Zander Kraz2 days agoI love that proof of why multiplying fractions works the way it does!!

requemao2 days agoI usually enjoy Vsauce videos, but I just can't bear to watch a video where one circle on top of another is said to be a number eight.

Qwerty3 days agoI feel like I should have a drink in my hand for this episode...

Tom Holtz3 days agoWhy is a rational number only one that can be expressed as a ratio of two integers? Just curious

Mariano3 days agoThats what we define a rational number as

Samson5erb Rocks3 days ago*O H M Y G O O D N E S S G R A C I O U S !*

Riley Jano4 days agodumb question, but at 7:06, why don't you pull out 4c?

Mariano3 days agoBecause the idea is leaving the expression in the form 2*(...)+1 to show that it is an odd number

If you factorized 4c out, you would get 4c(c+1)+1, and then you would have to pull a 2 out anyways to get 2*(2c(c+1))+1

WatchMeSayStuff4 days ago1/1.5 BOOM

SyncXer4 days ago^{+1}*_”or is it?”_*

tau4 days agoDoesn't -12 + 1 = -11? You said -13. This also creates two equations to find the two odd numbers surrounding an even number:

2 * even + 1

2 * even - 1

tau4 days agoThanks for correcting me, I wasn't really paying attention and just heard something along the lines of "-12 + 1 [...] -13", I'm also not a native so some of what he said might have been left up to how I interpreted it.

Yeah, I was totally wrong on this... I guess what I meant to say this.

"This also creates two equations for finding an even number and its two surrounding odd numbers:

2 * num + 1

2 * num - 1"

This creates a scenario where any number can be made even and its surrounding odd numbers found. I don't know why I thought this even was a useful remark, just ignore the whole comment because you just disproved it I guess lol.

Mariano4 days agohe said 2*-7 +1, which is -14+1

Also, if you wanted the 2 odd numbers around an even number, it is just even+1 and even-1

For example, take 4, 2*4+1 is 9 and 2*4-1 is 7, not 5 and 3

Rishabh Sharma4 days ago^{+1}micheal is so single that when the whole was finding the `X' he was finding one to `C'

Ayy Lien4 days agoYou teach me more than my math teacher

Ringo Ferrer5 days agoWell my school taught us that square of two is irrational

André W5 days agoi understand nothing but i love listening while drawing

Ethanagor S-D5 days ago^{+2}the way Michael writes the number 8 is mildly infuriating

Paras T5 days ago^{+1}Hey VSauce, Michael here. *I just broke maths.*

KN Franz6 days agoSo what does this supose to do with your life?

Tristan Gillis6 days agoNow find the square root of -1!

Eduardo Terra6 days agoI'm not great at math, and I am probably wrong about this

But I find something odd

At 4:48 My boy Mike claims "13 can be expressed as 2x -7 + 1" wouldn't that be 15?

I'm probably gonna apear at some r/fuckingimbecil but anyways, very confused

Eduardo Terra4 days agoThank you guys, I realized that layer on in the video and forget to delete the coment hahah

It was late at night sorry

Vaporeonus5 days ago^{+1}You forgot that it's negative, -14 + 1 = -13

Mariano5 days ago^{+1}he says -13, not 13

2*-7=-14, and-14+1=13

Todor Kolev6 days agowow this is waaay too dumbed down

Fsy North6 days ago1:35 We’re learning the Pythagoras thing at my school now

OllieRoseB6 days ago0:14 *_YiKeS_*

Seth Sertain6 days agoDid y’all catch when he said 2 time negative 7 plus 1 is 13

Mariano5 days agohe says -13

Ok Danks8 days agoStop please

Gamingwithsungod8 days agoWhy do you write eights like that?

Rage Against My Hairline8 days ago^{+2}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.

Dulcis909 days agoTo be squared or not to be squared, that is the question.

Ken Mason9 days agoNot enough THC in your youth. There's your problem right there.

Christopher Kohler10 days agoThese kind of lessons would've made high school math so much more interesting.

Donna Allen11 days ago^{+1}0:13

YOu'll thank me later

Education Everything11 days agoIn Indian CBSE it is a simple and easy question to prove this.

German Carranza12 days ago^{+1}That looks like pseudo-math. Could you not follow the same logic with any other square root or number for that matter? What if I picked the square root of 4 and decided to follow the same logic to see if the square root of 4 was rational. Someone help me out because I’m not buying this.

Vaporeonus5 days agoReducto ad Absurdum (I think that's what it's called) only really works when there's a contradiction, the square root of 4 wouldn't do that

Mariano11 days ago^{+2}Reasoning by contradiction (Reductio ad Absurdum) is indeed very real math. It is in fact one of the most powerful tools we have to prove stuff. The idea is that if we assume something, and then based on that we reach something that we know for sure is false, then whatever we assumed to be true MUST be false.

Lets try to apply the same method in the video to sqrt(4). First, we assume sqrt(4) is rational,That means there must exist 2 co-prime integers "a" and "b" such that sqrt(4)= a/b. So:

4* b^2=a^2. So a is even, the a=2c, so:

4* b^2= (2c)^2

4*b^2=4*c^2

b^2=c^2

So we can conclude that a is even, but unlike with sqrt(2), we cannot conclude that b is even as well. There is no contradiction here, so we cannot claim that sqrt(4) is irrational.

Interestingly, we cant claim sqrt(4) is rational neither. We assumed something, but didnt reach any contradiction, so we didnt really get any information at all.

Now, if you really wanna think about reasoning by contradiction, i recommend you watch the video "The Flaw in Reductio Ad Absurdum". it is a really interesting watch

Jason Yaputra12 days agoi dont get it... so what if they share 2 and theyre not coprime?.. i mean 8/10 also both share 2 as a factor but earlier he says 8/10 is also rational number right?

Jason Yaputra11 days ago+Mariano well said!

Mariano11 days agoThere are 2 ways to look at this.

1) i believe this is what michael does in the video (i watched it a while ago and its pretty long so i might be wrong). We start by assuming sqrt(2)=a/b, and then reach the conclusion that both a and b must be even. Since they are both even, we can divide them both by 2 and get an expresion of the form sqrt(2)=c/d. Now we can apply the same process again to get something like sqrt(2)=e/f, and we can continue applying this process over and over again infinitely. So basically, both a and b must be numbers that are infinitely divisible by 2, which is a contradiction.

2) The way I prefer to look at it is by assuming an extra condition. Instead of only assuming that sqrt(2)=a/b, we also assume that a and b are co-primes to begin with. We can do this because just as michael said, any ratio between 2 integers can be reduced to a ratio between co-primes. So if sqrt(2) is rational, there must exist 2 integers "c" and "d" so that sqrt(2)=c/d, but because c and d are integers, we also know there must exist 2 integers a and b so that sqrt(2)=c/d=a/b, with a and b co-primes. Now we apply the same method, and reach the conclusion that both a and b are even, so they have a common factor of 2. However, we assumed that they didnt have any common factors other than 1, so there is our contradiction.

Either of these methods reach the same conclusion

Jason Yaputra12 days agoi missed something here, help me

Dubu Dubu12 days agoI’ve done this stuff so much in discrete and real analysis, I don’t want to watch this

Cat rymer12 days ago^{+4}This sort of Math leaves me way behind in understanding. I have never received the schooling in these subjects therefore I think it's safe to assume (Its not that I couldn't do it it just wasn't an avenue offered while young.)So when I listen to you I am blown away. I do not understand but I will take your word for it. Mind Blowing.

PeachyDay agoi dont understand a word of what he is saying

Vinny Devriesere12 days agoCan we get an F for -(square root of 2)?

Remix Station12 days ago^{+1}All of the people who just want the proof and not the basislc stuff , the real video begins at 13:20

Lord Beerus13 days ago^{+1}I think my brain threw up

IWillTakeAGuranteeOfBetterOverAPromiseOfPerfect13 days agoFor everyone with a solid math background who just wants a reminder of the proof from a well spoken person; it starts at 13:06.

Also, Michael, I would leave 1 out as a factor for beginners. It makes the reduction concept unclean imho. It creates an exception case when someone is trying to learn a new rule.

Otherwise, thank you as always!!

CherryRibble MyButt14 days agoBeautiful handwriting

trulucy14 days agoIf my math teachers explained math to me the way Michael has here I would have graduated high school back in 1985 with better grades.

rbcallmeblue14 days agoUwU is that 8th quadratics I see??? 6:48

João Vitor estudante14 days ago^{+8}Square root of 2 is an irrational number? Proving through the method of the absurd.

Suppose that the square root of two is rational, that is, the root of two can be written as p / q, where p and q are integers, since q is different from 0.

Square root of two = / 2

/ 2 = p/q

(/2)^2 = (p/q)^2

2 = p ^ 2 / q ^ 2

2.q ^ 2 = p ^ 2

p ^ 2 = 2.q ^ 2

p ^ 2 is an even number, since any integer, whether odd or even, when multiplied by 2, will result in an even number. In our case, q ^ 2 would be our whole number.

knowing that p ^ 2 = 2.q ^ 2

2 = p ^ 2 / q ^ 2

q^2 is it odd or even?

If it were even, the result of the division and an even number by an even number could be odd or even. If it were odd, it could result in odd or even numbers. Conclusion: q ^ 2 can be even, as can also be odd; which is an ABSURD, there is no such thing as a squared q number being even and odd at the same time. Or it is odd or even ...

2 = p ^ 2 / q ^ 2 is absurd, so / 2 = p / q is also absurd. Since p / q is rational and the root of two is NOT rational, so the root of two can only be irrational.

Edit: it's almost the same of the video hahaha

Congratulations from Brazil.

ananya maxx14 days ago^{+1}I literally had my maths exam today and I had to prove √3 is irrational. Guess what, I used this method

ananya maxx13 days ago+HawkeyedSentinel yeah lol :)

HawkeyedSentinel13 days ago^{+1}ananya maxx oh haha. What i said must have been gibberish to you :D

ananya maxx13 days ago+HawkeyedSentinel I'm in 10th grade lol

HawkeyedSentinel13 days ago^{+1}ananya maxx let me guess....2017 specification a level maths?

dodAEK fraktolon15 days agosaying your mind and the paper and the numbers are not in the universe? maybe you should get out of your box...

Oreo Pods15 days ago*AND HEAR I AM LEARNING MATH ON WINTER VACATION.*

Limbo Cat15 days ago. . . . . . __ . . . __ __ . Jk I’m fine

Limbo Cat15 days agoThe rectangle of nature.

Mr Me15 days agoWhat the Hell am i doin' here. I dont belong here!!

Imon Sanyal15 days agoIn india, we learn this in the eighth class

Imon Sanyal5 days ago+Vaporeonus done get me wrong... I'm not saying that Indian education system is superior to other countries...I'm just saying that why make a youtube video if everybody already it

Vaporeonus5 days agoYeah, that's pretty much the standard in all countries

-Noirsette16 days agoE Y O W Z A S

11:22

DokterKaj16 days ago11:57 coן

David Wales16 days agoyowzas

Harshit Gupta16 days ago^{+4}Why I am watching this video?

I know why √2 is irrational.

Augustus Sorci2 days agoHarshit Gupta why are you commenting on this video, you clearly know why 2^.5 is irrational

Jade Everest16 days agoBasically guys, if the square root of two was rational it would be able to be a fraction but if you try to write it as a fraction in lowest terms both the numerator and denominator have to be even and therefore should be reducible but that's impossible.

Jade Everest16 days agoWe went over this three months ago in my 9 enriched class

Vaporeonus5 days ago+Jade Everest Everyone in basically every country does this at some point in 8-10th grade, your teacher just had some great timing

Jade Everest16 days agoWait, this was made around then, what if my math teacher just did it because of this video! I mean, he is a youtuber: Plant Eats

Robin Falk16 days agoIf there's a smallest distance, wouldn't it mean that the hypothesis of the right triangle with sides 1, 1 and sqrt2 would be actually measurable? Doesn't that mean that sqrt2 is rational?

DifficultyP13dood16 days agoMy head hurts.

David Newbon16 days agoThe way you write 8... 😧

Swegolf17 days agoYausa

Swegolf17 days agoLmfao

Anony Mouse17 days ago"Negative 13 can be expressed by negative 7 times 2 and plus 1"

Mate really?

Anony Mouse3 days agoVaporeonus oh sorry my math was dead

Vaporeonus5 days ago-7*2 = -14, -14 + 1 = -13

LilGamerDude_Alex17 days agoMy brain hurts

Andrei Mureșan17 days agoIdk about USA but here in Romania this is 7th grade math

Rose Johnson17 days agoHim: a over b squared equals 17 times 100000000000000

Me: *watching it fly over my head*

Chris Fard17 days ago#4:38 but -12 isn't 2 x -6

Stetson_Newsie17 days agoY I K E S

Lancer17 days agoRATIaNAL

aNAL

ok

Makoto Yūki18 days agoAre we going to ignore that Michael's favorite thing is *_B I N O M I A L M U L T I P L I C A T I O N_* ?

FireStorm68018 days agoF is not First.... F is reserved for Skeppy and BadBoyHalo ONLY

James Gabor18 days ago*yowzers*

pls kill minions18 days agoi got that on my exam

MrGuy18 days agoI guess Vsauce is just for Mind Field now

MrGuy18 days agoI thought the square root of 2 was always irrational?

Ninad Kulkarni18 days ago^{+3}2:31 *Where is the classic Vsauce Music!!?*

A folding chair18 days agoam no smart enuf healp

Games and toys With the boys18 days agoAre we gonna ignore that when he multiplied by b^2 he only did it on one side

Aditya Garg19 days agotoo irrational?

Rohith Krithivasan19 days ago^{+3}4:30, -12+1 does not equal 13

FplusE TV Channel15 days agoNo he said (2*-7)+1 not -1

1noxaj15 days agoThat’s correct, but if you listen closely, Micheal says

(2*-7)-1=13

FplusE TV Channel18 days agoWho says that it is

Lucy Bear19 days ago^{+1}Yowzers!

Gabriela Alvarez19 days agoTo see square equals to be square.

MaK3s sEns3

NotTheRealJay McKee19 days ago3:58 look how he drew the 8

Gabriela Alvarez19 days agoI have this homework:

-b (-)(+) _/b^2 -4 (a)(c)

---------------------

2(a)

And I am instead watching this video when it is actually kind of the same... but this is fun, homework isn't

Avinash Shaheed19 days agoMy god. I love math and im good at it. But doing it in a different language than my native tongue makes it extremely hard on my brain to translate while trying to keep up your calculations... Taxing....

Not that i don't know these. But i just know them. I don't hav eto think about them. But actively having to think about them in english xD. Wtf

Nacomi20 days ago^{+2}2c=even 2x0=0 0=even

Joey Melo20 days agoThe only teacher I will ever listen to with so much focus.

Joey Melo20 days agoWhat did 2 say to 1 that resulted in a breakup?

wanna try 68?

Buster McMuffin20 days agovideo starts at 13:09