A Proof That The Square Root of Two Is Irrational


Comments • 8 311

  • JDYoonaddict
    JDYoonaddict 2 hours ago

    Why 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...

  • Lowkii
    Lowkii 16 hours ago

    It triggers me how you write the number 8

  • Pretty Girl Strks
    Pretty Girl Strks 16 hours ago

    Heres 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 Dedaj
    Eristofan Dedaj 17 hours ago

    Can michael be my teacher pls pls pls

  • Keith Thedford II
    Keith Thedford II 18 hours ago

    Outro = Lit

  • Jeremiah Gerald Luna
    Jeremiah Gerald Luna 22 hours ago

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

  • Grace Hsu
    Grace Hsu Day ago

    Love this episode

  • SpicyMelon
    SpicyMelon Day ago

    So 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.

  • TimmacTR
    TimmacTR Day ago +1


  • TimmacTR
    TimmacTR Day ago +2

    1/3 = 0.3333333... in 10 based system
    but 0.4 in 12 based system

  • slapensee2
    slapensee2 Day ago

    (√-1) Love You!!

  • Michael White
    Michael White Day ago

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

  • Justin Little
    Justin Little Day ago

    1 isn't prime, so 14/15 don't have co-primes?

  • Daniele untouchable


  • Zander Kraz
    Zander Kraz 2 days ago

    I love that proof of why multiplying fractions works the way it does!!

  • requemao
    requemao 2 days ago

    I 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.

  • Qwerty
    Qwerty 3 days ago

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

  • Tom Holtz
    Tom Holtz 3 days ago

    Why is a rational number only one that can be expressed as a ratio of two integers? Just curious

    • Mariano
      Mariano 3 days ago

      Thats what we define a rational number as

  • Samson5erb Rocks
    Samson5erb Rocks 3 days ago

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

  • Riley Jano
    Riley Jano 4 days ago

    dumb question, but at 7:06, why don't you pull out 4c?

    • Mariano
      Mariano 3 days ago

      Because 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

  • WatchMeSayStuff
    WatchMeSayStuff 4 days ago

    1/1.5 BOOM

  • SyncXer
    SyncXer 4 days ago +1

    *_”or is it?”_*

  • tau
    tau 4 days ago

    Doesn'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

    • tau
      tau 4 days ago

      Thanks 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.

    • Mariano
      Mariano 4 days ago

      he 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 Sharma
    Rishabh Sharma 4 days ago +1

    micheal is so single that when the whole was finding the `X' he was finding one to `C'

  • Ayy Lien
    Ayy Lien 4 days ago

    You teach me more than my math teacher

  • Ringo Ferrer
    Ringo Ferrer 5 days ago

    Well my school taught us that square of two is irrational

  • André W
    André W 5 days ago

    i understand nothing but i love listening while drawing

  • Ethanagor S-D
    Ethanagor S-D 5 days ago +2

    the way Michael writes the number 8 is mildly infuriating

  • Paras T
    Paras T 5 days ago +1

    Hey VSauce, Michael here. *I just broke maths.*

  • KN Franz
    KN Franz 6 days ago

    So what does this supose to do with your life?

  • Tristan Gillis
    Tristan Gillis 6 days ago

    Now find the square root of -1!

  • Eduardo Terra
    Eduardo Terra 6 days ago

    I'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 Terra
      Eduardo Terra 4 days ago

      Thank you guys, I realized that layer on in the video and forget to delete the coment hahah
      It was late at night sorry

    • Vaporeonus
      Vaporeonus 5 days ago +1

      You forgot that it's negative, -14 + 1 = -13

    • Mariano
      Mariano 5 days ago +1

      he says -13, not 13
      2*-7=-14, and-14+1=13

  • Todor Kolev
    Todor Kolev 6 days ago

    wow this is waaay too dumbed down

  • Fsy North
    Fsy North 6 days ago

    1:35 We’re learning the Pythagoras thing at my school now

  • OllieRoseB
    OllieRoseB 6 days ago

    0:14 *_YiKeS_*

  • Seth Sertain
    Seth Sertain 6 days ago

    Did y’all catch when he said 2 time negative 7 plus 1 is 13

  • Ok Danks
    Ok Danks 8 days ago

    Stop please

  • Gamingwithsungod
    Gamingwithsungod 8 days ago

    Why do you write eights like that?

  • Rage Against My Hairline

    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.

  • Dulcis90
    Dulcis90 9 days ago

    To be squared or not to be squared, that is the question.

  • Ken Mason
    Ken Mason 9 days ago

    Not enough THC in your youth. There's your problem right there.

  • Christopher Kohler
    Christopher Kohler 10 days ago

    These kind of lessons would've made high school math so much more interesting.

  • Donna Allen
    Donna Allen 11 days ago +1

    YOu'll thank me later

  • Education Everything
    Education Everything 11 days ago

    In Indian CBSE it is a simple and easy question to prove this.

  • German Carranza
    German Carranza 12 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.

    • Vaporeonus
      Vaporeonus 5 days ago

      Reducto 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

    • Mariano
      Mariano 11 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
      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 Yaputra
    Jason Yaputra 12 days ago

    i 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 Yaputra
      Jason Yaputra 11 days ago

      +Mariano well said!

    • Mariano
      Mariano 11 days ago

      There 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 Yaputra
      Jason Yaputra 12 days ago

      i missed something here, help me

  • Dubu Dubu
    Dubu Dubu 12 days ago

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

  • Cat rymer
    Cat rymer 12 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.

    • Peachy
      Peachy Day ago

      i dont understand a word of what he is saying

  • Vinny Devriesere
    Vinny Devriesere 12 days ago

    Can we get an F for -(square root of 2)?

  • Remix Station
    Remix Station 12 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 Beerus
    Lord Beerus 13 days ago +1

    I think my brain threw up

  • IWillTakeAGuranteeOfBetterOverAPromiseOfPerfect

    For 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 MyButt
    CherryRibble MyButt 14 days ago

    Beautiful handwriting

  • trulucy
    trulucy 14 days ago

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

  • rbcallmeblue
    rbcallmeblue 14 days ago

    UwU is that 8th quadratics I see??? 6:48

  • João Vitor estudante
    João Vitor estudante 14 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 maxx
    ananya maxx 14 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 maxx
      ananya maxx 13 days ago

      +HawkeyedSentinel yeah lol :)

    • HawkeyedSentinel
      HawkeyedSentinel 13 days ago +1

      ananya maxx oh haha. What i said must have been gibberish to you :D

    • ananya maxx
      ananya maxx 13 days ago

      +HawkeyedSentinel I'm in 10th grade lol

    • HawkeyedSentinel
      HawkeyedSentinel 13 days ago +1

      ananya maxx let me guess....2017 specification a level maths?

  • dodAEK fraktolon
    dodAEK fraktolon 15 days ago

    saying your mind and the paper and the numbers are not in the universe? maybe you should get out of your box...

  • Oreo Pods
    Oreo Pods 15 days ago


  • Limbo Cat
    Limbo Cat 15 days ago

    . . . . . . __ . . . __ __ . Jk I’m fine

  • Limbo Cat
    Limbo Cat 15 days ago

    The rectangle of nature.

  • Mr Me
    Mr Me 15 days ago

    What the Hell am i doin' here. I dont belong here!!

  • Imon Sanyal
    Imon Sanyal 15 days ago

    In india, we learn this in the eighth class

    • Imon Sanyal
      Imon Sanyal 5 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

    • Vaporeonus
      Vaporeonus 5 days ago

      Yeah, that's pretty much the standard in all countries

  • -Noirsette
    -Noirsette 16 days ago

    E Y O W Z A S

  • DokterKaj
    DokterKaj 16 days ago

    11:57 coן

  • David Wales
    David Wales 16 days ago


  • Harshit Gupta
    Harshit Gupta 16 days ago +4

    Why I am watching this video?
    I know why √2 is irrational.

    • Augustus Sorci
      Augustus Sorci 2 days ago

      Harshit Gupta why are you commenting on this video, you clearly know why 2^.5 is irrational

  • Jade Everest
    Jade Everest 16 days ago

    Basically 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 Everest
    Jade Everest 16 days ago

    We went over this three months ago in my 9 enriched class

    • Vaporeonus
      Vaporeonus 5 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 Everest
      Jade Everest 16 days ago

      Wait, 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 Falk
    Robin Falk 16 days ago

    If 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?

  • DifficultyP13dood
    DifficultyP13dood 16 days ago

    My head hurts.

  • David Newbon
    David Newbon 16 days ago

    The way you write 8... 😧

  • Swegolf
    Swegolf 17 days ago


  • Anony Mouse
    Anony Mouse 17 days ago

    "Negative 13 can be expressed by negative 7 times 2 and plus 1"
    Mate really?

    • Anony Mouse
      Anony Mouse 3 days ago

      Vaporeonus oh sorry my math was dead

    • Vaporeonus
      Vaporeonus 5 days ago

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

  • LilGamerDude_Alex
    LilGamerDude_Alex 17 days ago

    My brain hurts

  • Andrei Mureșan
    Andrei Mureșan 17 days ago

    Idk about USA but here in Romania this is 7th grade math

  • Rose Johnson
    Rose Johnson 17 days ago

    Him: a over b squared equals 17 times 100000000000000
    Me: *watching it fly over my head*

  • Chris Fard
    Chris Fard 17 days ago

    #4:38 but -12 isn't 2 x -6

  • Stetson_Newsie
    Stetson_Newsie 17 days ago

    Y I K E S

  • Lancer
    Lancer 17 days ago


  • Makoto Yūki
    Makoto Yūki 18 days ago

    Are 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_* ?

  • FireStorm680
    FireStorm680 18 days ago

    F is not First.... F is reserved for Skeppy and BadBoyHalo ONLY

  • James Gabor
    James Gabor 18 days ago


  • pls kill minions
    pls kill minions 18 days ago

    i got that on my exam

  • MrGuy
    MrGuy 18 days ago

    I guess Vsauce is just for Mind Field now

  • MrGuy
    MrGuy 18 days ago

    I thought the square root of 2 was always irrational?

  • Ninad Kulkarni
    Ninad Kulkarni 18 days ago +3

    2:31 *Where is the classic Vsauce Music!!?*

  • A folding chair
    A folding chair 18 days ago

    am no smart enuf healp

  • Games and toys With the boys

    Are we gonna ignore that when he multiplied by b^2 he only did it on one side

  • Aditya Garg
    Aditya Garg 19 days ago

    too irrational?

  • Rohith Krithivasan
    Rohith Krithivasan 19 days ago +3

    4:30, -12+1 does not equal 13

  • Lucy Bear
    Lucy Bear 19 days ago +1


  • Gabriela Alvarez
    Gabriela Alvarez 19 days ago

    To see square equals to be square.
    MaK3s sEns3

  • NotTheRealJay McKee
    NotTheRealJay McKee 19 days ago

    3:58 look how he drew the 8

  • Gabriela Alvarez
    Gabriela Alvarez 19 days ago

    I have this homework:
    -b (-)(+) _/b^2 -4 (a)(c)
    And I am instead watching this video when it is actually kind of the same... but this is fun, homework isn't

  • Avinash Shaheed
    Avinash Shaheed 19 days ago

    My 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

  • Nacomi
    Nacomi 20 days ago +2

    2c=even 2x0=0 0=even

  • Joey Melo
    Joey Melo 20 days ago

    The only teacher I will ever listen to with so much focus.

  • Joey Melo
    Joey Melo 20 days ago

    What did 2 say to 1 that resulted in a breakup?

    wanna try 68?

  • Buster McMuffin
    Buster McMuffin 20 days ago

    video starts at 13:09