A Proof That The Square Root of Two Is Irrational


Comments • 7 814

  • Alex Goedken
    Alex Goedken 5 hours ago


  • Nod Boss
    Nod Boss 8 hours ago +1

    4:50 check that math

  • Michael Sal
    Michael Sal 12 hours ago

    I may be asking something really obvious to most of you, but how can a side of a triangle be unmeasurable?? Does this mean that such a triangle cannot be created and if we do create it the sides or angles are approximately what we believe they are so that the hypotenuse is actually measurable??

  • That guy who always make strange noises at night

    You know, with 17 minutes you could've proven that the kth square of any natural number is either a natural number or irrational

  • Math 5 Group
    Math 5 Group Day ago

    thanks for ur effort

  • quinn dallet
    quinn dallet Day ago

    almost better then hamster pics on Google images...

  • thequinnman325
    thequinnman325 Day ago


  • EpicDuty
    EpicDuty Day ago

    I love binomische Formeln like we call it in Germany

  • EpicDuty
    EpicDuty Day ago

    I want to concentrate on the video but your way of writing an 8 makes me angry it looks like out of the movie signs

  • Mathe Logix
    Mathe Logix Day ago

    Michael I have a request. Can you please make a video on Buffon's Needle Problem?? I really want to understand to it. Plz make a video on this topic plz plz plz.

  • Lane Robertson
    Lane Robertson 2 days ago

    16:32 Yeah, well, neither am I!

  • Melker Jonsson 9B
    Melker Jonsson 9B 2 days ago

    4:50 wrong. -12 + 1= -11

    • Priest Plaxis
      Priest Plaxis 13 hours ago

      Melker Jonsson 9B 2 times negative seven

  • Bixls
    Bixls 2 days ago

    11:32 taught me how to teach simplifying

  • KoishiKomeiji11
    KoishiKomeiji11 2 days ago

    i really like michael's handwriting

  • stan Daesung
    stan Daesung 2 days ago

    i love michael so much im crying ????

  • Suffek
    Suffek 2 days ago


  • lost
    lost 2 days ago

    outro music?

  • Kailash Ramakrishnan

    Or we could just assume that root2 is a rational no. (P/q where q doesn't equal zero and p>q) and square on both sides
    2= p^2/q^2
    q^2= p^2/2. -(1)
    Since p^2 /2 gives an answer (q)
    p is a multiple of 2 so let p= some no.2k
    Now from equation 1 we know that 2q^2= p^2
    2q^2 = (2k)^2
    2q^2 = 4 k^2
    q^2 = 2k^2
    q= 2k
    We know that in the case of rational numbers p/q the hcf must be 1 but in the above case it was wrong therefore root 2 is irrational

    • wispa
      wispa Day ago

      That is exactly the proof that Michael did.

  • Dashua
    Dashua 2 days ago

    Rationalism is strong in this one

  • gumikebbap
    gumikebbap 2 days ago

    I really like the video, but I miss the demonstration proving that any ratio of integers must be reducible to the ratio between 2 co-prime integers.
    Maybe you made a video about. I'll check it out :)

  • Nofxthepirate
    Nofxthepirate 2 days ago

    I could honestly watch Michael explain mathematical proofs for hours. Why is this the only one!

  • AbyssWalker240 Gaming

    the square root of 2 is 1.4142135623730951 and thats rational. put it through a calculator.

  • Tony Turtle
    Tony Turtle 2 days ago

    lol he said 13 = 2 x 7 + 1 but 2 x 7 = 14 and 14 + 1 = 15. Michael forgot how to do simple math.

    • Priest Plaxis
      Priest Plaxis 13 hours ago

      Tony Turtle two times negative 7 plus 1 is negative 13

  • Someoneawesome123 Plays


  • Explosive Kitten
    Explosive Kitten 2 days ago

    Wanna tutor me in math when I start college? xD

  • Airon Jones
    Airon Jones 3 days ago


  • Divyansh Singhal
    Divyansh Singhal 3 days ago

    I did it when I used to study in 3rd class. I was damn better than him...

  • Gus TV
    Gus TV 3 days ago

    2b^2 or not 2b^2

  • Ricardo Guzman
    Ricardo Guzman 3 days ago

    (√2+1)^n approaches a whole number value as the integer n increases. That is nuts.

    • Ricardo Guzman
      Ricardo Guzman Day ago

      +wispa Yes. But for each integer n, it gets closer and closer to a whole number.
      when n=12, the answer is. 39201.999974491... when n=16, the answer is
      when n=24 the answer is
      and so on to infinity.

    • wispa
      wispa Day ago

      (√2+1)^n → ∞ as n → ∞

  • tabaks
    tabaks 3 days ago

    Michael, that's a long DONG you had here. How about Vsauce? They fired you?

  • 郭维佳
    郭维佳 3 days ago

    nobody gonna talk abt the way michael writes 8?

  • KingJacobz
    KingJacobz 3 days ago

    He said -13 is 2 x -7 instead of -6 + 1. Can he not do retakes????? :) time stamp 4:49

    • wispa
      wispa Day ago

      2 x -7 = -14
      -14 + 1 = -13

  • Jill Calkins
    Jill Calkins 3 days ago

    Hosted by MICORN Stevens?

  • Forbidden Lusts
    Forbidden Lusts 3 days ago

    What is this gibberish

  • yahya pandit
    yahya pandit 4 days ago

    Vsauce gets orgasm from math

  • Tobe Shinji
    Tobe Shinji 4 days ago +1

    Oh god no. I may have a irrational fear for ratios now.

  • Collin Flintjer
    Collin Flintjer 4 days ago

    how would or could u justify 9 being even if u would assume it's divisible by 4.5?

    • wispa
      wispa Day ago

      9 is not even. Even numbers are whole numbers which when divided by 2 leave no remainder.

  • Fire Shark
    Fire Shark 4 days ago

    Well wouldn't you just reduce a/b into a/2 over b/2. The square root of two can be expressed as two numbers multiplied by two, and those two resulting numbers are a and b.

  • Driss EL ALLIOUI
    Driss EL ALLIOUI 4 days ago

    You proved that: IF a is odd then square a is odd
    But you did not prouve: IF square a is odd then a is odd
    However you're using the second result !!

  • LiamLikesGames
    LiamLikesGames 4 days ago

    I sometimes watch these in case I ever get that pill from limitless.

  • Mr. Cub Fan 415
    Mr. Cub Fan 415 4 days ago

    Is zero an even number?

  • The Keb
    The Keb 4 days ago

    My brain almost got shut down almost 2c X 1 times now while watching the video because my brain never felt this.

  • Perry Schafer
    Perry Schafer 4 days ago

    Yeah, I'm that guy that finds some fault in everything. It's easy because nothing is perfect. But, this time, what I want to point out is important as relates to the philosophical foundations of science.
    I liked everything, but the very last bit about this proof allowing us to "discover something about our universe" isn't accurate. Math is a method of describing our universe, and it can be predictive if the description is accurate. This doesn't tell us anything about an actual right triangle. For instance, if we measure the same triangle with different units, imperial rather than metric, sides A and B wouldn't be 1 and the hypotenuse would not be the square of 2, or an irrational number. The triangle hasn't changed, but the rationality of the hypotenuse has changed. The fact that it is irrational does not pertain to the triangle, or the universe itself, but to the quirks of the system doing the describing.

  • Boris Chan
    Boris Chan 4 days ago

    so every right angel triangles with equal length have a irrational side!

  • F54DF4 A044
    F54DF4 A044 4 days ago

    9:20 just realized they played elevator music in the background

  • Priyanshu Goel
    Priyanshu Goel 4 days ago

    So if 2 owns a square tree its root is not logical

  • Nobita Nobi
    Nobita Nobi 4 days ago

    my reaction to every other meme

  • Nobita Nobi
    Nobita Nobi 4 days ago +1

    4:45 when I knew I was better at math than this guy

    • Mustafa Rahmati
      Mustafa Rahmati 4 days ago

      We are the only ones who saw the mistake 13 is not 2x 7+1

  • BENZDIK 3.0
    BENZDIK 3.0 4 days ago

    Math = broken

  • Unless & UNtil
    Unless & UNtil 4 days ago

    You forget to greet v sause

  • Austin Chia
    Austin Chia 5 days ago

    So yeah this whole thing looked professional and he said -13 is an odd number because 2x-7+1 = -13


    • wispa
      wispa Day ago

      If you can write a whole number as 2 x [whole number] + 1 then it is odd.
      -13 = -14 + 1 = (2 x -7) + 1
      -7 is a whole number and so -13 is odd.

  • Patrik Schnell
    Patrik Schnell 5 days ago

    pythagoras is wrong! c is the speed of light! you cant go faster then a square with side length of 1! ! !! 11 !

  • CSquared Films
    CSquared Films 5 days ago

    *_I am 2._*
    *_Thank you Micheal, I learned something about myself today._*

  • Daniel Grimes
    Daniel Grimes 5 days ago

    Why is Michael a better math teacher than the teachers at my school.

  • sheky
    sheky 5 days ago

    you really like to use variable c : )

  • sheky
    sheky 5 days ago

    7777th comment!

  • sheky
    sheky 5 days ago


  • sheky
    sheky 5 days ago


  • Gaurav Gautam
    Gaurav Gautam 5 days ago

    16:00 Since an even number squared creates an even number b must be even? How? A => B and B is true. This doesn't mean A is true. Look up page 4 in 'Probability Theory by E. T. Jaynes. The book is available as a PDF online for free. You don't even have to buy it. Such a long video and you break the proof at the end.

    • Gaurav Gautam
      Gaurav Gautam 4 days ago

      Nice catch man! or woman! If you are a girl then you are so hot to me right now. But just to put what you said in another way- A: x is odd, B: x² is odd. A => B, and B = false. Therefore A = false. For me this is clearer than making the statement ~B => ~A. For me the contrapositive of implication is a bit non-intuitive.

    • Sky Wing
      Sky Wing 5 days ago

      He didn't break the proof, it seems that he mistakenly confirms the converse. However the gist of the proof is there. He proved at 5:33 that if a number x is even, then x² is also even, BUT he also proved that if a number x is odd then x² is also odd. So while x is even => x² is even does not means the converse is true. If you look at the second statement "if x odd then x² is odd". Its contrapositive would have said: " if x² is even, then x is even" is true since "a number is odd" is the negation of the statement "a number is even", which he should have stated that in the video.
      So yes you're right on "x is even => x² is even" doesn't mean "x² is even => x is even". However "x is odd => x² is odd" is equivalent to saying "x² is even => x is even", which he used.

  • Gaurav Gautam
    Gaurav Gautam 5 days ago

    a paper and a pencil? Really?

  • Carlos Sierra
    Carlos Sierra 5 days ago

    gotta say, DONG is a pretty sweet name for a channel

  • Fired Up Toast
    Fired Up Toast 5 days ago

    2:30 ok I understanf it so far
    3:40 WTF!?!?!?!

  • Айден Николс


  • SuperStratosfear
    SuperStratosfear 5 days ago

    I don't always do math, but when I do, I prefer Vsauce. Stay curious my friends

  • Jaraban
    Jaraban 5 days ago

    I'm in year 11 and this maths is the stuff I was doing in year 9, where's the juicy stuff michael

  • TheMatrix987
    TheMatrix987 5 days ago

    Guess the Vsauce channel became too mainstream..

  • ImSoEz
    ImSoEz 5 days ago

    what, (2c+1) squared is 4c +2... I'm confused as hell. 4c squared is 16c. The square root of 4c is 2c.
    Is this a joke, I'm confused and why is this in my recommended.

    • ImSoEz
      ImSoEz 3 days ago

      +MikeRosoftJH Yeah, I brainfarted this pretty badly xD Took the logic from the entire thing being multiplication, but the ^2 depends on the outcome of 2c.

    • MikeRosoftJH
      MikeRosoftJH 3 days ago

      Since when is (2c+1) squared equal to (4c+2)? Let c=5; is 11*11 equal to 22? Likewise, square of 4c is not 16c; it's 16*(c^2) or 16*c*c. Try c=2; the square of 4*2=8 is 8*8=64, which is not the same as 16*c=16*2=32.

  • [MF] Sprucey.-
    [MF] Sprucey.- 5 days ago


  • Spore Hux
    Spore Hux 5 days ago

    This really shows how sucky my math teacher was.

  • glwa j
    glwa j 6 days ago


    • MikeRosoftJH
      MikeRosoftJH 3 days ago

      This doesn't prove anything. You start with equation a^x+b^y=c^z. You multiply it by an arbitrary value, and (after some more mathematical operations) you get that (aF)^x+(bG)^y=(cH)^z, where F, G, and H are some mathematical formulae. You observe that each of F, G, and H were divisible by the same prime number. But that doesn't prove that the original a, b, and c weren't coprime! (And, as far as I can tell, you didn't even prove that F, G, and H are rational numbers.)
      I can do this as well. Let a^2+b^2=c^2. Multiply both sides by 4; we get 4*(a^2+b^2)=4*c^2. Equivalently, 4*a^2+4*b^2=4*c^2 or (2*a)^2+(2*b)^2=(2*c)^2. Therefore, a^2+b^2=c^2 can only be true if a, b, and c are all even. (Of course, the proof is invalid, as evidenced by the Pythagorean 3^2+4^2=5^2; it only proves that if a, b, and c are solutions to the equation, then so is 2*a, 2*b, and 2*c. In fact, the equation has infinitely many solutions; even more, it has infinitely many elementary solutions when a, b, and c do not share a prime factor.)

  • Davin B
    Davin B 6 days ago

    Gimmie ur DONG

  • Chen Levin
    Chen Levin 6 days ago

    This is why he's got so much subscribers...

  • Tumbash Tumbash
    Tumbash Tumbash 6 days ago

    Well you too no different! First 2-3 minutes were ok after that things got boring!😛😝

  • AlmightyTriggerer
    AlmightyTriggerer 6 days ago

    B i n o m i a l M u l t i p l i c a t i o n

  • Menos The Name
    Menos The Name 6 days ago

    2c^2 could be odd if c=0.51
    2c^2+1 could be even if c=0
    Although I have a hunch that c^0 won't be considered as 1 in the future.

    • wispa
      wispa 5 days ago

      2 x 0.51² = 0.5202 (and c has to be a whole number anyway).
      2 x 0² + 1 = 0 + 1 = 1

  • Roshni. S
    Roshni. S 6 days ago

    First of all awesome video but at 6:35 you could've used the formula (a+b)^2=a^2+2ab+b^2
    We had this proof at school as well

  • ReflectiveVids
    ReflectiveVids 7 days ago

    Your face is irrational...

  • Andrew Hawkins
    Andrew Hawkins 7 days ago

    Good ol' proof by contradiction. If some of you have a hard time with mathematics, look into Morris Kline. He has a book called Mathematics for the Non-Mathematician, as well as other books such as one that goes quite indepth into the Calculus.

  • Multikontroll
    Multikontroll 7 days ago

    You are kicking in open doors.

  • Marie BCFHS
    Marie BCFHS 7 days ago

    7:03 ain't that the first constant equation? (a+b)^2 = a^2+2ab+b^2

  • Karolyn Issup
    Karolyn Issup 7 days ago

    Does everything need to be proven to this extent? If one times one is one, and two times two is 4, then of course the square root of two is irrational. I'm just saying this because I was dozing off when I saw this on the side. I dreamed I sent you a letter that said "No Brainer." It made me wake up and laugh. LOL

    • wispa
      wispa 7 days ago

      That only shows that 1

    • Priest Plaxis
      Priest Plaxis 7 days ago

      Karolyn Issup How is it obvious though? Sqrt(2.25) is is rational even though it lies between 1 and 4, and so is sqrt(0.0625). But then again, if you can prove that if p is a non-square integer, then sqrt(p) is irrationel, then it’s of course obvious.

  • umeng2002
    umeng2002 7 days ago

    starts @13:06

  • Shawn Jaison
    Shawn Jaison 7 days ago

    0:05 that's not how you write the number 8

  • Bodie Buckby
    Bodie Buckby 7 days ago


  • Taylor Gibson
    Taylor Gibson 8 days ago

    Mathematicians aren't just scientists anymore. They're philosophers.

  • ManWithBeard1990
    ManWithBeard1990 8 days ago

    Quite a cool proof. It appears to me as well that this can be generalized to all prime numbers and not just 2, correct?

    • Priest Plaxis
      Priest Plaxis 7 days ago

      ManWithBeard1990 Yes it can be generalized as long as in the original proof, you don’t think it as even, but just as divisible by 2, which can then be generalized to all primes, which can then be adjusted a bit to prove it for all nonsquare integers.

    • ManWithBeard1990
      ManWithBeard1990 8 days ago

      +wispa Do we need that though because that too can be generalized to divisible/not divisible by that prime number. Not in the same way but if a number does not contain a certain prime factor, neither will its square.

    • wispa
      wispa 8 days ago

      Unfortunately, this proof only works with 2 (because we need all the odd/even stuff from earlier in the video).
      However, the proof on this page is applicable to all prime numbers (but it's a bit less elegant): proofwiki.org/wiki/Square_Root_of_Prime_is_Irrational

  • Lewisstrikes Gaming channel

    Where's vsauce

  • Julie Stenberg Andersen

    me doing my best to tutor my friends so that they'll understand

  • mackan919
    mackan919 8 days ago


  • Swaggerlord2.0 21
    Swaggerlord2.0 21 8 days ago

    Isn’t infinity even, since the number before is infinity-1 which makes infinity=infinity-0 and since zero is even so is infinity

    • looney1023
      looney1023 5 days ago

      Infinity is not an integer, so the concept of even or odd has no meaning in describing it.
      Also, and this is kind of a calculus rabbit hole, but infinity - 1 = infinity

    • Priest Plaxis
      Priest Plaxis 7 days ago

      Swaggerlord2.0 21 There’s sadly no number before infinity though.

  • Aero
    Aero 8 days ago

    Thanks vsauce I was able to good score for my midterm proofs understand this as a simple concept helps

  • CoOl GaMeR BEST
    CoOl GaMeR BEST 8 days ago

    This topic is in my first chapter maths....

  • Filip Gaczkowski
    Filip Gaczkowski 8 days ago

    comic sans...

  • Charles Mcbain
    Charles Mcbain 8 days ago

    This was done on a phone

  • i like musicals
    i like musicals 9 days ago

    vsauce is ruined

  • Kevin F
    Kevin F 9 days ago

    Wouldn't an odd number be expressed as being equal to 2c ± 1? Rather than just plus 1?
    When squaring both conditions, the eqn distills to:
    Odd = 2(2c^2 ± 2c) + 1

    • wispa
      wispa 8 days ago

      An odd number can be expressed as 2c plus any odd number. You can use 2c+1, or 2c-1, or 2c-385, or 2c+4,663,207.
      +1 just tends to be used for simplicity.

  • TheSam54123
    TheSam54123 9 days ago


  • fauzi hassan
    fauzi hassan 9 days ago

    Yowzer ...what a geniuses ....

  • MuandDib
    MuandDib 9 days ago

    what about complex numbers? :o

    • wispa
      wispa 8 days ago

      Rationality/Irrationality is something that only applies to real numbers

  • Logan Byrne
    Logan Byrne 9 days ago

    The answer is 1