A Proof That The Square Root of Two Is Irrational

  • Published on Oct 13, 2018
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Comments • 9 122

  • joshie oshie
    joshie oshie 14 minutes ago

    My fear of gimps is erational.

  • LosingLuke
    LosingLuke Hour ago

    “A proof”

  • LosingLuke
    LosingLuke 2 hours ago

    **writes 8 as 2 0’s**

  • Park Justin
    Park Justin 8 hours ago

    Because they will always be even even after factoring out 2. That makes it go on forever.

  • Veteran Mainiac
    Veteran Mainiac 10 hours ago

    Draws an 8
    *ooooh man*
    Draws 0.8

  • Aisha
    Aisha 11 hours ago

    The big bubble numbers and clean writing is so soothing to my eyes

  • wkf1114
    wkf1114 12 hours ago

    I think you probably need to add more explanations to be more comprehensive. In the first half part, you proved an even number's square is still an even number. However, you didn't prove the reverse case. Which means, the square root of an even number has to be an even number, or integer (such as 2, 6, 8, 10, 12, 14 etc.). Thus, at 14:36, square of "a" is equal to two times square of "b", which is an even number. But, you need some discussions that "a" could be a not-integer or an even number. Both cases violate the assumption of "a" and "b" are coprime integers.

  • Jimin Is A Living Mochi

    I don’t understand a word he’s saying but this is fascinating

  • omarPIMP DADDY
    omarPIMP DADDY Day ago

    This is all the math I learned in my life in 17 minutes

  • Greyson Miller
    Greyson Miller Day ago


  • ZacPlayzGamer
    ZacPlayzGamer Day ago

    I just learned the Pythagorean Theorem in pre al today XD

  • Crossdownstreet
    Crossdownstreet Day ago

    lol just use a calculator

  • WiktorRapBattle
    WiktorRapBattle Day ago


  • jgms
    jgms 2 days ago


  • Andrew PLas
    Andrew PLas 2 days ago

    PLUS OR MINUS the square root of two

    GAMING ST 2 days ago

    1 min problem extended to 20 min and I was still enjoying that.😂😂

  • Superbrains Video Game Lord

    Michael should teach math more often

  • Elijah G
    Elijah G 3 days ago +10

    2:32 he didn't use the true Vsause music this is an imposter

  • Evan Guzman
    Evan Guzman 3 days ago

    This is Bob.
    Bob said hi.
    This is Bob when the bus goes by.

  • Jye-Ming Serres
    Jye-Ming Serres 3 days ago


    βΣΨΩΠD RΣΔSΩΠ 3 days ago +1

    Be my math teacher.

  • Gerrit Schimmelpenninkc Mosse

    Dude makes maths interesting

  • Fancy in the House
    Fancy in the House 3 days ago

    -12+1 is -11 not -13

    • Mariano
      Mariano Day ago

      Yes but 7*-2 is-14 not -12

  • qimaru
    qimaru 3 days ago

    a proof?

  • Pyrosmantia
    Pyrosmantia 3 days ago


    • Mariano
      Mariano Day ago

      Why -12 tho, 2*-7 is -14

  • Bijoy Lakshmi
    Bijoy Lakshmi 4 days ago


  • Mi 28
    Mi 28 4 days ago +11

    Dong status: fully expanded.

  • Konstantinos Koufakis

    The maths in this video are literally primary school stuff

    • hey7328
      hey7328 2 days ago

      most people don't even know primary school maths

  • IamSaki
    IamSaki 4 days ago


  • Dalton Freeman
    Dalton Freeman 4 days ago

    please tell me there's a "Yeowzahs!" gif somewhere

  • unlucky7rusty boy
    unlucky7rusty boy 4 days ago

    Wait is every fraction technically one if you simplify it enough.

  • Christine Gago
    Christine Gago 4 days ago

    Wait what? I lost you at 0:00

  • Cabecool10 / Skelemaster767

    *Y O W Z A S*

  • dom cusano
    dom cusano 4 days ago

    I like how perfectly he draws his 8s

  • yoyoman
    yoyoman 5 days ago

    7:16 😂😂yausaaaa look at this results😂😂

    Rjfh ubfofhfuxifohfhfhfjfjghdih. Kxgcj jckfb jpcocjoccoh jvhxk np n nlvncochovjco b nckcn oj hlccgcpnvnlconvnlcohcjfbfohvnovhclhch jp lhfocj j jokfgfoh ojvjlcoh jdldjcpjdhfpjvj j k

    JOY IN RANDOM 5 days ago +9

    How long did you spend trying to find a loop hole

  • Arpit Soni
    Arpit Soni 5 days ago +1

    I am an Indian and I already studied it in my 10 standard

  • mychemicaljulia
    mychemicaljulia 5 days ago

    why did i take functions

  • foil
    foil 5 days ago

    oh hey, apparently im a math term now aha

  • Dv3v
    Dv3v 5 days ago

    i wish you were my math teacher

  • Bladed Sinicle
    Bladed Sinicle 5 days ago

    Dink donk

  • Fire Oblivious
    Fire Oblivious 5 days ago +2

    Why is his room so dark?

    • Not Gay
      Not Gay 4 days ago

      Oblivious 606 lighting

  • MontyTheGod
    MontyTheGod 5 days ago


  • Kaan Soytaşoğlu
    Kaan Soytaşoğlu 5 days ago

    17 minutes of nothing

  • MMGeri
    MMGeri 5 days ago

    this is actually recquired for graduation in my country

  • Sophia Ziegler
    Sophia Ziegler 5 days ago

    4:51 *-6*

  • fulcrum 29
    fulcrum 29 5 days ago

    I am more confused more than ever

  • V E K T O R
    V E K T O R 5 days ago

    Dy Michael!!
    I drew 8 in my childhood just like you!!!

  • IB
    IB 5 days ago

    We can prove that √2 is irrational by assuming that √2 is rational and and √2 = a/b where , a and b are +ve integer and co-prime number easy pisy 😋

  • Cory Baxter
    Cory Baxter 6 days ago

    who's Micorn Stevens?

  • Cory Baxter
    Cory Baxter 6 days ago

    1.41421356237 x 1.41421356237 = 2

  • Dude Mc Broham
    Dude Mc Broham 6 days ago

    Ive never been so invested in proofs before as i was just now


    isn't this proof supposed to be simple

  • Trying2 Understand
    Trying2 Understand 6 days ago

    I would love to see the proof of :
    p/(p-1) = 1/p^0 + 1/p^1 + 1/p^2 + 1/p^3... for all p>1

  • Alterate
    Alterate 6 days ago

    literally my intro to discrete math class

  • ranimations
    ranimations 6 days ago +1

    I just adore how you, when you were ubderlining even and odd multiplications at just before 17:21 you underlined the even part twice (even) and the odd once(odd)

  • Shae Perkins
    Shae Perkins 6 days ago

    Hey, DONG! Michael here.

  • Arhum Mudassir
    Arhum Mudassir 6 days ago

    I had a good understanding of math but now I have a lot of questions about the universe

  • Noel Crist
    Noel Crist 6 days ago +1

    If you divide 2 by 2 which are both even then it leads to a odd 1.

  • UY Scuti
    UY Scuti 6 days ago

    Its not hard to believe

  • MeTheGuy 12
    MeTheGuy 12 6 days ago +2

    Michael was losing an argument so he made this video

  • Kızıl Link
    Kızıl Link 6 days ago

    0.12 two of them is 4 lmao

  • nithin raaja
    nithin raaja 7 days ago

    why can't he use the identity (a+b)^2=a^2+2ab+b^2

    DREAMER SINCE 19XX 7 days ago

    if 2c+1 is odd...then what about 2c-1

  • Blake McConnell
    Blake McConnell 7 days ago

    1 is an “awd” number -Michael

  • TotallyHorsed
    TotallyHorsed 7 days ago +3

    you can't discover anything in mathematics without looking "at the universe itself".
    Your brain exists in the universe already, so by extension the thoughts within the brain are within the universe. This is the universe discovering itself through your brain.

    • hey7328
      hey7328 2 days ago

      aaand you missed the point.

  • TheGamingOstrich
    TheGamingOstrich 7 days ago

    I should be in bed rn

  • Bob The Boss Saget
    Bob The Boss Saget 7 days ago

    Kickin it w vsause doin up them algebraic proofs like some savages.

  • Alec Ray
    Alec Ray 7 days ago

    I learned more in this video than I ever did in Algebra of Calculus

  • Reid LeBlanc
    Reid LeBlanc 7 days ago

    oh boy binomial multplication

  • Glumbor Tango
    Glumbor Tango 7 days ago

    y o w z a !

  • I want die
    I want die 7 days ago


  • Fudmottin
    Fudmottin 7 days ago +7

    √2 was known to be irrational prior to algebra. So there should be a geometric proof. Can you construct one?

    • Cyberplayer5
      Cyberplayer5 4 days ago

      thexvid.com/video/yk6wbvNPZW0/video.html I think this video show how to do that.

  • I want die
    I want die 7 days ago

    I watched it, you happy yet youtube?

  • Peter Griffin
    Peter Griffin 7 days ago +6

    This did not help me for my tomorrow's maths exam

  • TheSmartyPants
    TheSmartyPants 7 days ago

    1st chapter grade 9 ncert CBSE India 🤦‍♂

  • Brad
    Brad 8 days ago

    I thought this was going to be some new proof, but nope. Just the same proof used for thousands of years.

  • Jerakk30
    Jerakk30 8 days ago

    My favorite number is 27. Why is 27 my favorite number? Because of some beautiful proofs about it. 27 is the cube root of 19683. The sum of the digits in 19683 equals 27. Both numbers are divisible by 9 which, without dividing, can be proven because the final sum of all numbers in both sets equals 9. Any number number multiplied by 9 has an answer where the final sum of that answer is always 9.

  • Jeff Sandt
    Jeff Sandt 8 days ago +12

    That was a strange effect when he says yikes at 0:13

    • Ger -
      Ger - 3 days ago

      It’s like the rolling shutter in the distortions video he made

  • Death Gaming
    Death Gaming 8 days ago +2

    I had to prove this for my Discrete Maths class

  • Ailesh Amatya
    Ailesh Amatya 8 days ago +3

    why do i feel like this is like Vsauce

  • Nick Falk
    Nick Falk 8 days ago


  • Igor Matias
    Igor Matias 8 days ago

    This gave me goosebumps.

  • Ádám Győrfi
    Ádám Győrfi 8 days ago

    Love the way of explaination;)

  • Juliane Lust
    Juliane Lust 8 days ago +1

    Why should the square root of 2 be rational?

    • I want die
      I want die 7 days ago

      This video just proves it, because like he says, we can't assume.

  • lil volu
    lil volu 8 days ago


  • James Cabrera
    James Cabrera 9 days ago


  • David Cely
    David Cely 9 days ago

    I think the title is wrong. He proved that square root of two is not rational. To proof that it is irrational is harder, it has something to do with the least-upper-bound axiom. See Apostol Calculus 2ed (section I3.13)

    • Paul 123098
      Paul 123098 8 days ago

      Nope, it's a valid proof. If we talk about only the Reals as the metric space, proving square root of two is not rational proves that it is in fact irrational .

  • Edits JAG
    Edits JAG 9 days ago +6

    I learned this on sophomore year of high school when reading principles of mathematical analysis by walter rudin, I wish that I could have seen this video instead of reading the proof, which was very confusing

  • zoiyoknd
    zoiyoknd 9 days ago

    Dong, but if you apply same to sqr(4) it may seem it is irrational. I think you need first to proof sqr(2) is less than 2 and greater than 1.

    • Mariano
      Mariano 8 days ago

      Not really, you can apply the same line of reasoning to sqrt(4) and you will never reach a contradiction

  • thesnusmumric
    thesnusmumric 9 days ago

    Oh come on! if A is even then A^2 is even - was prove. But if A^2 is even then A is even - actually was not prove, but was used

  • möhre möhre
    möhre möhre 9 days ago +1

    Anyone Else noticed how he write an 8?😂

  • Adam Hunt
    Adam Hunt 9 days ago

    Quite a fun proof, did it in maths

  • Ahmad Kashmar
    Ahmad Kashmar 10 days ago

    Seriously, why don't you go and prove it yourself instead of watch this?

  • Vincent Li
    Vincent Li 10 days ago

    But even numbers like 2 can be expressed as 2/1(And because 2 is definitely rational), so when a/b=c and c is even, you can express it as c/1. I mean, you don’t have to go on forever, if you take the actual answer of the square root of 2, it is possible to express it as a fraction, therefore it is rational.

    • Vincent Li
      Vincent Li 8 days ago

      Mariano Oh... oops
      My bad

    • Mariano
      Mariano 8 days ago

      +Vincent Li He never said anything even close to "the square root of 2 is even", you must have heavily missunderstood some part of the video

    • Vincent Li
      Vincent Li 8 days ago

      Jodie Waits I’m talking about the square root of 2 here, which according to Michael is even. In fact, natural numbers, integers, and fractions are all rational. So if the square root of 2 is even, the fraction is simply n/1. That’s rational.

    • Jodie Waits
      Jodie Waits 9 days ago

      Also, even/odd are concepts that only apply to integers. You'd have to prove that a/b reduces to an integer before you can call a/b = c is either even or odd to begin with.

    • Jodie Waits
      Jodie Waits 9 days ago

      For instance, 2/2 = 1.

  • Pablo Longoria
    Pablo Longoria 10 days ago

    Watching this video gave me flashbacks to middle school, when my teacher used to get mad with me on tutoring because I didn’t understood anything of what she was saying.

  • Gentle Path
    Gentle Path 10 days ago +1


  • Gawain_Goat
    Gawain_Goat 10 days ago

    I am now going to use this as a warm up before math tests.

  • corcorwhite
    corcorwhite 10 days ago

    Bro I cannot get over how you write your 8s lol

  • PatDoesThings - Games
    PatDoesThings - Games 10 days ago

    May I ask, who tf cares