# A Proof That The Square Root of Two Is Irrational

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Alex Goedken5 hours ago*YIKES*

Nod Boss8 hours ago^{+1}4:50 check that math

Michael Sal12 hours agoI 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 night21 hour ago^{+1}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 GroupDay agothanks for ur effort

quinn dalletDay agoOOOOOOOOOOOOH BOOTIFUL 8

almost better then hamster pics on Google images...

thequinnman325Day agoOOOOOOOOOH THAT HANDWRITING

UNGH

EpicDutyDay agoI love binomische Formeln like we call it in Germany

EpicDutyDay agoI 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 LogixDay agoMichael 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 Robertson2 days ago16:32 Yeah, well, neither am I!

Melker Jonsson 9B2 days ago4:50 wrong. -12 + 1= -11

Priest Plaxis13 hours agoMelker Jonsson 9B 2 times negative seven

Bixls2 days ago11:32 taught me how to teach simplifying

KoishiKomeiji112 days agoi really like michael's handwriting

stan Daesung2 days agoi love michael so much im crying ????

Suffek2 days ago*Yowza!*

lost2 days agooutro music?

Kailash Ramakrishnan2 days agoOr 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

wispaDay agoThat is exactly the proof that Michael did.

Dashua2 days agoRationalism is strong in this one

gumikebbap2 days agoI 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 :)

Nofxthepirate2 days agoI could honestly watch Michael explain mathematical proofs for hours. Why is this the only one!

AbyssWalker240 Gaming2 days agothe square root of 2 is 1.4142135623730951 and thats rational. put it through a calculator.

Tony Turtle2 days agolol he said 13 = 2 x 7 + 1 but 2 x 7 = 14 and 14 + 1 = 15. Michael forgot how to do simple math.

Priest Plaxis13 hours agoTony Turtle two times negative 7 plus 1 is negative 13

Someoneawesome123 Plays2 days ago^{+1}YOWZAZ

Explosive Kitten2 days agoWanna tutor me in math when I start college? xD

Airon Jones3 days agoyowzie

Divyansh Singhal3 days agoI did it when I used to study in 3rd class. I was damn better than him...

Gus TV3 days ago2b^2 or not 2b^2

Ricardo Guzman3 days ago(√2+1)^n approaches a whole number value as the integer n increases. That is nuts.

Ricardo GuzmanDay ago+wispa Yes. But for each integer n, it gets closer and closer to a whole number.

Examples---

when n=12, the answer is. 39201.999974491... when n=16, the answer is

1331713.999999249088....

when n=24 the answer is

1536796801.9999999993492...

and so on to infinity.

wispaDay ago???

(√2+1)^n → ∞ as n → ∞

tabaks3 days agoMichael, that's a long DONG you had here. How about Vsauce? They fired you?

郭维佳3 days agonobody gonna talk abt the way michael writes 8?

KingJacobz3 days agoHe said -13 is 2 x -7 instead of -6 + 1. Can he not do retakes????? :) time stamp 4:49

wispaDay ago2 x -7 = -14

-14 + 1 = -13

Jill Calkins3 days agoHosted by MICORN Stevens?

Forbidden Lusts3 days agoWhat is this gibberish

yahya pandit4 days agoVsauce gets orgasm from math

Tobe Shinji4 days ago^{+1}Oh god no. I may have a irrational fear for ratios now.

Collin Flintjer4 days agohow would or could u justify 9 being even if u would assume it's divisible by 4.5?

wispaDay ago9 is not even. Even numbers are whole numbers which when divided by 2 leave no remainder.

Fire Shark4 days agoWell 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 ALLIOUI4 days agoYou 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 !!

LiamLikesGames4 days agoI sometimes watch these in case I ever get that pill from limitless.

Mr. Cub Fan 4154 days agoIs zero an even number?

wispaDay agoYep!

The Keb4 days agoMy brain almost got shut down almost 2c X 1 times now while watching the video because my brain never felt this.

Perry Schafer4 days agoYeah, 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 Chan4 days agoso every right angel triangles with equal length have a irrational side!

F54DF4 A0444 days ago9:20 just realized they played elevator music in the background

Priyanshu Goel4 days agoSo if 2 owns a square tree its root is not logical

Nobita Nobi4 days agomy reaction to every other meme

Nobita Nobi4 days ago^{+1}4:45 when I knew I was better at math than this guy

Mustafa Rahmati4 days agoWe are the only ones who saw the mistake 13 is not 2x 7+1

BENZDIK 3.04 days agoMath = broken

Unless & UNtil4 days agoYou forget to greet v sause

Austin Chia5 days agoSo yeah this whole thing looked professional and he said -13 is an odd number because 2x-7+1 = -13

Wut

wispaDay agoIf 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 Schnell5 days agopythagoras is wrong! c is the speed of light! you cant go faster then a square with side length of 1! ! !! 11 !

CSquared Films5 days ago*_2._*

*_I am 2._*

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

Daniel Grimes5 days agoWhy is Michael a better math teacher than the teachers at my school.

sheky5 days agoyou really like to use variable c : )

sheky5 days ago7777th comment!

sheky5 days ago7776

sheky5 days ago7775

Gaurav Gautam5 days ago16:00 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 Gautam4 days agoNice 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 Wing5 days agoHe 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 Gautam5 days agoa paper and a pencil? Really?

Carlos Sierra5 days agogotta say, DONG is a pretty sweet name for a channel

Fired Up Toast5 days ago2:30 ok I understanf it so far

3:40 WTF!?!?!?!

Айден Николс5 days agoYOWZAH!

SuperStratosfear5 days agoI don't always do math, but when I do, I prefer Vsauce. Stay curious my friends

Jaraban5 days agoI'm in year 11 and this maths is the stuff I was doing in year 9, where's the juicy stuff michael

TheMatrix9875 days agoGuess the Vsauce channel became too mainstream..

ImSoEz5 days agowhat, (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.

ImSoEz3 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.

MikeRosoftJH3 days agoSince 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.-5 days agoDONG

Spore Hux5 days agoThis really shows how sucky my math teacher was.

glwa j6 days agowww.elakiri.com/forum/showthread.php?t=1858587&highlight=beal+conjecture

MikeRosoftJH3 days agoThis 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 B6 days agoGimmie ur DONG

Chen Levin6 days agoThis is why he's got so much subscribers...

Tumbash Tumbash6 days agoWell you too no different! First 2-3 minutes were ok after that things got boring!😛😝

AlmightyTriggerer6 days agoB i n o m i a l M u l t i p l i c a t i o n

Menos The Name6 days ago2c^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.

wispa5 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. S6 days agoFirst 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

ReflectiveVids7 days agoYour face is irrational...

Andrew Hawkins7 days agoGood 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.

Multikontroll7 days agoYou are kicking in open doors.

Marie BCFHS7 days ago7:03 ain't that the first constant equation? (a+b)^2 = a^2+2ab+b^2

Karolyn Issup7 days agoDoes 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

wispa7 days agoThat only shows that 1

Priest Plaxis7 days agoKarolyn 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.

umeng20027 days agostarts @13:06

Shawn Jaison7 days ago0:05 that's not how you write the number 8

Bodie Buckby7 days ago*YIKES*

Taylor Gibson8 days agoMathematicians aren't just scientists anymore. They're philosophers.

ManWithBeard19908 days agoQuite a cool proof. It appears to me as well that this can be generalized to all prime numbers and not just 2, correct?

Priest Plaxis7 days agoManWithBeard1990 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.

ManWithBeard19908 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.

wispa8 days agoUnfortunately, 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 channel8 days agoWhere's vsauce

Julie Stenberg Andersen8 days agome doing my best to tutor my friends so that they'll understand

mackan9198 days agoYIKES

Swaggerlord2.0 218 days agoIsn’t infinity even, since the number before is infinity-1 which makes infinity=infinity-0 and since zero is even so is infinity

looney10235 days agoInfinity 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 Plaxis7 days agoSwaggerlord2.0 21 There’s sadly no number before infinity though.

Aero8 days agoThanks vsauce I was able to good score for my midterm proofs understand this as a simple concept helps

CoOl GaMeR BEST8 days agoThis topic is in my first chapter maths....

Filip Gaczkowski8 days ago11:57

comic sans...

Charles Mcbain8 days ago1.4141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141414141..

This was done on a phone

i like musicals9 days agovsauce is ruined

Kevin F9 days agoWouldn'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

wispa8 days agoAn 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.

TheSam541239 days ago11:22 YOWWWWZAAAS

fauzi hassan9 days agoYowzer ...what a geniuses ....

MuandDib9 days agowhat about complex numbers? :o

wispa8 days agoRationality/Irrationality is something that only applies to real numbers

Logan Byrne9 days agoThe answer is 1