# A Proof That The Square Root of Two Is Irrational

Share
Embed
• Published on Oct 13, 2018
• Like my shirt? Well, you already own it if you're a Curiosity Box subscriber. If you're not, don't miss out again! Get exclusive math and science stuff from me delivered right to your door four times a year: www.curiositybox.com
Twitter: twitter.com/DoOnlineNowGuys
Instagram: instagr.am/doonlinenowguys

Vsauce PO Box: PO Box 33168
L.A. CA 90033

***CREDITS***
Hosted by
Micorn Stevens
Edited by
Hannah Canetti
Curiosity Box VFX by
Eric Langlay
(thexvid.com/user/ericlanglay)
Music by Jake Chudnow and from AudioNetwork
(thexvid.com/user/jakechudnow)
(www.audionetwork.com)
***VSAUCE LINKS***
Vsauce1: thexvid.com/user/vsauce1
Vsauce2: thexvid.com/user/vsauce2
Vsauce3: thexvid.com/user/vsauce3
• Entertainment

## Comments • 9 122

• joshie oshie 14 minutes ago

My fear of gimps is erational.

• LosingLuke Hour ago

“A proof”

• LosingLuke 2 hours ago

**writes 8 as 2 0’s**

• Park Justin 8 hours ago

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

• Veteran Mainiac 10 hours ago

Draws an 8
*ooooh man*
Draws 0.8
*realeases*

• Aisha 11 hours ago

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

• 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.

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

• omarPIMP DADDY Day ago

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

• Greyson Miller Day ago

*Y**i*kE*s*

• ZacPlayzGamer Day ago

I just learned the Pythagorean Theorem in pre al today XD

• Crossdownstreet Day ago

lol just use a calculator

• WiktorRapBattle Day ago

YIKES

• jgms 2 days ago

*YAAOZAS*

• 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.😂😂

• Michael should teach math more often

• Elijah G 3 days ago +10

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

• Evan Guzman 3 days ago

👀
👃🏻
👄
👕
👖
👞
This is Bob.
Bob said hi.
This is Bob when the bus goes by.
XX
👃🏻
👄
👕
👖
👞

• Jye-Ming Serres 3 days ago

yiikkkesss

• βΣΨΩΠD RΣΔSΩΠ 3 days ago +1

Be my math teacher.

• Dude makes maths interesting

• Fancy in the House 3 days ago

4:30
-12+1 is -11 not -13

• Mariano Day ago

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

• qimaru 3 days ago

a proof?

• Pyrosmantia 3 days ago

-12+1=/=-13

• Mariano Day ago

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

• Bijoy Lakshmi 4 days ago

Thanks

• Mi 28 4 days ago +11

Dong status: fully expanded.

• The maths in this video are literally primary school stuff

• hey7328 2 days ago

most people don't even know primary school maths

• IamSaki 4 days ago

*YAAWZAAA*

• Dalton Freeman 4 days ago

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

• unlucky7rusty boy 4 days ago

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

• Christine Gago 4 days ago

Wait what? I lost you at 0:00

• *Y O W Z A S*

• dom cusano 4 days ago

I like how perfectly he draws his 8s

• 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
Sorry.

• JOY IN RANDOM 5 days ago +9

How long did you spend trying to find a loop hole

• Arpit Soni 5 days ago +1

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

• mychemicaljulia 5 days ago

why did i take functions

• foil 5 days ago

oh hey, apparently im a math term now aha

• Dv3v 5 days ago

i wish you were my math teacher

• Bladed Sinicle 5 days ago

Dink donk

• Fire Oblivious 5 days ago +2

Why is his room so dark?

• Not Gay 4 days ago

Oblivious 606 lighting

• MontyTheGod 5 days ago

“Yikes”

• Kaan Soytaşoğlu 5 days ago

17 minutes of nothing

• MMGeri 5 days ago

this is actually recquired for graduation in my country

• Sophia Ziegler 5 days ago

4:51 *-6*

• fulcrum 29 5 days ago

I am more confused more than ever

• V E K T O R 5 days ago

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

• 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 6 days ago

who's Micorn Stevens?

• Cory Baxter 6 days ago

1.41421356237 x 1.41421356237 = 2

• 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 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 6 days ago

literally my intro to discrete math class

• 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 6 days ago

Hey, DONG! Michael here.

• 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 6 days ago +1

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

• UY Scuti 6 days ago

Its not hard to believe

• MeTheGuy 12 6 days ago +2

Michael was losing an argument so he made this video

• Kızıl Link 6 days ago

0.12 two of them is 4 lmao

• 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 7 days ago

1 is an “awd” number -Michael

• 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 2 days ago

aaand you missed the point.

• TheGamingOstrich 7 days ago

I should be in bed rn

• Bob The Boss Saget 7 days ago

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

• Alec Ray 7 days ago

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

• Reid LeBlanc 7 days ago

oh boy binomial multplication

• Glumbor Tango 7 days ago

y o w z a !

• I want die 7 days ago

11:22

• 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 4 days ago

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

• I want die 7 days ago

I watched it, you happy yet youtube?

• Peter Griffin 7 days ago +6

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

• TheSmartyPants 7 days ago

1st chapter grade 9 ncert CBSE India 🤦‍♂

• 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 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 8 days ago +12

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

• Ger - 3 days ago

It’s like the rolling shutter in the distortions video he made
thexvid.com/video/mQ0hS7l9ckY/video.html

• Death Gaming 8 days ago +2

I had to prove this for my Discrete Maths class

• Ailesh Amatya 8 days ago +3

why do i feel like this is like Vsauce

• Nick Falk 8 days ago

THOSE AREN'T EIGHTS THEY'RE SNOWMEN! STOP IT!!

• Igor Matias 8 days ago

This gave me goosebumps.

• Ádám Győrfi 8 days ago

Love the way of explaination;)

• Juliane Lust 8 days ago +1

Why should the square root of 2 be rational?

• I want die 7 days ago

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

• lil volu 8 days ago

YIKES

• James Cabrera 9 days ago

DISCREET MATH

• 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 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 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 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 8 days ago

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

• 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 9 days ago +1

Anyone Else noticed how he write an 8?😂

• Adam Hunt 9 days ago

Quite a fun proof, did it in maths

• Ahmad Kashmar 10 days ago

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

• 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 8 days ago

Mariano Oh... oops
My bad

• 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 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 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 9 days ago

For instance, 2/2 = 1.

• 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 10 days ago +1

YAUZAS!

• Gawain_Goat 10 days ago

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

• corcorwhite 10 days ago

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

• PatDoesThings - Games 10 days ago

May I ask, who tf cares