Proof: Square Root of 3 is Irrational

We prove the square root of 3 is irrational. Proving some numbers are irrational is a real pain, but it doesn't always have to be so hard! To prove sqrt(3) is irrational, we can use the proof by contradiction strategy famously used to prove the square root of 2 is irrational. We'll assume that the square root of 3 is NOT irrational, and thus is rational. Then we'll write it as a ratio of integers in fully reduced form, and find a contradiction!

★DONATE★

Thanks to Robert Rennie, Barbara Sharrock, and Lyndon for their generous support on Patreon!

Follow Wrath of Math on...

[BenjiShock]

I read this proof in "Understanding Analysis" and he did not mention anything about Euclids Lemma so I was confused. Apparantly Euclids Lemma is something to be taken for granted for Mathematicians but for myself it was actually the most important part of the proof. Thanks so much for explaining it in full length! And I also learned about Euclids Lemma at the same time!

[mikeychrisanthus]

So, this also proves that the square root of any integer that isn’t a perfect square is irrational. Wow.

[jaya.]

The production quality is amazing and you explain so clearly, keep it up!

[tewahedootube]

Damn u sound like Sheldon fr
Good explanation tho

[JLovesDisney]

My math teacher is great, but she had about 30 seconds to explain this so your video came in very handy. Thank you, it's a neat proof

[Xebtria]

Does that mean, that this proof by contradiction basically works for every square root of n if n is a prime number? Because we always will get to the the same exact point that a and b must be integer multiples of n, contradicting the original assumption of having a gcd of one, and therefore all square roots of primes must be irrational.

[punditgi]

Very well explained! So, can you show us the proof that the square root of any prime number is irrational? We have to use the Fundamental Theorem of Arithmetic?

[rossk]

Hey, I like this explanation but it’s bugging me that it seems if you plug root 4 instead of root 3 at the beginning, you will end up at the same result; 4 divides a and divides b, so gcd assumption is violated, so root4 is irrational(which it’s clearly not). Am I messing up somewhere?

Edit: I was missing Euclids lemma, which means you can’t get 4 divides a from 4 divides a^2 bc 4 isn’t prime(example a^2=36, a=6)

[jussef]

Right now was the perfect time for me to watch this video. Pre christmas vibes are kicking in and I have to do exactly this prove for my discrete mathematics class. Thanks a lot!

[vinodb]

Kindly clarify if 3 divides p square then p divides p also

[abdullahshaikh]

Sir! it was very well explained, we have a request can you explain that (2+2 root 5) is an irrational number

[GomezUnplugged]

Man u r soo god I watched 7 videos but only now u made me understand thaks

[Chimangonyirenda-os]

Clarify where you explained about 3k for some integer k

[timwhite]

You didn't need to go to all this trouble as I would have taken your word for it...

[nayefmsh]

This video actually felt fun listening to

[kuei]

Thanks for the explanation, but I have a question.I think this proof is a little be weird since you can also prove that square root of 4 9 16 25... is irrational with the very same technique, which is obviously wrong.

[mudarisrinibas]

It would be easy enough if we use rational zeros law.

[noobchickensupper]

Without assuming at first a and b are co prime, can we say at the end that since a and b are always in the form divisible by 3, they are not integers as the ratio of two integers can always be reduced to the ratio of two co prime numbers.

[madeleinezahn]

Great video! Youtube is what's carrying me through my discrete math class this quarter 😂

[lipichandra]

Could you explain polynomial and trigonometry topics as well? 🥺🥺