Proof: Square Root of 2 is Irrational

In this math lesson we go over a nice and easy proof that the square root of 2 is irrational. We suppose for the sake of contradiction that the square root of 2 is rational and we write it as a fraction in lowest terms. From here, we can prove that if the square root of 2 does in fact equal this fraction, then both the numerator and denominator are even. Thus, it contradicts our original assumption that the fraction is in lowest terms (and thus, they have no common factors and hence cannot both be even), thus our original assumption must be wrong and therefore the opposite must be true. So the square root of 2 is indeed irrational.

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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[CAMOGOD]

Proof so good it got Hippasus killed by Pythagoras himself

[celeste]

Fantastic proof. I also really appreciate how you replied to a lot of the questions in the comments. If it wasn't for your replies, I wouldn't have understood it. Thanks a ton!

[minsapint]

I understood that. So either it was a very clear and lucid explanation or I am a genius. I am going to go with the latter. Seriously though, great video.

[asenazaleas]

I don't understand, aren't we assuming two things:-
1. Square root of 2 can be expressed as a ratio of two integers, a and b
2. The ratio is in lowest terms

Why does it have to be in lowest terms, wouldn't the number still remain the same even if the fraction is not in lowest terms?

[jamikojonen]

Hello!
Is it generalizing things too much if I just thought that if a^2/b^2 = 2 is true, then the absolute value of a has to be exactly 2 times the absolute value of b for the fraction to be two.

So then we have 4b^2=2b^2 which is false.

Your proof is great, leaves nothing unexplained but I suppose I was on the right tracks at least, I kind of proved that they must have a common factor of 2 but I was not clever enough to think of it that way!

I will definitely watch more of your stuff and subscribe, my studies are not that math heavy but I suck at it since I didn't take high school seriously so I am trying to learn some proof writing on my own. Thanks for the video!

[kendoolin]

Maybe I misunderstood something. But the initial hypotheis was it is a fraction AND in its lowest terms. You derived a contradiction. This means the initial statement is false ie it is not a fraction OR not in its lowest terms OR both. It doesn't necessarily mean it is not a fraction - it may simply not be in its lowest terms. I'm sure I got something wrong ...

[vkrnfsu]

After watching many videos， finally there is one that I can understand. 😭
Tq for not assuming we know anything to begin with. 🙏

[AubreyForever]

Did we just prove that the square root of 2 cannot be a rational number? I watched some other video that I didn't quite understand that show that the square root of 2 exists using some kind of bounded theorem. Can you od this proof so we can understand? Thank you

[sanjursan]

The bit about being in lowest terms bothers me. I mean given the fact that there are an infinitude of pairs of values to check, how can we be so certain that we have lowest terms?

[VitumbikoMtafya-ncwp]

What if you have been given √5 prove that it's irrational?

[rikkardo]

Why exactly can't a and b have any common factors? I tried the proof with n instead of 2 and it seems to work, "proving" that there are no rational numbers at all... My math is most certainly wrong, but please tell me how

[amissanime]

Here's a simpler way.
✓2 = p/q
=> 2 = (p/q)^2
=> 2q = p^2/q

Since 2q is natural
And there's no common factor between p and q,
Therefore p/q or p^2/q can't be natural.

Hence, 2q = p^2/q can't be true.

[anuragsoni]

i wanna question the man who created this que did he really didnt know root 2 is irrational why does he need to create such an illogical question, nice explation btw

[dyva]

I still don’t understand why a and b can’t be even and can’t share the same factor?would you mind tell? I’m new

[mohamedelzoheiry]

@mohamedelzoheiry1413
0 seconds ago
the solution of square root 2 in nearest fraction is
1871/1323 or 1970/1393 or 3124/2209 or 3363/2378 or 5234/3701 or 7105/5024 or 8119/5741 or 9611/6796

the solution of square root 2 is
25/18 + 1/42 + 1/660 = 19601/13860 = 1.4142136
1.4142136 = (2)^(1/2)

the solution of square root 2 is
11/8 + 3/83 + 6/1955 = 1835819/1298120 = 1.4142136
1.4142136 = (2)^(1/2)

[paullloyd]

I don't understand. If a/b is not in it's lowest terms at the end it's still rational so there's no contradiction. Can you explain this? Thanks

[Goddamn_heisenberg]

Read about this in Stephen hawking’s book but didn’t understand, thank you

[kondwanimvula]

Can you please prove if the square of 2 - 1 is irrational

[leviackermann]

By that logic isnt 2/4 also irrational?

[mollystudent]

Got it in my math textbook when I was in 8 th grade.