Prove Square Roots of Non-Perfect Squares are Irrational

Real Analysis Proofs, Video #1. Use the Fundamental Theorem of Arithmetic (Unique Prime Factorization).

(0:00) Real Analysis Proofs, Video #1
(0:53) Notation and problem statement
(1:19) We are assuming the square root of n exists
(2:10) This is a proof by contradiction
(3:15) Then x=a/b, where, WLOG, a and b are relatively prime (gcd(a,b)=1)
(5:38) Square and use the Fundamental Theorem of Arithmetic
(8:24) Use the fact that n is not a perfect square
(9:57) Use nb^2=a^2 to say pj divides a
(12:07) pj divides b
(13:45) This is a contradiction
(14:36) The original assumption must be wrong and x is not rational

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[shangavim.]

How do you know that Pj divides into b^2? Its possible that Pj has an odd power within n greater than the even power within a^2. For example, what if Pj divides into n with a power of 3 and Pj divides into a^2 with a power of 2?

[coycatrett]

could you also observe that b can’t equal 1, and then n would not be an integer?

[joshualee]

Hey there thanks for the great video. One question though, why would, applying this method, show that root 25 is irrational for example. Can't this method be applied, and shown that root 25 is irrational?

[vinmierlak]

Can we exploit the fact that n is a natural number? Would it not be true that if n = a^2/b^2 and n is natural number, then b^2 = 1? If so, then we would get n = a^2, which must be a perfect square (since a is also a natural number), thereby contradicting the assumption that n is not a perfect square.

[saumyatripathi]

Is it correct that there arises one more contradiction that
In the equation square(a) =n *square(b)
The rhs has an even power of pj in its prime factorization while the lhs has (odd +even)=odd power of pj in its prime factorization

[ksmyth]

But can't you just say n = a.a / b.b. a and b are relative prime. So they have no common factors (b can't be 1 by assumption that n^2 is not perfect). Therefore a.a/b.b cannot be an integer. But n is an integer, therefore there is a contradiction. QED.

[hja]

hi. really the only good video i can find on proof that all nonperfect squares are irrational. one thing i dont understand is how come if pj divides a(squared) and so pj divides a, why does that mean a(squared) has an even power of pj in prime factorisation.