Formal vs. informal proof that sqrt(2) is irrational

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Reply Wed 23 Jul, 2008 10:50 am
What's the difference b/w a formal and an informal proof, in mathematics? I have an informal proof that sqrt(2) is irrational:

1. Assume sqrt(2)=x/y, where x and y are nonzero integers and relatively prime
2. ysqrt(2)=x
3. 2y^2=x^2
4. x=2z, since x^2 is even by this, since it is able to be divided by 2, and it and the lhs are equal
5. 2y^2=(2z)^2=4z^2
6. y^2=2z^2
7. According line 4, since x is even, y must also be even
8. But then they must both share a factor, 2. But this contradicts our assumption that x and y do not share a common factor
9. Therefore, sqrt(2) is irrational
Btw, is there a less messy version of this?

This is an informal proof, but what's the formal proof?

Thanks!
 
Quatl
 
Reply Thu 7 Aug, 2008 01:22 pm
@Protoman2050,
EDIT: never mind my brain is dysfunctional today
 
Zetetic11235
 
Reply Thu 7 Aug, 2008 09:52 pm
@Quatl,
that is the formal proof. n/m=2^(1/2) where n,m are elements of Z the set of integers. n/m must be irreducible and thus n & m must not both be even.

(n/m)^2 = 2 --> n^2=2(m)^2 thus n^2 must be divisible by 2 and thus it is divisible by 4 as any even square must be divisible by 4 as one of the prime divisors of its root must be two. Thus two can be divide out of n^2 and it will still be even thus so will m^2 since m^2=(n^2)/2 and n^2/2 is even thus so is m^2. A contradiction. I cannot write it in FOL due to symbol notation not being availible.

The case of an irrational times a rational being irratoin is as follows:

N=irrational P= rational p=n/m where n,m are in z (set of integers)
N*n/m=(Nn)/m. Assume NP is rational NP=n'/m' thus (Nn)/m=n'/m'
Thus N=(n'm)/(m'n) where n'm and m'n are in z thus N is rational, a contradiction.

I can also give one for a rational plus an irrational is always irrational. The first is from the Rudin texts so you will have a tough time finding a more formal proof. I suppose you could argue form the peano axioms, but it would only involve showing everything from propositions derived from the axioms, which are stated in the rudin book Principles of Mathematical Analysis. Proofs like this ammount to pretty simple algebraic manipulations combined with propositions.
 
triclino
 
Reply Thu 18 Aug, 2011 04:08 pm
@Protoman2050,
Protoman2050 wrote:

What's the difference b/w a formal and an informal proof, in mathematics? I have an informal proof that sqrt(2) is irrational:

1. Assume sqrt(2)=x/y, where x and y are nonzero integers and relatively prime
2. ysqrt(2)=x
3. 2y^2=x^2
4. x=2z, since x^2 is even by this, since it is able to be divided by 2, and it and the lhs are equal
5. 2y^2=(2z)^2=4z^2
6. y^2=2z^2
7. According line 4, since x is even, y must also be even
8. But then they must both share a factor, 2. But this contradicts our assumption that x and y do not share a common factor
9. Therefore, sqrt(2) is irrational
Btw, is there a less messy version of this?

This is an informal proof, but what's the formal proof?

Thanks!

Are you still interested in a formal proof ,because a lot of time has elapsed since you wrote this post .

In case you are still interested i can write a formal proof for you.

Also ,if i may ask ,have you ask this question in other philosophical or mathematical forums ,whether of advanced or ordinary mathematical or philosophical level??
 
 

 
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