
Re: HELP  SQRT(3)
Posted:
Nov 21, 1997 3:30 PM


In article <346CC27B.398@netvision.net.il>, Emanuel Binnun <e_binnun@netvision.net.il> wrote: >Hi > >Please help me to proove that Square root of 3 is an irrational number >
Adding a few extra lines to the standard proof, we can get a "constructive" proof of the irrationality of sqrt(3): for every (positive) rational number x, we can find another positive rational number r such that abs(x  sqrt(3)) > r.
Consider x = p/q where p, q are positive integers, which we can choose without a common factor (greater than 1). We first show tat
(*) abs(p^2  3 * q^2) >= 1.
If to the contrary, abs(p^2  3 * q^2) < 1 then p^2 = 3 * q^2.
So, discussing the cases p=3n, p=3n+1, p=3n+2 separately, we find that p must be divisible by 3: p=3n. Then we cancel 3's and obtain q^2 = 3 * n^2, and q would be divisible by 3. We would see that p and q are both divisible by 3, contrary to our assumption (no common factor).
Here the standard proof concludes that p^2/q^2 is never 3. We go further: Rationalize at the right moment and use sqrt(3) < 2 to get
abs(p/q  sqrt(3)) = abs((p  q * sqrt(3))/q) = abs((p^2  3 * q^2) / (q * (p + q * sqrt(3)))) > 1/(q*(p+2*q))
Now r = 1/(q*(p+2*q)) satisfies the claim.
(We showed that p/q is computably too far from sqrt(3) to have a chance to be equal to it.)
(For example, test p=5042, q=2911: your calculator with 8digit display shows p/q approximately 1.7320508, same as the 8digit approximation of sqrt(3). But in fact,
abs(5042/2911  sqrt(3)) > 1/(2911*10864) = 1/31625104.)
Cheers, ZVK (Slavek).

