[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
In case the (ir)rationality of sqrt(2)^sqrt(2) is bugging you... + Constructive vs. Existential math.
- To: Rao Kambhampati <rao@asu.edu>
- Subject: In case the (ir)rationality of sqrt(2)^sqrt(2) is bugging you... + Constructive vs. Existential math.
- From: Subbarao Kambhampati <rao@asu.edu>
- Date: Thu, 9 Apr 2009 13:47:00 -0700
- Dkim-signature: v=1; a=rsa-sha256; c=relaxed/relaxed; d=gmail.com; s=gamma; h=domainkey-signature:mime-version:sender:received:date :x-google-sender-auth:message-id:subject:from:to:content-type; bh=R/H1Vh4mR4vAbeX8RBQ6BIBi7MOerOlrzJHaSyvy9Oc=; b=x5gdrtjDn9WpppFkDdKF378vqM0xj+/MBJzj+5Cr4OcYupwYcKjdMmSz0hbQuurEwh GQZkuYTitpR98UmPQ8JWys++vkXolW4NztuUXuCUOg7WTruo7BNGeb/GEStQyuSAdbwR AwH/4LKpn+KWgMhVZW11S7wYiIXxOL6ncWH7k=
- Domainkey-signature: a=rsa-sha1; c=nofws; d=gmail.com; s=gamma; h=mime-version:sender:date:x-google-sender-auth:message-id:subject :from:to:content-type; b=mGyE3rf2oXz/N5ga4NuandB45h3rjiwFwaSzmLbxE+4FUX+oOI0ZQRoGHYzmfwiN45 tEgGqfu0wfiHbh1zdglJ3JrM8sxELucM4MbrZ2p2pVX73xxLbo7nXRL6eTHzN7w1Q1cd idXo32ZLN/bhtENDqrZHsLFrGYjCF7q8VHpKw=
- Sender: subbarao2z2@gmail.com
..In case you are dying to know whether sqrt(2)^sqrt(2) is rational or irrational, you can be rest assured
that it is irrational (actually transcendental? (*)). So a constructive proof for
our theorem is with p=sqrt(2)^sqrt(2) and q=sqrt(2)
see http://www.math.hmc.edu/funfacts/ffiles/10004.3-5.shtml
(which also points out a more general and easy to understand constructive proof. Consider
? e^{log_e q} for any transcendental number e and rational number
q--which will be q. All you need to show is log_e(q) is irrational and
you can show this easily (If log_e(q) = m/n with integers m and n
without common factors, then
q = e^{m/n}. This would mean that e is the root of an algebraic
equation x^m - q^n = 0. But the definition of transcendental number is
that it cannot be the root of any algebraic equation!).
Rao
(*)
By the way, transcendental => irrational but not vice versa. In
particular, transcendentals are those irrational numbers that cannot be
roots of any algebraic equation. Two famous examples of course are e
and pi.? Notice that proving that a number e *is* transcendental
involves showing that e^r for any rational number r cannot be rational
(since if it is, then e will be the root of an algebraic equation).
Thus, proving transcendentality is not all that easy.
(ps 2:
Check out
http://digitalphysics.org/Publications/Cal79/html/cmath.htm
for a nice discussion on the Constructive vs. Classical mathematics--and
how during Hilbert's time there was a pretty big controversy in mathematics--with mathematicians such as Brouer insisted
that all math that depended on existential proofs be thrown out.Papa Hilbert had to come to rescue--pretty heady stuff.
You might also look at
http://plato.stanford.edu/entries/mathematics-constructive/
which also talks about the slick "irrational power irrational can be
rational proof..."