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        riddles >> putnam exam (pure math) >> help irrational number that is not transcendental 
        
(Message started by: apsurf on Apr 8^th, 2005, 11:36am)

Title: help irrational number that is not transcendental
Post by apsurf on Apr 8^th, 2005, 11:36am

What is an example of an irrational number that is not transcendental?

Title: Re: help irrational number that is not transcenden
Post by THUDandBLUNDER on Apr 8^th, 2005, 11:48am

on 04/08/05 at 11:36:43, apsurf wrote:

What is an example of an irrational number that is not transcendental?

sqrt(2)

Title: Re: help irrational number that is not transcenden
Post by Ricardo on Dec 3^rd, 2005, 1:26am

could it be slightly more general to say that the square root of any prime is irrational but not transcendental? (i dont know if 1 classifies as a prime, but w/e)

Title: Re: help irrational number that is not transcenden
Post by Icarus on Dec 3^rd, 2005, 8:07am

"transcendental" means that the number is not the root of a polynomial with integer coefficients. So any root of any polynomial which is not rational, is an example.

sqrt(n) is a root of x² - n. For it to be an example of a non-transcendental (i.e., algebraic) irrational number, it merely has to be irrational.

So suppose (r/s)² = n, or r² = nsⁿ, where r and s are integers, which we may take to be relatively prime. But if p is any prime dividing s, then p divides ns² = r², and so p divides r, contrary to our choice of r, s. So s has no prime divisors, and therefore s = 1, and n = r².

I.e., if n is an integer >=0, then n is a perfect square, or sqrt(n) is irrational.

So, sqrt(2), sqrt(3), sqrt(5), sqrt(6), sqrt(7), sqrt(8), sqrt(10), ... are all irrational algebraic numbers.

More generally, if a, b, c, d are all integers with a, b relatively prime, and with c, d relatively prime, then (a/b)^c/d is always algebraic, and is rational iff both a and b are perfect d powers (a = n^d and b = m^d for some integers n, m).

Title: Re: help irrational number that is not transcenden
Post by Ricardo on Dec 5^th, 2005, 12:15pm

Icarus, your argument seems to boil down to a generalization of what i said. All that changes is the fact that the square root of any integer greater than 0 (to ignore imaginary numbers) can be boiled down to a perfect square plus some error, which then turns out to be prime, and hence irrational.
Thanks for clarifying that Icarus.
Also, is there a possibility of there being a rational transcendental number, real or imaginary? or am i just mixing up the definitions? (i realize that you cannot turn pi into a fraction, but that doesnt exhaust all possibilities...)

Title: Re: help irrational number that is not transcenden
Post by rmsgrey on Dec 5^th, 2005, 12:19pm

on 12/05/05 at 12:15:01, Ricardo wrote:

All rational numbers are algebraic - p/q is a root of qx-p

Title: Re: help irrational number that is not transcenden
Post by Icarus on Dec 5^th, 2005, 6:34pm

on 12/05/05 at 12:15:01, Ricardo wrote:

All that changes is the fact that the square root of any integer greater than 0 (to ignore imaginary numbers) can be boiled down to a perfect square plus some error, which then turns out to be prime, and hence irrational.

No. The difference does not necessarily turn out to be prime, nor is primeness required to be irrational.

The square root of any positive integer is either an integer (the number was a perfect square), or else it is irrational, regardless of primality of any associated values.