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Topic: help irrational number that is not transcendental (Read 1714 times) |
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apsurf
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help irrational number that is not transcendental
« on: Apr 8th, 2005, 11:36am » |
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What is an example of an irrational number that is not transcendental?
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ThudnBlunder
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Re: help irrational number that is not transcenden
« Reply #1 on: Apr 8th, 2005, 11:48am » |
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on Apr 8th, 2005, 11:36am, apsurf wrote:| What is an example of an irrational number that is not transcendental? |
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sqrt(2)
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Ricardo
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Re: help irrational number that is not transcenden
« Reply #2 on: Dec 3rd, 2005, 1:26am » |
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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)
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Icarus
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Re: help irrational number that is not transcenden
« Reply #3 on: Dec 3rd, 2005, 8:07am » |
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"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 = nd and b = md for some integers n, m).
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| « Last Edit: Dec 3rd, 2005, 8:07am by Icarus » |
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Ricardo
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Re: help irrational number that is not transcenden
« Reply #4 on: Dec 5th, 2005, 12:15pm » |
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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...)
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rmsgrey
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Re: help irrational number that is not transcenden
« Reply #5 on: Dec 5th, 2005, 12:19pm » |
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on Dec 5th, 2005, 12:15pm, Ricardo wrote: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...) |
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All rational numbers are algebraic - p/q is a root of qx-p
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Icarus
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Re: help irrational number that is not transcenden
« Reply #6 on: Dec 5th, 2005, 6:34pm » |
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on Dec 5th, 2005, 12:15pm, 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. |
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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.
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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