Square roots and such
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Square roots and such
Is there any proof that the square (or cube etc) roots of numbers that don't have integer square (or cube etc) roots are irrational?
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Re: Square roots and such
Nope, because that isn't true. sqrt(9/4)=3/2.
Aside from that type of case, I think I have seen a proof, but I can't think of it now. If the numerator and denominator are not perfect squares, it will be irrational.
Aside from that type of case, I think I have seen a proof, but I can't think of it now. If the numerator and denominator are not perfect squares, it will be irrational.
Re: Square roots and such
Thanks for clarifying. I think I meant "square (or cube etc) roots of integers", but it would extend to fractions with perfect square/cube/etc numerators ad denominators. Positive, of course.
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 jestingrabbit
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Re: Square roots and such
It is true that "if n is an integer and not a perfect square, then [imath]\sqrt{n}[/imath] is irrational." To prove it, have a good look at the proof for [imath]\sqrt{2}[/imath] and carefully generlise it.
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Re: Square roots and such
As it turns out, I am not familiar with such a proof. I am only a freshman in high school, so I not had much opportunity to encounter higher math.
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 jestingrabbit
 Factoids are just Datas that haven't grown up yet
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Re: Square roots and such
Fair enough. Any of the first three proofs here should be pretty easily generalisable.
However, you might want to try writing a proof that the square root of 2 is irrational by yourself first, then see how yours looks compared to the other proofs, just to try getting a start with proving things, a skill that will come in handy if you pursue further study in maths later.
However, you might want to try writing a proof that the square root of 2 is irrational by yourself first, then see how yours looks compared to the other proofs, just to try getting a start with proving things, a skill that will come in handy if you pursue further study in maths later.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
Re: Square roots and such
Definitely try and prove it yourself. It is a pretty nifty proof.
Hint:
Hint:
Spoiler:

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Re: Square roots and such
It's something you might encounter in first year mathematics in university. You're thinking ahead!
I found it pretty cool... especially at the end where it's like "that doesn't work... all the math is right... that must mean we messed up the start! QED".
I found it pretty cool... especially at the end where it's like "that doesn't work... all the math is right... that must mean we messed up the start! QED".
 Yakk
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Re: Square roots and such
So, a useful results for proving this:
Unique Factorization: Let n be a positive integer. Then n = product p_{i}^{ni} (possibly an empty product, which by definition equals 1), where each p_{i} is a unique prime and n_{i} is a positive integer. Furthermore, this entire product is unique up to reordering.
Basically, this says that any positive integer can be expressed as a unique product of prime numbers.
Unique Factorization: Let n be a positive integer. Then n = product p_{i}^{ni} (possibly an empty product, which by definition equals 1), where each p_{i} is a unique prime and n_{i} is a positive integer. Furthermore, this entire product is unique up to reordering.
Basically, this says that any positive integer can be expressed as a unique product of prime numbers.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Last edited by JHVH on Fri Oct 23, 4004 BCE 6:17 pm, edited 6 times in total.
Re: Square roots and such
Another fun thm is the rational roots test. It isn't difficult to prove the root of primes are irrational using it.
Spoiler:
Last edited by polymer on Tue Feb 01, 2011 6:44 pm UTC, edited 11 times in total.
Re: Square roots and such
polymer wrote:Another fun thm is the rational roots test. It isn't difficult to prove primes are irrational using it.Spoiler:
You do not quite have a proof there.
Spoiler:
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Re: Square roots and such
polymer wrote:It isn't difficult to prove primes are irrational using it.
I assume you mean 'roots of primes'?
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Re: Square roots and such
You could generalize something along the lines of the square root of two is irrational, or you could turn it into a conditional statement and see what that tells us. So another way of asking the question would be to examine the conditional statement "Given that m is an integer, if m is not a perfect square, then the square root of m is irrational." From conditional statements, we know that P implies Q is logically equivalent to ~Q implies ~P. In other words, if we can show that the statement "If the square root of m is rational, then m is a perfect square" to be true, then we have proven what we want to know. This is a much easier approach in my opinion.
Re: Square roots and such
yea...you caught me before I corrected myself. This is why one shouldn't do math at 12:30...I'll correct it so the post isn't confusing.
Oops >_<, you're right my proof doesn't make sense given the definition of divides. I'll rewrite it, I was just hoping to leave one trick for the reader to try and solve.
Qaanol wrote:
You do not quite have a proof there.Spoiler:
Oops >_<, you're right my proof doesn't make sense given the definition of divides. I'll rewrite it, I was just hoping to leave one trick for the reader to try and solve.
Last edited by polymer on Tue Feb 01, 2011 6:29 pm UTC, edited 1 time in total.
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