Posted: 7/15/2005 8:13:00 PM EDT
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I was explaining about the DUm-DUm’s to a relative when the phrase “Irrational Numbers” popped into my brain. Back in school when I first learned about Irrationals I once had an idea: Most Irrational Numbers come from getting the square root of a number. Pi and e being two obvious exceptions. But I wondered: Is there an Irrational Number where its Square Root is a Rational Number? Example: Pi is an Irrational Number. Its Square Root is an Irrational Number. But what if the Square Root of Pi is a Rational Number? My theory would be proven. But can anybody out there prove it? And how would you prove it? Can the Arfcom Mathematical geniususususus Hive Mind Prove or Disprove my Theory? And my Math Teachers often wondered why my eyes often glazed over. |
irrational means it can't be written as a quotient of two integers proof by contradiction: assume you have a number like you describe, call it x then x^(1/2) would be a rational number, so x^(1/2) = a/b for some integers a and b then (x^(1/2))^2 = (a/b)^2 -> x = a^2/b^2 the square of an integer is an integer (not going to try to prove that), so what you have above is that if such an x exists, it can be written as the quotient of two integers, a^2/b^2 but x was assumed to be an irrational number, so this is a contradiction. so there is no such x |
Proof by contradiction brings back horrible memories of Discrete Math in college.  Can you prove it by contraposition? 
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i generally prefer proof by googleization. |
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By definition: An Irrational number is any number that cannot be expressed as some Fraction (ie. Rational Number). There are 2 types of Irrational Numbers: (a.) Algebraic Irrational Numbers and (b.) Transcendental Irrational Numbers. Algebraic Irrational Numbers arise from finding the roots of a polynomial (examples are square root of two, cube root of 5, etc.) Transcendental Irrational Numbers are all Irrational Numbers which cannot arise Algebraically, Examples are PI, e, etc. The proof for proving that the square root of 2 is an Irrational Number is quite trivial. Hint: do a proof by Contridiction. Similarly, If I took a Transcental Irrational Number and took its square root, I would still end up with a Irrational Number. Use the same method for proving the above: ie. proof by contradiction. |
Couldn't you just write an irrational number as the fraction: irrational number ------------------ 1 ? |
Great. Now we are getting back into the great ".9bar" debate... 
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no, rational means it is the ratio of two integers (i.e. two "whole" numbers positive or negative) |
+ .9bar 
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As I have only recently really fully learned the proper mathematical proof methods (somehow lacking in my undergrad math preparation), I was very happy to read your proof and understand exactly what you were doing, and why. And to think I used to hate math. Now I wish I had double-majored... Jim |
No they didn't. The decimal point is still on opisite sides. PERIOD! Sgat1r5 PS I'll take ham & cheese...provolone. |
