Proof - irrational numbers

  • Thread starter kmeado07
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Homework Statement



Prove that if x^2 is irrational then x must be irrational.

Homework Equations





The Attempt at a Solution



Maybe do proof by contradiction. I'm not really sure where to start.
 

Answers and Replies

  • #2
Office_Shredder
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Proof by contradiction sounds good. What if x is rational?
 
  • #3
HallsofIvy
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"Suppose x is rational. Then x= __________"
 
  • #4
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so i let x= a/b

then obviously x^2 = a^2/b^2

im not sure how to continue to reach the contradiction
 
  • #5
Dick
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Assume x^2 is irrational and x is NOT irrational. You've shown that in such a case, x^2 is rational. That's a contradiction. x^2 can't be both rational and irrational. Therefore x must be irrational.
 
  • #6
HallsofIvy
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so i let x= a/b

then obviously x^2 = a^2/b^2

im not sure how to continue to reach the contradiction

It is not enough just to say "let x= a/b" without saying what a and b are. A number is rational if and only if it can be expressed as a ratio of integers: a/b where a and b are integers (and b is not 0). If x is rational, the x= a/b where a and b are integers. You arrive at the fact that x2= a2/b2, also a ratio of integers. That itself contradicts the hypothesis, that x2 is irrational.
 

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