Is x Irrational if x² Is Irrational?

[kmeado07]

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.

 

[Office_Shredder]

Proof by contradiction sounds good. What if x is rational?

 

[HallsofIvy]

"Suppose x is rational. Then x= __________"

 

[kmeado07]

so i let x= a/b

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

im not sure how to continue to reach the contradiction

 

[Dick]

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.

 

[HallsofIvy]

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.