Proving something is irrational

  • #1
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Homework Statement


Prove that for all x ∈ ℝ, at least one of √3 - x and √3 +x is irrational.

Homework Equations




The Attempt at a Solution


I understand how to prove that √3 is a irrational number by proof by contradiction. However I am not sure how to prove this one.

Would I have to equate √3 - x = a/b and √3 + x = a/b and prove by contradiction?

Thank you.
 

Answers and Replies

  • #2
PeroK
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Homework Statement


Prove that for all x ∈ ℝ, at least one of √3 - x and √3 +x is irrational.

Homework Equations




The Attempt at a Solution


I understand how to prove that √3 is a irrational number by proof by contradiction. However I am not sure how to prove this one.

Would I have to equate √3 - x = a/b and √3 + x = a/b and prove by contradiction?

Thank you.
That would be a good start. Although the two numbers can't both be equal to the same ##a/b##.
 
  • #3
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If you see something with ##a-b## and ##a+b## it is always an idea to multiply them. Try a proof by contradiction, i.e. assume both were rational.
 
  • #4
PeroK
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If you see something with ##a-b## and ##a+b## it is always an idea to multiply them. Try a proof by contradiction, i.e. assume both were rational.
Or add them!
 
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  • #5
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If you see something with ##a-b## and ##a+b## it is always an idea to multiply them. Try a proof by contradiction, i.e. assume both were rational.
So I'd multiply (√3-x)(√3+x) and then equate it to a/b?
 
  • #7
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@PeroK's hint is the better idea: what do you get if you add / subtract the two?
Okay so when I add then I obtain √3 + √3 or 2√3 and then this is what we equate to a/b? Thank you.
 
  • #9
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The proof starts with - as you already suggested: Assume ##x-\sqrt{3}## and ##x+\sqrt{3}## were both rational. Then their sum and difference ....
 

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