Proving something is irrational

Homework Statement

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

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

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PeroK
Science Advisor
Homework Helper
Gold Member

Homework Statement

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

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##.

fresh_42
Mentor
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.

PeroK
Science Advisor
Homework Helper
Gold Member
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!

fresh_42
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?

fresh_42
Mentor
So I'd multiply (√3-x)(√3+x) and then equate it to a/b?
@PeroK's hint is the better idea: what do you get if you add / subtract the two?

@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.

Or add them!
Thank you. I'd get 2√3 = a/b?

fresh_42
Mentor
The proof starts with - as you already suggested: Assume ##x-\sqrt{3}## and ##x+\sqrt{3}## were both rational. Then their sum and difference ....