The problem involves proving that for all x ∈ ℝ, at least one of the expressions √3 - x and √3 + x is irrational. The subject area pertains to number theory, specifically the properties of rational and irrational numbers.
The discussion is active, with participants sharing hints and exploring different approaches. Some guidance has been offered regarding the use of addition and multiplication of the expressions, but no consensus has been reached on a specific method.
Participants are working under the assumption that they need to prove the irrationality of at least one of the expressions without providing complete solutions. There is a focus on the implications of assuming both expressions are rational.
ver_mathstats said: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.
Or add them!fresh_42 said: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 said: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.
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.fresh_42 said:@PeroK's hint is the better idea: what do you get if you add / subtract the two?
Thank you. I'd get 2√3 = a/b?PeroK said:Or add them!