Prove that number `(log)_2 7` is an irrational number.

Every irrational number

(i) Prove that sqrt2 is an irrational number. (ii) Prove that sqrt3 is an irrational number.

Prove that, (sqrt(3) - sqrt(5)) is an irrational number.

Prove that (2sqrt3+sqrt5) is an irrational number. Also check whether (2sqrt 3+ sqrt5)(2sqrt 3- sqrt5) is rational or irrational.

Are the following pairs of statements negations of each other: The number x is a rational number. The number x is an irrational number.

Prove that, log_(3)6 is an irrational number.

Are the following pairs of statements negations of each other: The number x is not a rational number. The number x is not an irrational number.

Prove the following by contradiction : "The sum of a rational and irrational number is an irrational number ."

Define an irrational number and show that sqrt(5) is an irrational number.

(a) (i) By taking any two irrational numbers, prove that their sum is a rational number. (ii) By taking any two irrational numbers, prove that their difference is a rational number. (b) Insert in between 1/7 and 2/7 (i) a rational number : (ii) an irrational number.