Prove that 2 0 for any rational z SolutionFirst we prove tha

Solution

First we prove that if x is a rational number, then x ² 0. The product of two positive numbers is always positive, i.e., if a 0 and b 0, then ab 0. In particular if a/b 0 then (a/b) ² = x · x 0. If a/b is negative, then x is positive, hence (a/b)² 0. But we can conduct the following computation by the associativity and the commutativity of the product of rational numbers:

x² 0

= (a/b)(a/b)

= (a²/b²)

=(a/b)>0

let if a/b is possitive then x² greater than 0

(1/2)²=1/4>0

let if a/b is negative then x² greater than 0

(-1/2)²=1/4>0

let if a/b is zero then x² equal to 0

so ,,x²>0 for any rational x