sum or product of two irrational numbers is
rational or irrational
g enerally irrational.
Proof:
Let a and b be two irrational numbers. We assume that their sum (a+b) is rational. Then, there exist two integers p and q (q≠0) such that a+b=p/q.
Rearranging the equation, we get a=p/q-b. Since p and q are integers and b is irrational, p/q-b is also irrational. This contradicts our assumption. Therefore, the sum of two irrational numbers is always irrational.
Similarly, let us assume that the product (ab) of two irrational numbers a and b is rational. Then, there exist two integers p and q (q≠0) such that ab=p/q. Rearranging the equation, we get a=p/(qb).
Since p and qb are integers and b is irrational, p/(qb) is also irrational. This contradicts the assumption that ab is rational. Therefore, the product of two irrational numbers is always irrational.
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