Question

Let a be an irrational number and r a nonzero rational number. prove that if s is a real number, then either ar s or ar - s is irrational.

Answer 1

To prove that if a is an irrational number and r is a nonzero rational number, then either ar + s or ar - s is irrational, we can use proof by contradiction.

Step-by-step explanation:

To prove that if a is an irrational number and r is a nonzero rational number, then either ar + s or ar - s is irrational, we can use proof by contradiction.

1. Assume that both ar + s and ar - s are rational.
2. Since a is irrational and r is rational, then ar is irrational.
3. By adding and subtracting a rational number from an irrational number, we cannot obtain two rational numbers.
4. This contradicts our assumption, so either ar + s or ar - s must be irrational.