Rational and Irrational Numbers — Important Questions

0.333..., √16, 22/7

Step 1: Check each number. 0.333... = 1/3 (recurring decimal = rational). √16 = 4 (perfect square = rational). √7 is non-terminating non-recurring (irrational). 22/7 is already in p/q form (rational). π = 3.14159... is non-terminating non-recurring (irrational).

True

Step 1: Let r be rational and i be irrational. Step 2: Assume r + i = q (rational). Step 3: Then i = q - r. Since q and r are both rational, q - r is rational. Step 4: This contradicts that i is irrational. Therefore, r + i must be irrational.

Non-terminating, non-recurring decimals, Cannot be expressed as p/q, Product with non-zero rational is irrational

Step 1: Check each property. Non-terminating, non-recurring decimals ✓. Cannot be expressed as p/q ✓. Sum of two irrationals: (2+√3) + (2-√3) = 4 (rational) ✗. Product with non-zero rational: 2×√3 = 2√3 (irrational) ✓.