Real Numbers — Important Questions

0.75

Step 1: Divide 3 by 4 using long division. Step 2: 3 ÷ 4 = 0.75 (since 30 ÷ 4 = 7 remainder 2, then 20 ÷ 4 = 5). Step 3: The division terminates, giving us a terminating decimal. Step 4: Therefore, 3/4 = 0.75.

2/3

Step 1: Let x = 0.666... Step 2: Multiply both sides by 10: 10x = 6.666... Step 3: Subtract the original equation: 10x - x = 6.666... - 0.666... Step 4: Simplify: 9x = 6, so x = 6/9 = 2/3. Therefore, 0.6̄ = 2/3.

{√2, √3, √5}

Step 1: Check each set for rationality. Step 2: {√4, √9, √16} = {2, 3, 4} are all rational. Step 3: {1/2, 3/4, 5/6} are all rational fractions. Step 4: {π, 22/7, √4} contains 22/7 (rational) and √4 = 2 (rational). Only {√2, √3, √5} contains numbers that are not perfect squares, making them all irrational.

Terminating

Step 1: Check the prime factorization of the denominator: 8 = 2³. Step 2: For a rational number p/q in lowest terms, the decimal terminates if and only if q has only factors of 2 and 5. Step 3: Since 8 = 2³ has only factor 2, the decimal will terminate. Step 4: Indeed, 1/8 = 0.125 (terminating decimal).