Proof-based questions with complete step-by-step explanations
Prove that √5 is an irrational number.
Step 1: Assume √5 is a rational number.
Then it can be written as:
√5 = a / b, where a and b are integers having no common factor and b ≠ 0.
Step 2: Squaring both sides:
5 = a² / b²
Step 3: Multiply both sides by b²:
5b² = a²
Step 4: This shows a² is divisible by 5, so a is divisible by 5.
Let a = 5k
Step 5: Substitute back:
5b² = 25k² ⇒ b² = 5k²
This implies b is also divisible by 5.
Contradiction: a and b cannot both be divisible by 5.
Therefore, √5 is irrational.
Prove that (3 + 2√5) is irrational.
Step 1: Assume 3 + 2√5 is rational.
Then:
3 + 2√5 = a / b
Step 2: Subtract 3 from both sides:
2√5 = (a / b) − 3
The right-hand side is rational.
Step 3: Divide both sides by 2:
√5 is rational
This contradicts the fact that √5 is irrational.
Therefore, 3 + 2√5 is irrational.
Prove that the following numbers are irrational:
(i) 1/√2 (ii) 7√5 (iii) 6 + √2
(i) 1/√2
Assume 1/√2 is rational.
Then √2 becomes rational, which is false.
Hence, 1/√2 is irrational.
(ii) 7√5
Assume 7√5 is rational.
Dividing both sides by 7 gives √5 rational.
This contradicts the fact that √5 is irrational.
Hence, 7√5 is irrational.
(iii) 6 + √2
Assume 6 + √2 is rational.
Subtract 6 from both sides:
√2 is rational — which is false.
Hence, 6 + √2 is irrational.