Number Systems — Class 9 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Number Systems" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 9 Mathematics chapter "Number Systems" — 8 important questions with detailed answers for CBSE board exam p…
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Key Questions Covered:
- Classify √2 as rational or irrational. Justify your answer.
- Express 2.3̄ (2.333...) as a fraction in lowest terms.
- Rationalize the denominator: 1/(√5 - √3)
- + 5 more questions in the full chapter
Solutions Summary:
| Question | Status |
|---|---|
| Classify √2 as rational or irrational. Justify your answer. | ✓ Solved |
| Express 2.3̄ (2.333...) as a fraction in lowest terms. | ✓ Solved |
| Rationalize the denominator: 1/(√5 - √3) | ✓ Solved |
Showing 3 of 8 questions
Q1: Classify √2 as rational or irrational. Justify your answer.
√2 is irrational.
Justification (Proof by contradiction):
Suppose √2 is rational. Then √2 = p/q where p and q are coprime integers (gcd(p,q) = 1).
Squaring both sides: 2 = p²/q²
Therefore: p² = 2q²
This means p² is even, so p must be even. Let p = 2m.
Substituting: (2m)² = 2q²
4m² = 2q²
2m² = q²
Th...
Q2: Express 2.3̄ (2.333...) as a fraction in lowest terms.
Let x = 2.3̄ = 2.333...
Step 1: Write x = 2 + 0.333...
Let y = 0.333...
Step 2: Multiply y by 10
10y = 3.333...
Step 3: Subtract to eliminate the repeating part
10y - y = 3.333... - 0.333...
9y = 3
y = 3/9 = 1/3
Step 4: Find x
x = 2 + 1/3 = 6/3 + 1/3 = 7/3
Answer: 2.3̄ = 7/3
Q3: Rationalize the denominator: 1/(√5 - √3)
Expression: 1/(√5 - √3)
Step 1: Identify the conjugate
Conjugate of (√5 - √3) is (√5 + √3)
Step 2: Multiply numerator and denominator by the conjugate
= 1/(√5 - √3) × (√5 + √3)/(√5 + √3)
= (√5 + √3)/[(√5 - √3)(√5 + √3)]
Step 3: Apply difference of squares formula (a - b)(a + b) = a² - b²
= (√5 + ...
Showing 3 of 8 questions. Visit the full page for complete solutions.