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.

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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…

Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated Jul 23, 2026

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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)
If a = 1/(2 - √3), find the value of a² - 2a.
Locate √5 on the number line using the Pythagorean theorem.
Simplify: (5 + 2√3)/(7 + 4√3) by rationalizing the denominator.
+ 2 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
If a = 1/(2 - √3), find the value of a² - 2a.	✓ Solved
Locate √5 on the number line using the Pythagorean theorem.	✓ Solved
Simplify: (5 + 2√3)/(7 + 4√3) by rationalizing the denomi…	✓ Solved

Showing 6 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 This means q² is even, so q is even. But if both p and q are even, they have a common factor of 2, contradicting our assumption that gcd(p,q) = 1. Therefore, √2 cannot be rational. Conclusion: √2 is irr...

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 + √3)/[(√5)² - (√3)²] = (√5 + √3)/(5 - 3) = (√5 + √3)/2 Answer: (√5 + √3)/2

Q4: If a = 1/(2 - √3), find the value of a² - 2a.

Step 1: Rationalize a a = 1/(2 - √3) × (2 + √3)/(2 + √3) = (2 + √3)/[(2 - √3)(2 + √3)] = (2 + √3)/(4 - 3) = 2 + √3 Step 2: Calculate a² a² = (2 + √3)² = 4 + 4√3 + 3 = 7 + 4√3 Step 3: Calculate 2a 2a = 2(2 + √3) = 4 + 2√3 Step 4: Find a² - 2a a² - 2a = (7 + 4√3) - (4 + 2√3) = 7 + 4√3 - 4 - 2√3 = 3 + 2√3 Answer: a² - 2a = 3 + 2√3

Q5: Locate √5 on the number line using the Pythagorean theorem.

Step 1: Express √5 using Pythagorean theorem √5 = √(1² + 2²) This is the hypotenuse of a right triangle with legs 1 and 2. Step 2: Construction on number line - Mark point 0 (origin) and point 2 on the number line. - At point 2, draw a perpendicular line of length 1 unit. - Using compass, measure the hypotenuse from 0 to the endpoint of the perpendicular. - This distance equals √5. Step 3: Locate √5 - With center at 0 and radius equal to the hypotenuse length, draw an arc on the number line. -...

Q6: Simplify: (5 + 2√3)/(7 + 4√3) by rationalizing the denominator.

Expression: (5 + 2√3)/(7 + 4√3) Step 1: Multiply by conjugate of denominator Conjugate of (7 + 4√3) is (7 - 4√3) = (5 + 2√3)/(7 + 4√3) × (7 - 4√3)/(7 - 4√3) Step 2: Calculate denominator using (a + b)(a - b) = a² - b² = (5 + 2√3)(7 - 4√3)/[(7)² - (4√3)²] = (5 + 2√3)(7 - 4√3)/(49 - 48) = (5 + 2√3)(7 - 4√3)/1 Step 3: Expand numerator = 5(7) + 5(-4√3) + 2√3(7) + 2√3(-4√3) = 35 - 20√3 + 14√3 - 8(3) = 35 - 20√3 + 14√3 - 24 = 11 - 6√3 Answer: (5 + 2√3)/(7 + 4√3) = 11 - 6√3

Showing 6 of 8 questions. Visit the full page for complete solutions.