How many practice questions are available for Number Systems?

SparkEd Math offers 60+ practice questions for Number Systems across 3 difficulty levels (Easy, Medium, Hard) plus a timed test mode. Each question comes with visual step-by-step solutions and AI coaching hints.

Is Number Systems content free on SparkEd Math?

Yes! All Number Systems content on SparkEd Math is 100% free, including practice questions, visual solutions, lessons, and AI coaching. No credit card required.

How does SparkEd Math teach Number Systems differently from textbooks?

SparkEd Math uses visual step-by-step solutions for every Number Systems problem, making complex concepts easy to understand. The platform includes 3 difficulty levels for progressive learning, a Super Power Help AI coach for instant doubt-clearing, and interactive practice rather than passive reading.

Is SparkEd's Number Systems content aligned with the CBSE syllabus?

Yes, all Number Systems questions and lessons on SparkEd Math are fully aligned with the Central Board of Secondary Education (CBSE) syllabus for Class 9. The content follows the official curriculum structure and exam pattern.

Number Systems — Class 9 CBSE (2025-26)

About Number Systems — Class 9 CBSE

Classify numbers into naturals, integers, rationals, and irrationals; represent on number line. This topic is part of the CBSE Class 9 mathematics syllabus (chapter: Chapter 1). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Number Systems

Introduction to Number Systems
Rational Numbers
Irrational Numbers
Real Numbers and Their Properties
Laws of Exponents for Real Numbers

Interactive lesson · about 25 minutes · checkpoint question after every unit

Number Systems — solved examples for Class 9 CBSE

Example 1easy

Which of the following statements is TRUE?

A)Every irrational number is a real number.
B)Every real number is an irrational number.
C)Every rational number is an integer.
D)Every integer is a whole number.

Step-by-step solution

Real numbers are composed of both rational and irrational numbers. Therefore, every irrational number is a real number.
Option B is false because rational numbers are also real numbers.
Option C is false because fractions like 1/2 are rational but not integers.
Option D is false because negative integers (e.g., -3) are integers but not whole numbers.

Answer: Every irrational number is a real number.

Example 2medium

Identify the irrational number among the following:

A)√100
B)0.121212...
C)3.14
D)√7

Step-by-step solution

Option A: √100 = 10, which is a rational number (can be written as 10/1).
Option B: 0.121212... is a non-terminating recurring decimal, which can be expressed as a rational number (12/99).
Option C: 3.14 is a terminating decimal, which can be expressed as a rational number (314/100).
Option D: √7 cannot be simplified to an integer or a fraction, and its decimal expansion is non-terminating and non-recurring. Therefore, √7 is an irrational number.

Answer: √7

Example 3hard

If x = (√3 + 1) / (√3 - 1) and y = (√3 - 1) / (√3 + 1), what is the value of x² + y²?

A)14
B)16
C)12
D)18

Step-by-step solution

Rationalize x: x = [(√3 + 1) / (√3 - 1)] × [(√3 + 1) / (√3 + 1)] = (√3 + 1)² / (3 - 1) = (3 + 1 + 2√3) / 2 = (4 + 2√3) / 2 = 2 + √3.
Rationalize y: y = [(√3 - 1) / (√3 + 1)] × [(√3 - 1) / (√3 - 1)] = (√3 - 1)² / (3 - 1) = (3 + 1 - 2√3) / 2 = (4 - 2√3) / 2 = 2 - √3.
Now, calculate x² and y²: x² = (2 + √3)² = 4 + 3 + 4√3 = 7 + 4√3. y² = (2 - √3)² = 4 + 3 - 4√3 = 7 - 4√3.
Finally, sum them: x² + y² = (7 + 4√3) + (7 - 4√3) = 14.

Answer: 14

Practice questions on Number Systems

Q1.easy
Ravi states that the decimal expansion of 7/8 is 0.875, and the decimal expansion of 1/3 is 0.333... (non-terminating repeating). Based on this, he concludes that 7/8 is a rational number and 1/3 is an irrational number. Is Ravi's conclusion correct?
1. A)Both conclusions are correct.
2. B)7/8 is rational, but 1/3 is also rational.
3. C)7/8 is irrational, but 1/3 is rational.
4. D)Both numbers are irrational.
Show answer
Answer: 7/8 is rational, but 1/3 is also rational.
Hint: Remember the definition of a rational number in terms of its decimal expansion.
Q2.easy
To represent √5 on the number line, a student first draws a right-angled triangle OAB with OA = 2 units and AB = 1 unit, where A is on the x-axis and O is the origin. What is the next logical step?
1. A)Draw an arc with center A and radius OA to intersect the number line.
2. B)Draw an arc with center O and radius OB to intersect the number line.
3. C)Draw an arc with center B and radius AB to intersect the number line.
4. D)Draw an arc with center O and radius AB to intersect the number line.
Show answer
Answer: Draw an arc with center O and radius OB to intersect the number line.
Hint: Use the Pythagorean theorem to find the length of OB, and then transfer that length to the number line using the origin as the center.
Q3.easy
If 'x' is a rational number and 'y' is an irrational number, which of the following statements is ALWAYS TRUE?
1. A)x + y is an irrational number.
2. B)x × y is a rational number.
3. C)y × y is an irrational number.
4. D)x / y is a rational number.
Show answer
Answer: x + y is an irrational number.
Hint: Recall the properties of combining rational and irrational numbers through different operations.
Q4.medium
What is the result when 0.127 recurring (0.12777...) is expressed in the form p/q, where p and q are integers and q ≠ 0?
1. A)127/999
2. B)115/900
3. C)127/990
4. D)115/990
Show answer
Answer: 115/900
Hint: Let x be the given number. Multiply x by powers of 10 to shift the decimal point so that the repeating part aligns, then subtract the equations.
Q5.medium
Simplify the expression: (√5 + √3)²
1. A)8
2. B)8 + 2√15
3. C)8 + √15
4. D)15
Show answer
Answer: 8 + 2√15
Hint: Remember the algebraic identity (a + b)² = a² + 2ab + b². Apply this identity where 'a' and 'b' are square roots.
Q6.medium
Rationalise the denominator of 1 / (3 + √2).
1. A)(3 - √2) / 7
2. B)(3 + √2) / 7
3. C)(3 - √2) / 11
4. D)(3 + √2) / 11
Show answer
Answer: (3 - √2) / 7
Hint: Multiply both the numerator and denominator by the conjugate of the denominator to eliminate the radical in the denominator.
Q7.hard
Consider a non-zero rational number 'a' and an irrational number 'b'. Which of the following statements is always true?
1. A)a + b is always rational.
2. B)a × b is always rational.
3. C)b / a is always irrational.
4. D)a - b is always rational.
Show answer
Answer: b / a is always irrational.
Hint: Explore each option by considering the definitions of rational and irrational numbers. Remember that the set of irrational numbers is not closed under all operations.
Q8.hard
Simplify: (81/16)^(-3/4) × [(25/9)^(-3/2) ÷ (5/2)^(-3)]
1. A)1
2. B)16/81
3. C)8/27
4. D)27/8
Show answer
Answer: 1
Hint: First, deal with negative exponents by taking reciprocals. Then, express bases as powers of smaller numbers (e.g., 81 = 3⁴, 16 = 2⁴) to simplify fractional exponents.
Q9.hard
If (√5 + √3) / (√5 - √3) = a + b√15, what is the value of a + b?
1. A)8
2. B)5
3. C)4
4. D)2
Show answer
Answer: 5
Hint: Rationalize the left-hand side of the equation. Once it's in the form A + B√C, compare it to a + b√15 to find the values of 'a' and 'b'.

These are 9 of the 60 questions available for Number Systems. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.