NCERT Solutions Class 9 Maths Chapter 1 helps you master Number Systems through detailed solutions covering rational and irrational numbers, real numbers, and their decimal expansions. You’ll learn how to represent numbers on the number line, rationalize denominators, apply laws of exponents with rational powers, and prove whether numbers are rational or irrational using step-by-step methods. These fundamental concepts are essential for algebra, geometry, and advanced mathematics in higher classes.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 9 Maths Chapter 1 Number Systems – Complete Guide
NCERT Class 9 Chapter 1 Number Systems forms the foundation of your mathematics journey in secondary school. This chapter carries 10 marks weightage in CBSE board exams and is considered very high importance because it introduces concepts that you’ll use throughout Classes 9 and 10. You’ll explore the evolution of numbers—from natural numbers and whole numbers to integers, rational numbers, and finally irrational numbers—understanding how each set expands to solve mathematical limitations.
📊 CBSE Class 9 Maths Chapter 1 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 10 Marks |
| Unit Name | Number Systems |
| Difficulty Level | Medium |
| Importance | Very High |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 2-3 questions |
You’ll learn to identify rational and irrational numbers, represent them on the number line, and perform operations like addition, subtraction, multiplication, and division on them. A crucial skill you’ll develop is rationalizing denominators, which frequently appears in CBSE board exams as 2-3 mark questions. You’ll also master the laws of exponents for real numbers, understanding how to simplify expressions with rational and irrational bases.
This chapter is highly relevant for competitive exams and builds critical thinking about number properties. Expect a mix of MCQs (1 mark), short answer questions (2-3 marks) on rationalization and number line representation, and long answer questions (4-5 marks) involving proofs about irrational numbers. The chapter connects directly to Chapter 2 (Polynomials) and later to coordinate geometry and algebra.
Quick Facts – Class 9 Chapter 1
| 📖 Chapter Number | Chapter 1 |
| 📚 Chapter Name | Number Systems |
| ✏️ Total Exercises | 5 Exercises |
| ❓ Total Questions | 25 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering Number Systems gives you confidence in handling complex mathematical expressions and develops your logical reasoning skills. With consistent practice of NCERT solutions and understanding the theoretical concepts, you’ll find this chapter scoring well in both school exams and board examinations.
NCERT Solutions Class 9 Maths Chapter 1 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| EXERCISE 1.1 | Complete step-by-step solutions for 4 questions | 📥 Download PDF |
| EXERCISE 1.2 | Complete step-by-step solutions for 4 questions | 📥 Download PDF |
| EXERCISE 1.3 | Complete step-by-step solutions for 9 questions | 📥 Download PDF |
| EXERCISE 1.4 | Complete step-by-step solutions for 5 questions | 📥 Download PDF |
| EXERCISE 1.5 | Complete step-by-step solutions for 3 questions | 📥 Download PDF |
Number Systems – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Rational Number Between Two Numbers \(\frac{a+b}{2}\) | Finds a rational number between two given numbers a and b. Note: This formula always gives a rational number that lies exactly in the middle of a and b. You can repeat this process to find more rational numbers. | When asked to find a rational number between two given numbers. Useful for inserting rational numbers between two given rationals. |
| Finding ‘n’ Rational Numbers \(d = \frac{b-a}{n+1}\), then \(a+d, a+2d, a+3d, …, a+nd\) | Finds n rational numbers between two given numbers a and b. Note: Make sure ‘a’ is less than ‘b’. ‘d’ is the common difference. This method ensures equally spaced rational numbers. | When you need to insert a specific number (‘n’) of rational numbers between two given numbers. ‘a’ and ‘b’ are the two given numbers. |
| Converting p/q to Decimal (Terminating) \(\frac{p}{2^m 5^n}\) | A rational number p/q will have a terminating decimal expansion if q can be expressed in the form 2m5n where m and n are non-negative integers. Note: First, simplify the fraction to its lowest form. Then, check if the denominator can be expressed as a product of powers of 2 and 5 only. | To determine if a rational number will have a terminating decimal expansion without actually performing the division. |
| Identifying Irrational Numbers \(\sqrt{p}\) | Square root of any prime number ‘p’ is irrational. Note: Also, non-terminating, non-repeating decimals are irrational numbers. For example, 0.1010010001… is irrational. | To quickly identify irrational numbers. |
| Representing \(\sqrt{x}\) on Number Line Geometric Construction using Pythagoras Theorem | Method for locating the value of an irrational number like \(\sqrt{x}\) on the number line. Note: Draw a line segment AB = x units. Extend it to BC = 1 unit. Find the midpoint of AC (call it O). With O as center and OC as radius, draw a semi-circle. Draw a perpendicular BD at point B. The length of BD is \(\sqrt{x}\). | When the question specifically asks to represent \(\sqrt{x}\) geometrically on the number line. |
| Law of Exponents: Multiplication (Same Base) \(a^m \cdot a^n = a^{m+n}\) | When multiplying powers with the same base, add the exponents. Note: The base ‘a’ must be the same for both terms. | Simplifying expressions involving multiplication of numbers with the same base. |
| Law of Exponents: Division (Same Base) \(\frac{a^m}{a^n} = a^{m-n}\) | When dividing powers with the same base, subtract the exponents. Note: The base ‘a’ must be the same for both terms and a ≠ 0. | Simplifying expressions involving division of numbers with the same base. |
| Law of Exponents: Power of a Power \((a^m)^n = a^{mn}\) | When raising a power to another power, multiply the exponents. Note: Be careful with the order of operations. | Simplifying expressions where a power is raised to another power. |
| Law of Exponents: Same Exponent \(a^m \cdot b^m = (ab)^m\) | When multiplying powers with the same exponent, multiply the bases and keep the exponent. Note: The exponent ‘m’ must be the same for both terms. | Simplifying expressions involving multiplication of numbers with the same exponent. |
| Law of Exponents: Zero Exponent \(a^0 = 1\) | Any number (except 0) raised to the power of 0 is equal to 1. Note: a ≠ 0. 00 is undefined. | Simplifying expressions where a number is raised to the power of 0. |
| Rationalizing the Denominator (Single Term) \(\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}\) | Removing the square root from the denominator of a fraction. Note: Multiply both numerator and denominator by the square root term in the denominator. | When you have a single square root term in the denominator. |
| Rationalizing the Denominator (Two Terms) \(\frac{1}{a + \sqrt{b}} = \frac{a – \sqrt{b}}{a^2 – b}\) | Removing the square root from the denominator of a fraction when the denominator has two terms. Note: Multiply numerator and denominator by the conjugate of the denominator. The conjugate of a + \(\sqrt{b}\) is a – \(\sqrt{b}\). | When the denominator is of the form a + \(\sqrt{b}\) or a – \(\sqrt{b}\) or \(\sqrt{a}\) + \(\sqrt{b}\) or \(\sqrt{a}\) – \(\sqrt{b}\). Multiply both numerator and denominator by the conjugate of the denominator. |
| Negative Exponent \(a^{-n} = \frac{1}{a^n}\) | Expressing a number with a negative exponent as a fraction with a positive exponent. Note: a ≠ 0 | Simplifying expressions with negative exponents. |
Frequently Asked Questions – NCERT Class 9 Maths Chapter 1
📚 Related Study Materials – Class 9 Maths Resources
| Resource | Access |
|---|---|
| NCERT Class 9 Mathematics Textbook | Download Book |
| NCERT Class 9 Science Solutions | View Solutions |
| RD Sharma Class 9 (Updated 2025-26) | View Solutions |
| NCERT Class 9 English (Beehive) | Download Book |