Number Systems
Explore rational and irrational numbers, their decimal expansions, and representation on the number line. Understand operations on real numbers and laws of exponents — as per the NCERT 2026–27 updated textbook.
NCERT 2026–27 Updated Content
This chapter is based on the latest NCERT textbook for session 2026–27 aligned with NEP 2020. Content reflects the most current syllabus issued by NCERT for this academic year.
Chapter Summary
Chapter 1 of the NCERT Class 9 Mathematics (2026–27) introduces students to the complete system of real numbers. You will learn to distinguish rational from irrational numbers, understand their decimal expansions, and locate irrational numbers on the number line using geometric constructions.
The 2026–27 syllabus retains all core concepts while focusing on conceptual clarity. Key skills: identifying number types, performing operations with surds, rationalising two-term denominators, and applying laws of exponents for real numbers.
💡 Key Concepts
Natural Numbers (N)
Counting numbers starting from 1: 1, 2, 3, 4, ... NCERT 2026–27 emphasizes their use in real-life counting and ordering contexts.
Whole Numbers (W)
All natural numbers along with zero: 0, 1, 2, 3, ... Zero was introduced to represent "nothing" — a key concept in the NCERT 2026–27 approach.
Integers (Z)
Includes all whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ... Used to represent temperatures, altitudes, and financial transactions.
Rational Numbers (Q)
Numbers expressible as p/q where p, q are integers and q ≠ 0. NCERT 2026–27: Decimal expansion is either terminating or non-terminating repeating.
Irrational Numbers
Cannot be expressed as p/q. Decimal expansion is non-terminating and non-repeating. Examples: √2, √3, π, e. NCERT 2026–27 includes locating these on number line.
Real Numbers (R)
Union of rational and irrational numbers. NCERT 2026–27: Focus on operations, laws of exponents for real numbers, and rationalisation of denominators.
⭐ Important Points to Remember
NCERT 2026–27: Every rational number has a terminating or non-terminating repeating decimal expansion.
Every irrational number has a non-terminating, non-repeating decimal expansion.
If p is a prime number, then √p is irrational — a key theorem in the NCERT 2026–27 syllabus.
To rationalise 1/(√a + √b), multiply by (√a − √b)/(√a − √b) using the conjugate.
Laws of exponents for real numbers: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ.
NCERT 2026–27: Representing irrational numbers on the number line using the Pythagoras theorem (spiral of irrationals).
Between any two distinct real numbers, there exist infinitely many real numbers.
📘 Key Definitions
Irrational Number
A real number that cannot be expressed as p/q (p, q integers, q ≠ 0). Its decimal expansion is non-terminating and non-repeating.
Terminating Decimal
A decimal that ends after a finite number of digits. E.g., 0.25, 1.375. Rational numbers with denominator having only 2 and/or 5 as prime factors.
Non-terminating Repeating
A decimal that never ends but has a repeating block. E.g., 0.333... = 1/3, 0.142857142857... = 1/7.
Rationalisation
Eliminating irrational numbers from the denominator by multiplying with the conjugate. NCERT 2026–27 includes two-term denominators.
Conjugate
For (√a + √b), the conjugate is (√a − √b). Their product (a − b) is rational, used in rationalisation.
Number Line
A straight line representing all real numbers. NCERT 2026–27: Irrational numbers like √2, √3 can be located geometrically using right triangles.
💡 NCERT 2026–27 Exam Tip
In the 2026–27 syllabus, questions on rationalising two-term denominators like 1/(√3 + √2) are important. Always use the conjugate and the identity (a+b)(a−b) = a²−b².