CBSE • Class 9 • Mathematics • Chapter 1 (Number Systems) • Exercise 1.1

Exercise 1.1: Number Systems — NCERT Solutions

Rational vs irrational numbers — closure, density on the line, and locating simple surds.

Aligned to the latest NCERT 2024-25 edition • 4 questions in this exercise • Free plan, no credit card

Open this exercise in the AI tutorBack to Chapter 1

What this exercise covers

Rational numbersIrrational numbersNumber line density

Step-by-step solutions — Exercise 1.1

2 solved questions • Each solution includes a Socratic hint, full working and a common-mistake callout • Last reviewed 2026-09-03

Exercise 1.1 Q1 • 2 marks

Is zero a rational number? Can you express it in the form pq\frac{p}{q} where pp and qq are integers and q≠0q \neq 0?
Hint (Socratic — try this first)▾
Can you find at least one pair of integers whose ratio equals zero?
Step-by-step solution▾

Definition: A number is rational if it can be written as pq\frac{p}{q} where p,qp, q are integers and q≠0q \neq 0.

Checking zero: 0=01=02=05=…0 = \frac{0}{1} = \frac{0}{2} = \frac{0}{5} = \dots

Here p=0p = 0 and qq can be any non-zero integer. Both are integers and the denominator is not zero.

Conclusion: Yes, zero is a rational number. It can be expressed in infinitely many ways as pq\frac{p}{q}.

Common mistake:
Students think zero cannot be rational because 'you can't divide by zero' — confusing the numerator being zero with the denominator being zero.
Open this question in the AI tutor →

Exercise 1.1 Q2 • 3 marks

Find three rational numbers lying between 25\frac{2}{5} and 35\frac{3}{5}.
Hint (Socratic — try this first)▾
What happens to the gap between two fractions if you make their denominators larger by multiplying top and bottom by the same number?
Step-by-step solution▾

Idea: To find rationals between two fractions, give them a larger common denominator so there is room in between.

Multiply numerator and denominator by 1010: 25=2050,35=3050\frac{2}{5} = \frac{20}{50}, \qquad \frac{3}{5} = \frac{30}{50}

Now choose numbers between 2050\frac{20}{50} and 3050\frac{30}{50}: 2150,2550,2750\frac{21}{50}, \quad \frac{25}{50}, \quad \frac{27}{50}

Conclusion: Three rational numbers between 25\frac{2}{5} and 35\frac{3}{5} are 2150,2550,2750\frac{21}{50}, \frac{25}{50}, \frac{27}{50} (many other answers are possible).

Common mistake:
Students say there are only a finite number of rationals between two fractions, when in fact there are infinitely many.
Open this question in the AI tutor →

How to approach Exercise 1.1

  1. Re-read the chapter summary first. Open Number Systems and refresh the key concepts: Rational numbers, Irrational numbers, Decimal expansions, Laws of exponents.
  2. Try each problem yourself before opening the solution. Ten focused minutes per problem usually beats reading two finished solutions.
  3. Use Guru AI for the questions you get stuck on. The Socratic AI tutor walks you through it with hints instead of dictating the answer.
  4. Mark the questions you got wrong and revisit them after 24 hours — the spacing is what locks the method into long-term memory.

All exercises in Number Systems

  1. Exercise 1.1 — Rational vs irrational numbers — closure, density on the line, and locating simple surds.
  2. Exercise 1.2 — Decimal expansions of rationals — terminating and non-terminating recurring forms.
  3. Exercise 1.3 — Real numbers — operations and representability; approximating irrational numbers.
  4. Exercise 1.4 — Laws of exponents extended to rational and real bases and exponents.
  5. Exercise 1.5 — Rationalising denominators with surds and revisiting identities with roots.
  6. Exercise 1.6 — Representing irrational numbers like √x on the number line using constructions.

Exercise 1.2 →

Decimal expansions of rationals — terminating and non-terminating recurring forms.

Solve Exercise 1.1 with AI guidance

Free plan. No credit card. Works on any device.

Start Free