Exercise 1.2 — Irrational Numbers

What Are Irrational Numbers?

Exercise 1.2 introduces one of the most important ideas in Class 9 Mathematics — irrational numbers. You already know that rational numbers are expressible as p/q. An irrational number is simply one that cannot be written that way. The clearest entry point is √2: since no integer squares to give exactly 2, √2 cannot be a fraction, and its decimal expansion — 1.41421356… — continues forever without any repeating pattern. Numbers with non-terminating, non-recurring decimal expansions are irrational, and they are denoted by Q′ or S. This topic is part of the Class 9 syllabus for CBSE, Telangana, and Andhra Pradesh boards.

Recognising Irrational Numbers — The Key Rules

• Square roots of non-perfect squares are irrational. √2, √3, √5, √6, √7, √8 are all irrational. But √1 = 1, √4 = 2, √9 = 3 are rational because 1, 4, and 9 are perfect squares.
• Non-terminating, non-recurring decimals are irrational. A decimal that goes on forever without any repeating block cannot be written as p/q.
• π is irrational. Although we use 22/7 in calculations, that is only an approximation. The true value of π = 3.14159265… is non-terminating and non-recurring. C/d always involves at least one irrational measurement, so π itself is irrational.
• If n is a natural number that is not a perfect square, then √n is irrational. This single rule covers infinitely many cases.

Classifying Numbers — Question 1

Question 1 asks you to classify six numbers as rational or irrational. The reasoning for each follows directly from the rules above.

• √27 — 27 is not a perfect square, so √27 is irrational.
• √441 — 441 = 21², a perfect square, so √441 = 21 is rational.
• 30.232342345… — non-terminating and non-repeating, so irrational.
• 7.484848… — non-terminating but repeating (48 repeats), so rational.
• 11.2132435465 — terminates, so rational.
• 0.3030030003… — the gaps between 3s keep growing; it never repeats, so irrational.

Finding Irrational Numbers Between Two Values — Questions 3 and 4

Just as there are infinitely many rational numbers between any two values, there are also infinitely many irrational numbers. To write one, convert both given numbers to their decimal forms, then invent a decimal between them that is clearly non-repeating.

For Question 3, between 5/7 = 0.714285̄ and 7/9 = 0.7̄, a valid irrational number is 0.72723724… — the digits do not follow any repeating pattern. For Question 4, between 0.70 and 0.77, two irrational numbers are 0.71712713… and 0.74865498… There is no unique answer; any non-terminating, non-repeating decimal in the range works.

The formula method also applies: if a and b are positive rationals such that a × b is not a perfect square, then √(ab) is an irrational number between a and b. For example, the irrational number between 3 and 4 using this method is √(3 × 4) = √12 = 2√3.

Finding Square Roots by Long Division — Questions 5 and 6

Questions 5 and 6 ask students to compute √5 and √7 to several decimal places using the long division method. This is a standard board exam technique. The key steps are: group digits in pairs from the decimal point outward, find the largest integer whose square fits, double the current quotient to form the next divisor, and bring down the next pair. Working carefully:

• √5 ≈ 2.236 (to 3 decimal places)
• √7 ≈ 2.645751 (to 6 decimal places)

Both results confirm irrational behaviour — the digits continue without repeating. Practising this method builds number sense and is directly examined in CBSE and state board papers.

Locating Irrational Numbers on the Number Line — Question 7

Question 7 asks you to mark √10 on the number line. The geometric approach uses the Pythagorean theorem: express the target as a sum of two squares. Since 10 = 9 + 1 = 3² + 1², draw a right triangle with one leg of length 3 (along the number line from 0) and a perpendicular leg of length 1. The hypotenuse has length √10. Using a compass to transfer this length onto the number line marks the exact position of √10 — approximately between 3 and 4, closer to 3.

Real Numbers — Putting It All Together

The set of real numbers (R) is simply the union of all rational and all irrational numbers. Every number you will encounter in Class 9 and 10 is a real number. Question 9 consolidates this with true/false statements — the important ones to remember are: every irrational is real (true), every rational is real (true), but not every real number is rational (true), and the claim that all real numbers are irrational (false — 7/5 is real and rational).

What This Exercise Prepares You For

Understanding irrational numbers is essential for the rest of Chapter 1, including surds and rationalisation of denominators. These ideas also connect to Exercise 1.1 on rational numbers and forward to Class 10 Real Numbers where the irrationality of √2 and √3 is formally proved. The number line construction technique from Question 7 reappears in Coordinate Geometry.