Exercise 1.1 — Rational Numbers Revision

What Exercise 1.1 Covers

Exercise 1.1 from Chapter 1 — Real Numbers is the first hands-on practice set in Class 9 Mathematics. It tests whether students can correctly identify rational numbers across different number sets, locate them on the number line, find rational numbers between two given values, convert fractions to decimals, and reverse the process. These skills are examined directly in CBSE, Telangana, and Andhra Pradesh board papers and form the bedrock of the entire Real Numbers chapter.

Identifying Rational Numbers — Questions 1 and 2

A rational number is any number that can be written as p/q where p and q are integers and q ≠ 0. This definition is broader than most students initially expect — it includes negative numbers, zero, whole numbers, decimals, and repeating decimals, all at once. Question 2 sharpens this understanding by asking for examples that satisfy one condition but not another.

• Rational but not an integer: 0.35, 9/5, −7.232323… — these cannot be written without a fractional part.
• Whole number but not a natural number: only 0 fits this description.
• Integer but not a whole number: any negative integer such as −3, −7, or −10.
• Natural, whole, integer, and rational all at once: any positive integer such as 2, 7, or 205.
• Integer but not a natural number: 0 or any negative integer like −3 or −125.

Finding Rational Numbers Between Two Values — Questions 3 and 4

To find n rational numbers between two values in a single step, rewrite both as fractions with denominator (n + 1), then list the fractions in between.

For Question 3, finding five rationals between 1 and 2: use denominator 6 (= 5 + 1). Write 1 = 6/6 and 2 = 12/6. The five numbers in between are 7/6, 8/6, 9/6, 10/6, and 11/6.

Question 4 involves fractions rather than integers: finding three rationals between 3/5 and 2/3. First make the denominators equal — 3/5 = 9/15 and 2/3 = 10/15. These are already adjacent, so multiply both by 4 to get 36/60 and 40/60. Now three numbers sit in between: 37/60, 38/60, and 39/60.

Number Line Representation — Question 5

Question 5 asks students to mark 8/5 and −8/5 on the number line. The method is to convert to a mixed number first. Since 8/5 = 1 + 3/5, it lies between 1 and 2, at the third of five equal divisions after 1. For −8/5 = −1 − 3/5, the same position is mirrored on the negative side, between −1 and −2. Dividing the unit interval into exactly 5 parts (matching the denominator) and counting carefully is the key technique.

Converting Fractions to Decimals — Question 6

Question 6 is split into two groups. The first group produces terminating decimals — the long division ends cleanly. The second produces non-terminating recurring decimals — a block of digits repeats forever, shown with a bar over the repeating part.

• 242/1000 = 0.242  (terminating)
• 354/500 = 0.708  (terminating)
• 2/5 = 0.4  (terminating)
• 115/4 = 28.75  (terminating)
• 2/3 = 0.6̄  (recurring)
• −25/36 = −0.694̄  (recurring)
• 22/7 = 3.̄142857  (recurring — the famous approximation of π)
• 11/9 = 1.2̄  (recurring)

Converting Decimals Back to p/q Form — Questions 7 and 8

Question 7 covers terminating decimals, which convert by placing the digits over the matching power of 10 and simplifying. For example, 10.25 = 1025/100 = 41/4.

Question 8 covers recurring decimals. The rules are:

• Purely recurring (e.g. 0.5̄): place the repeating block over the same number of 9s → 0.5̄ = 5/9.
• Recurring block of two digits (e.g. 0.3̄6̄): place over 99 → 0.3̄6̄ = 36/99 = 4/11.
• Mixed — some digits recur, some don't (e.g. 3.127̄): subtract the non-recurring part from the full number and place over the correct combination of 9s and 0s. Here, 3.127̄ = 3 + (127 − 12)/900 = 3 + 115/900 = 563/180.

Testing for Terminating Decimals Without Dividing — Question 9

Question 9 applies the key rule from the introduction: a fraction terminates if and only if its denominator (in lowest terms) has no prime factors other than 2 and 5.

• 3/25: denominator = 5² = 2⁰ × 5² → terminating
• 11/18: denominator = 2 × 3² → contains 3, so non-terminating recurring
• 13/20: denominator = 2² × 5 → terminating
• 41/42: denominator = 2 × 3 × 7 → contains 3 and 7, so non-terminating recurring

What This Exercise Prepares You For

Exercise 1.1 builds the rational number fluency needed before the chapter introduces irrational numbers in Exercise 1.2. The decimal-to-fraction conversion skill is tested again in later exercises and in Class 10. The number line technique here also links to Coordinate Geometry, where placing values precisely on axes is essential. For a solid foundation, revisit the Real Numbers Introduction before attempting this exercise.