Exercise 1.2 — Number Line

What Does Exercise 1.2 Cover?

Exercise 1.2 moves from the algebraic properties of rational numbers to their geometric representation — placing them on a number line and finding rational numbers between two given values. These skills are regularly tested in Class 8 board exams for CBSE, Telangana, and Andhra Pradesh, and they build intuition that is essential for understanding real numbers in Class 9.

Representing Rational Numbers on the Number Line

The key idea is to first convert an improper fraction to a mixed number so you know which two consecutive integers it falls between, then divide that gap into equal parts matching the denominator.

• 9/7 = 1 and 2/7 — lies between 1 and 2. Divide the segment from 1 to 2 into 7 equal parts. Mark the 2nd point to the right of 1 — that is 9/7.
• −7/5 = −1 and −2/5 — lies between −2 and −1. Divide the segment from −1 to −2 into 5 equal parts. Mark the 2nd point to the left of −1 — that is −7/5.

For Question 2, representing −2/13, 5/13, and −9/13 on the same number line: since all denominators are 13, divide both the (−1, 0) and (0, 1) segments into 13 equal parts. Then −9/13 and −2/13 are marked between −1 and 0, while 5/13 is marked between 0 and 1.

Finding Rational Numbers Between Two Values

There are infinitely many rational numbers between any two rational numbers. The standard method is to convert both numbers to like fractions (same denominator), then pick any fraction whose numerator lies between the two numerators. If you need more numbers, multiply both numerator and denominator by a larger factor to widen the gap.

Worked Examples — Questions 4, 5, and 6

Q4. Find 12 rational numbers between −1 and 2.
Convert both to tenths: −1 = −10/10 and 2 = 20/10. There are now 29 fractions between them. Picking any 12, for example: −9/10, −7/10, −5/10, −3/10, −1/10, 0, 2/10, 5/10, 9/10, 11/10, 15/10, 19/10.

Q5. Find a rational number between 2/3 and 3/4.
Convert to a common denominator: 2/3 = 16/24 and 3/4 = 18/24. The rational number between them is 17/24.

Q6. Find ten rational numbers between −3/4 and 5/6.
Convert to twelfths: −3/4 = −9/12 and 5/6 = 10/12. Ten values from the range, for example: −8/12, −7/12, −5/12, −3/12, −2/12, 0, 3/12, 5/12, 6/12, 8/12.

Common Mistakes to Avoid

• When representing a negative fraction on the number line, remember that moving left means moving in the negative direction — −7/5 is 2 steps left of −1, not 2 steps left of 0.
• There is no single "correct" set of rational numbers between two values — any valid fractions in the range are acceptable answers, so your answer may differ from the textbook and still be right.
• To find a rational number between two fractions quickly, you can also use the mean method: add the two numbers and divide by 2. For 2/3 and 3/4: (2/3 + 3/4) ÷ 2 = (8/12 + 9/12) ÷ 2 = 17/24.
• When asked for multiple rational numbers, always increase the denominator (multiply top and bottom) to create enough room between the two values.

What This Exercise Prepares You For

Representing numbers on the number line and finding values between two rationals directly feeds into the study of real numbers in Class 9, where irrational numbers like √2 are also placed on the number line. It also connects to comparing quantities and strengthens the fraction fluency needed throughout Class 8. For the theory behind rational number types, revisit the Introduction to Rational Numbers.