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MathematicsClass 9The Dawn of Mathematics: The Human Need to Count

The Dawn of Mathematics: The Human Need to Count | Class 9 Mathematics Notes

By ConceptScroll Team · Published on 17 July 2026 · 4 min read

The Dawn of Mathematics: The Human Need to Count – this guide gives you a concise, exam-ready overview of The Dawn of Mathematics: The Human Need to Count from Class 9 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.

3.4 FILLING THE SPACES: FRACTIONS AND RATIONAL NUMBERS

As societies became more complex, the need to represent parts of a whole arose, leading to fractions. Fractions are numbers expressed as p/q, where p and q are integers and q ≠ 0. Negative fractions also exist as additive inverses of positive fractions. Combining integers and fractions (both positive and negative) forms the set of rational numbers (ℚ). Rational numbers can be equal even if their fractional forms differ, as in equivalent fractions (e.g., -1/3 = -2/6). Rational numbers are closed under addition, subtraction, multiplication, and division (except division by zero). Arithmetic laws such as equality, addition, subtraction, multiplication, division, commutativity, and distributivity apply to rational numbers. Representing rational numbers on the number line involves dividing the unit interval into equal parts according to the denominator and moving p parts from zero to the right (positive) or left (negative). The absolute value of a rational number is its distance from zero on the number line. Rational numbers are dense, meaning between any two rational numbers, there is always another rational number.

📊 Diagram: Fig. 3.3 shows integers on the number line with zero as origin. Fig. 3.4 shows rational numbers like 1/2 and -3/4 between integers. Fig. 3.5 and 3.6 illustrate representing fractions greater than 1 on the number line.

🧪 Activity: Try representing fractions like 8/5 and -7/4 on the number line and find rational numbers between given numbers.

🔗 Connection: Sets the stage for understanding irrational numbers and the completeness of the number line.

Frequently asked questions

1. A merchant in the port city of Lothal is exchanging bags of spices for copper ingots. He receives 15 ingots for every 2 bags of spices. If he brings 12 bags of spices to the market, how many copper ingots will he leave with?

Given: 15 ingots for every 2 bags of spices.

To find: Number of copper ingots for 12 bags of spices.

Step 1: Find the number of ingots per bag = 15/2 = 7.5 ingots per bag. Step 2: For 12 bags, total ingots = 7.5 × 12 = 90 ingots.

Therefore, the merchant will leave with 90 copper ingots.

2. Look at the sequence of numbers on one column of the Ishango bone: 11, 13, 17, 19. What do these numbers have in common? List the next three numbers that fit this pattern.

The numbers 11, 13, 17, and 19 are all prime numbers. They are consecutive prime numbers starting from 11.

The next three prime numbers after 19 are 23, 29, and 31.

Therefore, the next three numbers in the pattern are 23, 29, and 31.

3. We know that Natural Numbers are closed under addition (the sum of any two natural numbers is always a natural number). Are they closed under subtraction? Provide a couple of examples to justify your answer.

Natural numbers are not closed under subtraction.

Example 1: 5 and 3 are natural numbers. 5 - 3 = 2, which is a natural number.

Example 2: 3 and 5 are natural numbers. 3 - 5 = -2, which is not a natural number.

Since subtraction of two natural numbers can result in a number that is not natural (like a negative number), natural numbers are not closed under subtraction.

4. Ancient Indians used the joints of their fingers to count, a practice still seen today. Each finger has 3 joints, and the thumb is used to count them. How many can you count on one hand? How does this relate to the ancient base-12 counting systems?

Each finger (except the thumb) has 3 joints. There are 4 fingers (index, middle, ring, little) on one hand.

Number of joints counted = 4 fingers × 3 joints each = 12 joints.

The thumb is used to point to each joint to count.

This method allows counting up to 12 on one hand, which relates to the ancient base-12 (duodecimal) counting system used by Indians and other ancient cultures.

Thus, the finger-joint counting method naturally supports the base-12 system.

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