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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 · 3 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.

3.3 INTEGERS: EXPANDING THE HORIZON

Brahmagupta extended the number system beyond zero by introducing negative numbers to represent debts (rīna), contrasting with positive numbers representing fortunes (dhana). He recognized that subtracting a larger number from a smaller one (e.g., 3 - 5) required numbers less than zero. This led to the formal introduction of integers (ℤ), which include positive numbers, negative numbers, and zero. Brahmagupta provided arithmetic rules for integers that remain valid today: adding fortunes or debts, subtracting zero, and multiplying fortunes and debts. For example, the product of two debts is a fortune (negative × negative = positive). This extension allowed mathematics to model real-world situations such as debts and losses, greatly expanding its scope.

📊 Diagram: Fig. 3.2 shows the number line extending to include negative numbers to the left of zero and positive numbers to the right.

🧪 Activity: Think about why a negative times a negative equals a positive by considering the removal of debts.

🔗 Connection: Prepares for the introduction of rational numbers and fractions to represent parts of a whole.

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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