NCERT Class 9 Ganita Manjari Chapter 03 The World of Numbers PDF Download

Chapter 3: The World of Numbers

3.1. The Dawn of Mathematics: The Human Need to Count

Long before humanity built cities, formulated laws, or studied the stars, there existed a fundamental, practical necessity: the need to keep count. Mathematics did not begin in a classroom with equations on a board; it began in the dirt, on the bark of trees, and on bones.

Imagine you are living thousands of years ago in a small agricultural settlement along the banks of the Saraswati river. You have a herd of cattle. Every morning, they go out into the dense forests to graze, and every evening, they return. How do you ensure that a calf has not wandered off? Without words for numbers, and without written symbols, early humans solved this through a concept called one-to-one correspondence.

For every cow that left the settlement, the herder might place one pebble in a clay pot. In the evening, for every cow that returned, one pebble was removed. If the pot was empty at the end of the day, the herd was safe. If pebbles remained, cows were missing. This simple act of matching one object to another was the birth of the Natural Numbers (\( \mathbb{N} = \{1, 2, 3, 4, \ldots \} \)).

Teacher's Note

One-to-one correspondence is the foundation of counting itself. You are not just matching objects; you are assigning order and quantity. This is why every complete count relies on pairing items with numbers in a one-to-one way - without gaps, without repeats. This idea underpins all of counting, from ancient herders to today.

3.1.1 A History Written in Bone

While the decimal place-value system we use today was perfected in the Indian subcontinent, the earliest physical evidence of humanity recording natural numbers takes us deep into the heart of Africa. The first mathematicians did not use paper; they used tally marks carved into bone.

The Lebombo Bone, discovered in the Lebombo Mountains between South Africa and Swaziland, dates back approximately 35,000 years. It is a bone featuring 29 distinct, deliberately-carved uniformly-sized notches. Anthropologists and mathematicians believe this was not just random scratching, but a tool used as a lunar phase counter or a menstrual calendar, indicating that early humans were tracking time through natural numbers.

Even more fascinating is the Ishango bone, found near the headwaters of the Nile River in the Democratic Republic of Congo, dating to around 20,000 BCE. This bone contains three columns of asymmetrical notches. What makes the Ishango bone a mathematical marvel is the specific grouping of the tallies. One of the columns groups notches into 11, 13, 17, and 19 - the prime numbers between 10 and 20. Another column seems to demonstrate the concept of multiplication by 2 (doubling). These artefacts indicate that the abstract concept of a 'number' is indeed tens of thousands of years old.

[Figure 3.1: Representation of the prime number tally groupings found on the Ishango bone, See in your textbook]

3.1.2 The Indian Context: Trade and Astronomy

As civilisations advanced, so did the need for larger numbers. In the ancient urban centers of the Indus Valley Civilisation, such as Lothal and Harappa, standardised weights and measures were crucial for trade. A merchant trading terracotta pottery, lapis lazuli, or cotton with Mesopotamia needed a robust system of accounting.

During Vedic times, Indian philosophers were fascinated by and deeply pondered large numbers. In the Vedas, which go back thousands of years, names were given to all powers of 10 up to \( 10^{12} \) (which was called parardha). In the Lalitavistara in the 4th century BCE, Buddha describes names up to \( 10^{53} \), which is called tallakshana.

Expressing quantities in terms of powers of 10 was explicitly used in the Rigveda, thus setting the stage for the number system based on powers of 10 to be developed in India in the ensuing years, and which we now use around the world today. The development of the Indian numeral system in terms of place values and powers of 10 also helped pave the way for what is perhaps the most important mathematical invention in human history: the concept of zero.

Exercise Set 3.1

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?
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.
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.
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?

3.2 The Revolution of Shunya: When Nothing Became Something

For millennia, the number line started at 1. If you had five apples and gave all five away, you did not have a number to represent your state; you simply had a void, a lack of apples. Civilisations like the Babylonians and Mayans used placeholders - symbols to indicate an empty column in a number - but they did not treat 'nothing' as a number that you could add, subtract, and multiply.

It was in the work of Brahmagupta (628 CE) that the void was formally transformed into a number, which truly transformed mathematics. This monumental leap was in turn inspired by Indian philosophical traditions.

Teacher's Note

Before zero existed as a number, you could not perform operations on "nothing." If you subtract 5 from 5, what do you have? Zero allows you to answer: zero. This is why Brahmagupta's recognition of zero as a number - not just a placeholder - was revolutionary. It made the number system complete and allowed algebra to work the way it does today.

3.2.1 From Philosophy to Mathematics: The Concept of Shunyata

In the Upanishads and in the vast Buddhist literature starting well before the 7th century BCE, the concept of Shunyata (emptiness or nothingness)

Key Points

• One-to-one correspondence was how ancient humans first understood counting: by matching each object in one group to exactly one object in another group, such as matching cattle to pebbles.
• The Lebombo Bone (35,000 years old) and Ishango Bone (20,000 BCE) show that early humans recorded numbers through tally marks and even understood concepts like prime numbers and multiplication thousands of years ago.
• Ancient Indians recognized the importance of large numbers and powers of 10, which led to the development of the place-value decimal number system we use today.
• Zero was not always treated as a true number; it took the work of Brahmagupta in 628 CE to transform zero from a placeholder into an actual number that could be used in calculations, inspired by the philosophical concept of Shunyata.

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