Summary
Chapter 3 of NCERT Class 9 Maths, "The World of Numbers", traces how number systems grew from natural numbers and zero to integers, rational numbers, irrational numbers and real numbers, covering Brahmagupta's rules, decimal expansions and the proof that sqrt(2) is irrational.
This chapter tells the story of how numbers evolved to meet human needs, from tally marks on the Lebombo and Ishango bones to the Indian invention of zero (Shunya), formalised by Brahmagupta in 628 CE. It introduces integers as 'fortunes' and 'debts', defines rational numbers as p/q (q not 0), and shows they are dense on the number line. It then explores irrational numbers like sqrt(2) and pi, the proof of irrationality by contradiction, and how decimal expansions distinguish rational (terminating or repeating) from irrational numbers, finally uniting them as the real numbers.
Key points & formulas
- 01Natural numbers arose from counting using one-to-one correspondence
- 02Brahmagupta formalised zero and its arithmetic rules in 628 CE
- 03Integers (Z) include positive numbers, zero and negative numbers (debts)
- 04Rational numbers are p/q with integers p, q and q not equal to 0
- 05Rational numbers are dense; one exists between any two via their average
- 06Proof by contradiction shows sqrt(2) is irrational; pi is also irrational
- 07Rational decimals terminate or repeat; irrational decimals never repeat
Frequently asked questions
01How did Brahmagupta define zero and negative numbers?
In his work the Brahmasphutasiddhanta (628 CE), Brahmagupta defined zero as the result of subtracting a number from itself (a - a = 0). He introduced negative numbers as 'debts' (rina) and positive numbers as 'fortunes' (dhana), giving rules such as the product of two debts is a fortune.
02What is a rational number in Class 9 Maths Chapter 3?
A rational number is any number that can be expressed in the form p/q, where p and q are integers and q is not equal to 0. Rational numbers include the natural numbers, whole numbers and integers.
03How does the chapter prove that sqrt(2) is irrational?
It uses proof by contradiction, first given by Hippasus (c. 400 BCE). Assuming sqrt(2) = p/q in simplest form leads to both p and q being even, which contradicts them sharing no common factors. So sqrt(2) cannot be written as a fraction and is irrational.
04How can you tell if a rational number has a terminating decimal?
Write the number in lowest terms and find the prime factors of the denominator. The decimal expansion terminates precisely when the prime factors of the denominator are only 2, only 5, or both 2 and 5, because the denominator can then be made a power of 10.
More chapters in Ganita Manjari
This is the complete Ganita Manjari Chapter 3 as published by NCERT — every diagram, solved example, and exercise included, free. Browse all NCERT Class 9 textbooks.
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