NCERT Solutions for Class 9th Maths Chapter 3 The World of Numbers

About this chapter

Natural numbers grew out of one-to-one correspondence; integers ℤ appear once Brahmagupta (628 CE) turns śhūnya into a number and reads negatives as debts (ṛiṇa) against fortunes (dhana). A rational number is any number of the form p/q with p, q integers and q ≠ 0. ℚ is closed under +, −, × and under ÷ except by zero. ℚ is dense: the average (a + b)/2 of any two rationals is a rational lying strictly between them, so between any two points there are infinitely many rationals. √2 is irrational. Hippasus' proof by contradiction assumes √2 = p/q in lowest terms, forces both p and q to be even, and contradicts the assumption. A rational number in lowest terms p/q has a terminating decimal exactly when q has no prime factor other than 2 and 5; otherwise a remainder must repeat, so the decimal r

• .1 The Dawn of Mathematics: The Human Need to Count
• .3.1 The Arithmetic of Integers
• .3 Integers: Expanding the Horizon
• .4 Filling the Spaces: Fractions and Rational Numbers
• .4.1 Representation of Rational Numbers on the Number Line
• .4.2 The Density of Rational Numbers
• .5 Irrational Numbers
• .5.1 The Proof of Irrationality of √2
• .5.2 Construction of Length √n
• .6.1 Rational Decimals: Terminating and Repeating