Chapter at a glance
The story of numbers from counting to real numbers: natural numbers, zero, integers, rationals and irrationals, and how decimals reveal which is which.
What this chapter covers
- 3.1 The Dawn of Mathematics: The Human Need to Count
- 3.3 Integers: Expanding the Horizon
- 3.4 Filling the Spaces: Fractions and Rational Numbers
- 3.5 Irrational Numbers
- 3.6 Real Numbers: Decimals and Cyclic Patterns
- 3.7 Conclusion: The Never-Ending Journey
Key points
- Zero (śhūnya) was brought into mathematics as a number by Brahmagupta (628 CE), who also gave rules for negative numbers (‘debts’ and ‘fortunes’).
- Rational numbers p/q (q ≠ 0) are dense: between any two there is always another.
- Irrational numbers such as √2 and π cannot be written as fractions; the first irrationality proof (of √2, by contradiction) is attributed to Hippasus.
- Rational numbers have terminating or repeating (cyclic) decimals; irrational numbers have non-terminating, non-repeating decimals.
Read the NCERT chapter
Read this chapter in the official NCERT textbook Ganita Manjari (2026-27 edition) for free on ncert.nic.in.
Source: NCERT, Ganita Manjari, Grade 9 (2026-27). NCERT chapter PDF. Spotted a mistake? Email edura.class9.yt@gmail.com. Last updated 11 October 2026.