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Schooling · Class 9 · Mathematics · The World of Numbers · Chapter on a page

CBSE Class 9 · Mathematics · NCERT Ganita Manjari

The World of Numbers: chapter on a page

A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).

About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.

This note's current text has no applied fact-check on record.

Mind map

Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.

Mind map of The World of Numbers: Key ideas, Formulas and facts, Watch out forKey ideasNatural numbersWhole numbersIntegersRational numbersIrrational numbersReal numbersThe decimal expansion of a rational…Formulas and factsrationalTerminating decimalsRepeating decimalsNon-terminating, non-repeating…√2, √3, √5, √7 and the square root…The sum or difference of a rational…The product or quotient of a…Watch out forThinking zero is not a whole numberBelieving all square roots are…Writing 22/7 is exactly equal to πAssuming a long decimal must be…Forgetting that integers are also…The World of Numbers
The mind map as a list
  • Key ideas
    • Natural numbers
    • Whole numbers
    • Integers
    • Rational numbers
    • Irrational numbers
    • Real numbers
    • The decimal expansion of a rational…
  • Formulas and facts
    • rational
    • Terminating decimals
    • Repeating decimals
    • Non-terminating, non-repeating…
    • √2, √3, √5, √7 and the square root…
    • The sum or difference of a rational…
    • The product or quotient of a…
  • Watch out for
    • Thinking zero is not a whole number
    • Believing all square roots are…
    • Writing 22/7 is exactly equal to π
    • Assuming a long decimal must be…
    • Forgetting that integers are also…

What this chapter is about

This chapter explores the vast landscape of numbers that you use in mathematics. Starting from the counting numbers you learned as a child, it builds up to the complete system of real numbers. You will understand how natural numbers, whole numbers, integers, rational numbers and irrational numbers relate to one another, and why each type was needed as mathematics grew.

Key ideas

  • Natural numbers are the counting numbers 1, 2, 3, 4, … that we use to count objects. They do not include zero or negative numbers.
  • Whole numbers are natural numbers together with zero: 0, 1, 2, 3, 4, … They allow us to represent "nothing" as a quantity.
  • Integers include all whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, … They let us represent quantities below zero, like temperatures or debts.
  • Rational numbers are numbers that can be written as p/q, where p and q are integers and q is not zero. Examples include 3/4, −2/5, 7 (which is 7/1), and 0.25 (which is 1/4).
  • Irrational numbers cannot be expressed as p/q for any integers p and q. Their decimal expansions go on forever without repeating. Examples include √2, √3, and π.
  • Real numbers comprise all rational and irrational numbers together. Every point on the number line corresponds to a real number, and every real number corresponds to a point on the number line.
  • The decimal expansion of a rational number either terminates (like 0.75) or repeats in a pattern (like 0.333… or 0.142857142857…).
  • Every natural number is a whole number, every whole number is an integer, every integer is a rational number, and every rational number is a real number. This forms a nested hierarchy.

Formulas and facts to remember

  • A number is rational if and only if it can be written as p/q where p, q are integers and q ≠ 0.
  • Terminating decimals have a finite number of digits after the decimal point; they are always rational.
  • Repeating decimals have a block of digits that repeats forever; they are also rational.
  • Non-terminating, non-repeating decimals are irrational.
  • √2, √3, √5, √7 and the square root of any prime number are irrational.
  • The sum or difference of a rational and an irrational number is always irrational.
  • The product or quotient of a non-zero rational number and an irrational number is irrational.

Common mistakes

  • Thinking zero is not a whole number → Zero is included in whole numbers; it represents the absence of quantity.
  • Believing all square roots are irrational → Square roots of perfect squares like √16 = 4 or √25 = 5 are rational.
  • Writing 22/7 is exactly equal to π → 22/7 is only an approximation; π is irrational and equals 3.14159265…
  • Assuming a long decimal must be irrational → Check whether the digits eventually repeat; 0.166666… is rational (1/6).
  • Forgetting that integers are also rational → Every integer n can be written as n/1, making it rational.

Quick revision

  • Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real; Irrational numbers fill the gaps.
  • Rational means expressible as p/q with integers p, q and q ≠ 0.
  • Terminating or repeating decimal → rational; non-terminating non-repeating → irrational.
  • √(prime) is always irrational.
  • Every point on the number line is a real number.