Schooling · Class 8 · Mathematics · A Story of Numbers · Chapter on a page
CBSE Class 8 · Mathematics · NCERT Ganita Prakash Part-I
A Story 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.
The mind map as a list
- Key ideas
- Natural numbers
- Whole numbers
- Integers
- Rational numbers
- density
- Every integer is a rational number…
- Rational numbers can be placed…
- Formulas and facts
- Fact or rule
- Standard form of a rational number
- Watch out for
- Thinking 0 is not a whole number
- Writing a rational number with…
- Believing there is no number between…
- Confusing equivalent fractions with…
- Placing negative fractions to the…
What this chapter is about
This chapter takes you on a journey through the world of numbers — exploring how numbers are classified, what makes each type special, and how they relate to one another. You have already worked with whole numbers, integers and fractions in earlier classes. Now you will see a bigger picture: the number system as a whole, including rational numbers and a first glimpse at numbers that cannot be written as fractions.
Key ideas
- Natural numbers are the counting numbers: 1, 2, 3, 4, … They begin at 1 and go on without end.
- Whole numbers include all natural numbers plus zero: 0, 1, 2, 3, …
- Integers extend whole numbers to include negative numbers: …, −3, −2, −1, 0, 1, 2, 3, …
- Rational numbers are numbers that can be written as p/q where p and q are integers and q is not zero. Examples: 3/4, −5/2, 7 (which is 7/1).
- Between any two rational numbers, you can always find another rational number — in fact, infinitely many. This property is called density.
- Every integer is a rational number (write it with denominator 1), but not every rational number is an integer.
- Rational numbers can be placed exactly on the number line; their position depends on both numerator and denominator.
- Some numbers, like the square root of 2, cannot be written as p/q. These are called irrational numbers and hint at an even larger number system you will study later.
Formulas and facts to remember
- Fact or rule: Natural numbers ⊂ Whole numbers ⊂ Integers ⊂ Rational numbers · Meaning: Each set is contained inside the next larger set.
- Fact or rule: A rational number is p/q with q ≠ 0 and p, q integers · Meaning: Defines the form; q cannot be zero because division by zero is undefined.
- Fact or rule: To find a rational number between a and b: (a + b)/2 · Meaning: The average of two rationals is also rational and lies between them.
- Fact or rule: Equivalent rational numbers: p/q = (p × k)/(q × k) for any non-zero integer k · Meaning: Multiplying numerator and denominator by the same number keeps the value unchanged.
- Fact or rule: Standard form of a rational number · Meaning: Denominator is positive, numerator and denominator share no common factor other than 1.
Common mistakes
- Thinking 0 is not a whole number → 0 is the first whole number; it just is not a natural number.
- Writing a rational number with denominator 0 → Division by zero is undefined, so q must never be 0.
- Believing there is no number between two very close fractions → There are always infinitely many rational numbers between any two distinct rationals.
- Confusing equivalent fractions with equal fractions → 2/4 and 1/2 are the same rational number, not two different numbers.
- Placing negative fractions to the right of 0 on the number line → Negative numbers always lie to the left of 0.
Quick revision
- Natural ⊂ Whole ⊂ Integer ⊂ Rational — each set fits inside the next.
- Rational number = p/q where p, q are integers and q ≠ 0.
- Between any two rationals lie infinitely many more rationals.
- To locate p/q on the number line, divide the unit into q equal parts and count p parts.
- Every integer can be written as a rational number with denominator 1.