Number Systems | Maths Grade 9 CBSE

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The families of numbers

Numbers come in nested families — each one bigger than the last, until you reach the real numbers that fill the whole line. Tap each term to see what it contains and an example.

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The three big ideas

The families nest inside one another: natural ⊂ whole ⊂ integers ⊂ rational. Each step just adds more numbers.
Natural 1, 2, 3, … → add 0 to get whole → add the negatives to get integers → allow p/q fractions to get rationals.
The irrationals (like √2 and π) sit outside the rationals. Put both groups together and you get the real numbers — everything on the number line.

Rational — can be written as p/q (q ≠ 0). Its decimal either terminates (0.75) or repeats (0.333…).
Irrational — its decimal is non-terminating and non-repeating, so it cannot be written as p/q. e.g. √2, √3, π.

Worked example. Is 0.3333… rational? Is √2?

0.3333… repeats, and 0.333… = 1/3 — it can be written as p/q, so it is rational.

√2 = 1.41421356… never ends and never repeats, so it cannot be written as p/q — it is irrational.

Common mistake: thinking 22/7 is π. 22/7 is only an approximation of π — it is itself a rational number and is not equal to π. The real π is irrational, its digits go on forever without repeating.

Every real number is a unique point on the number line, and every point is a real number — a perfect one-to-one match.
Even an irrational like √2 has its own exact spot: build a right triangle with both legs 1 unit, and its diagonal length √2 can be marked off on the line with a compass.
So the line has no gaps — rationals and irrationals together fill it completely.

Where you'll meet it

Irrationals in the real world

√2 — the diagonal of a square

Make a square 1 unit on each side and its diagonal is exactly √2 units. Any time a carpenter checks a corner is square, or a screen size is quoted, this √2 diagonal is doing the work — an irrational number you can actually draw.

π — in every circle

Every wheel, pipe, plate and clock face uses π to link a circle's diameter to its circumference and area. π is irrational, so we round it (3.14 or 22/7) for sums — but the true value never ends.

Check yourself

Competency quiz

Modelled on the competency-based pattern — MCQ, assertion–reason and a case study, testing whether you can use the ideas, not just recall them.

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