Number system · संख्या पद्धति
Type p/q — see if it terminates or repeats, and why. Then practise every Class 9–10 number skill.
Number system games
Decimal expansion tool · संख्या पद्धति
Decimal expansion of p/q
The number family
| Natural (N) | 1, 2, 3, … |
|---|---|
| Whole (W) | 0, 1, 2, 3, … |
| Integers (Z) | …, −2, −1, 0, 1, 2, … |
| Rational (Q) | p/q with q ≠ 0 — decimals that terminate or repeat: 3/4, −2, 0.333… |
| Irrational | non-terminating, non-recurring: √2, √3, π, 0.1010010001… |
| Real (R) | all rational and irrational numbers — every point on the number line |
Key facts
- p/q (lowest terms) terminates exactly when q = 2ⁿ × 5ᵐ; otherwise it is non-terminating recurring.
- Recurring decimal → p/q: x = 0.272727…, 100x = 27.2727…, 99x = 27, x = 27/99 = 3/11.
- Rationalise: multiply by √a, or by the conjugate (√a − √b).
- Class 10: every composite number is a product of primes in one way (Fundamental Theorem of Arithmetic); for two numbers, HCF × LCM = product.
- √2, √3, √5 are irrational (proved by contradiction in NCERT Class 10).
- 22/7 is rational (3.142857142857…) — it is only an approximation of π, which is irrational.
More: rational and irrational · HCF and LCM · factor tree · exponents
For Class 9 (NCERT Mathematics 9, “Number Systems”) and Class 10 (“Real Numbers”). Every answer is computed and checked by a program.
A one-page chart to stick on the wall, and a worksheet to practise.
Number system
N ⊂ W ⊂ Z ⊂ Q ⊂ R · p/q terminates ⇔ q = 2ⁿ5ᵐ · HCF × LCM = product
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Number system — worksheet
Name: ____________________ Class: ______ Date: __________
1. Classify: −7, 0, 3/5, √7, 2.25, 0.121121112…
2. Without dividing, say which terminate: 7/16, 5/12, 11/125, 4/21
3. Write 0.474747… and 1.3333… as p/q.
4. Rationalise: 1/√7 and 1/(√5 + √2)
5. HCF(84, 126) = 42. Find their LCM.
6. Prove that √3 is irrational.
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