Number Systems — Mathematics Class 9 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Number Systems (Class 9) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Number Systems (CBSE & HBSE)
Foundation of the entire Class 9 syllabus. CBSE focuses on representing real numbers on the number line, rationalisation, and laws of exponents. HBSE leans on direct conversion (recurring decimal ↔ fraction) and surd simplification.
Rational, Irrational and Real Numbers
Classification of Numbers
| Set | Symbol | Members |
|---|---|---|
| Natural | ℕ | 1, 2, 3, ... |
| Whole | W | 0, 1, 2, 3, ... |
| Integers | ℤ | ..., −2, −1, 0, 1, 2, ... |
| Rational | ℚ | p/q form (q ≠ 0) |
| Irrational | T | cannot be written as p/q |
| Real | ℝ | all rationals + irrationals |
Rational Numbers — Key Facts
- Between any two rationals, infinitely many rationals exist (use average n times).
- Decimal form: either terminating (e.g. 0.75) or non-terminating recurring (e.g. 0.333…).
- Inserting rationals: between a and b take (a+b)/2.
Irrational Numbers
- Non-terminating non-recurring decimals: √2, √3, π, e.
- π is irrational, but 22/7 is a rational approximation (NOT equal to π).
Real Numbers
ℝ = ℚ ∪ T. Every real number corresponds to a unique point on the number line and vice-versa (Completeness Property).
Successive Magnification (Representing Decimals)
To locate 3.7654 on the number line:
- First divide 3 → 4 into 10 parts; pick 3.7 → 3.8.
- Divide 3.7 → 3.8 into 10 parts; pick 3.76 → 3.77.
- Continue until the required precision.
CBSE Tip: Always state whether each step is a magnification by 10 — examiners look for this phrase.
Decimal Expansion & Conversion
Decimal Expansion of Rationals
| Type | Condition on p/q (lowest terms) |
|---|---|
| Terminating | q = 2ⁿ × 5ᵐ |
| Non-terminating recurring | q has any prime factor other than 2 or 5 |
Converting Recurring Decimal → Fraction
Pure recurring (0.7̄): let x = 0.777… → 10x = 7.777… → 9x = 7 → x = 7/9.
Mixed recurring (0.23̄): let x = 0.2333… → 10x = 2.333…, 100x = 23.333… → 90x = 21 → x = 21/90 = 7/30.
General trick: subtract the non-recurring part once; denominator = number of 9s (one per recurring digit) followed by zeros (one per non-recurring digit after decimal).
Representing √x on the Number Line
Spiral construction: to plot √n start from 1, draw a unit perpendicular at 1, hypotenuse = √2; then perpendicular = 1 at top of √2, hypotenuse = √3; continue.
Direct method for √n (n natural):
- Mark AB = n on the line; extend by 1 unit to C so AC = n + 1.
- Draw a semicircle with AC as diameter.
- At B draw a perpendicular meeting the semicircle at D.
- BD = √n. (Use Pythagoras on triangle inside the semicircle to verify.)
HBSE Tip: Always label your construction with arc names and centres — marks are given for the steps, not just the final answer.
Operations on Surds & Laws of Exponents
Operations on Real Numbers
- Rational + Rational = Rational
- Rational × Rational = Rational
- Rational + Irrational = Irrational (always)
- Rational × Irrational (≠ 0) = Irrational
- Irrational ± Irrational may be rational or irrational (e.g. (√2) + (−√2) = 0 rational).
Surds — Key Properties
For a, b > 0:
- √a × √b = √(ab)
- √a / √b = √(a/b)
- (√a + √b)(√a − √b) = a − b
- (a + √b)(a − √b) = a² − b
Rationalising the Denominator
Multiply numerator and denominator by the conjugate.
| Denominator | Multiply by |
|---|---|
| √a | √a |
| a + √b | a − √b |
| √a + √b | √a − √b |
Example: 1 / (√3 + √2) × (√3 − √2)/(√3 − √2) = (√3 − √2)/(3 − 2) = √3 − √2.
Laws of Exponents (real bases, rational exponents)
For a > 0 and rational p, q:
- aᵖ · aᵍ = aᵖ⁺ᵍ
- (aᵖ)ᵍ = aᵖᵍ
- aᵖ / aᵍ = aᵖ⁻ᵍ
- (a·b)ᵖ = aᵖ · bᵖ
- a⁰ = 1 (a ≠ 0)
- a⁻ᵖ = 1/aᵖ
- aᵖ/ᵍ = (a^(1/q))ᵖ = ᵍ√(aᵖ)
CBSE 1-mark trap: evaluate (16)^(3/4) — apply (a^(1/q))ᵖ ⇒ (2)³ = 8.
Frequently asked questions
Are these Number Systems notes free?
Yes — the Number Systems notes for Mathematics (Class 9) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Number Systems notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Number Systems chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Number Systems.