Number Systems Class 9 Notes: Rational, Irrational Numbers and Laws of Exponents

Number Systems is the first chapter of Class 9 Maths. It looks simple but the exam questions on rationalisation and exponents trap many students.

Types of numbers

• Natural (1, 2, 3…), Whole (0, 1, 2…), Integers (…−1, 0, 1…)
• Rational: p/q form, q ≠ 0; decimal terminates or repeats (0.75, 0.333…)
• Irrational: cannot be written as p/q; decimal is non-terminating, non-repeating (√2, π, 0.101001…)
• Real numbers = rational + irrational

Key facts

• Between any two rationals there are infinitely many rationals (use (a + b)/2).
• √n is irrational when n is not a perfect square.
• Sum/product of a rational and an irrational is irrational (except × 0).
• Sum of two irrationals may be rational: (2 + √3) + (2 − √3) = 4.

Rationalising the denominator

Multiply top and bottom by the conjugate. Example: 1/(√5 − 2) = (√5 + 2)/(5 − 4) = √5 + 2.

Example: if x = 2 + √3, find x + 1/x: 1/x = 2 − √3 → x + 1/x = 4.

Laws of exponents (a > 0)

• aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ/aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; aᵐbᵐ = (ab)ᵐ
• a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(1/n) = ⁿ√a
• Example: 64^(2/3) = (4³)^(2/3) = 4² = 16; 2^(1/2) × 8^(1/2) = 16^(1/2) = 4

Common mistakes

• Calling 22/7 irrational (it is rational; π is irrational)
• Adding exponents when bases are different
• Multiplying by the same expression instead of its conjugate

Related: Polynomials Class 9.

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Frequently asked questions

Are these Number Systems Class 9 Notes notes enough for the exam?

They cover every formula, definition and question type that appears in school and board papers for this chapter. Use them with the NCERT examples, exercise questions and the last five years' papers to be fully prepared.

How should I revise Number Systems Class 9 Notes?

Revise the notes after 1 day, 7 days and 30 days. Solve five questions each time instead of only reading, and keep an error list of the steps you get wrong.

What if I get stuck on a number systems class 9 question?

Type or photograph the exact question on DoubtPay AI, choose your class and language, and get a step-by-step answer with a diagram. You can also ask for a hint or have your own attempt checked. Packs start at ₹9 for 15 answers.