Chapter 2 of 9

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Tamil Nadu (TNBSE)Class 9 Mathematics★ 6-9 marks in the TN Class 9 annual exam through MCQs, surd simplification and rationalising problems

Real Numbers

Samacheer Kalvi (TN State Board) Class 9 Maths Chapter 2, Real Numbers. Rational and irrational numbers, decimal representation, surds and operations on surds, rationalising the denominator, and scientific notation. Notes with formulas, worked examples, common mistakes, book-back practice and quick revision.

⏱ 20 min readMedium difficulty✓ Free · NCERT-aligned

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By the end of this chapter you'll be able to…

1Distinguish rational and irrational numbers
2Classify decimals by their representation
3Perform operations on surds
4Rationalise denominators using the conjugate
5Express numbers in scientific notation

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Why this chapter matters

Real Numbers builds the number system used throughout mathematics. Classifying numbers, simplifying surds and rationalising denominators are reliable scoring skills in the TN Class 9 exam.

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Class 8 — Rational numbers

Fractions, integers and decimals.

Real Numbers — Class 9 Maths (Samacheer Kalvi)

TN State Board (Samacheer Kalvi) Class 9 Mathematics, Chapter 2. Rational and irrational numbers, surds and scientific notation.

1. About this chapter

This chapter covers rational and irrational numbers, their decimal representation, surds and operations on them, rationalising the denominator, and scientific notation.

2. Rational and irrational numbers

Rational number: can be written as p/q (q ≠ 0, integers).
Irrational number: cannot be written as p/q (e.g., √2, π).
Decimal representation: rational numbers are terminating or non-terminating recurring; irrational numbers are non-terminating non-recurring.
Between any two rational numbers there are infinitely many rationals (denseness).

3. Surds

A surd is an irrational root like √2, ∛5. Order = the root index.
Operations: like surds can be added/subtracted; multiply/divide using √a × √b = √(ab) and √a ÷ √b = √(a/b).

4. Rationalising and scientific notation

Rationalising the denominator: multiply by a suitable factor to remove the surd, e.g. 1/√2 × √2/√2 = √2/2; for 1/(a + √b) multiply by the conjugate (a − √b).
Scientific notation: write a number as m × 10ⁿ with 1 ≤ m < 10 (e.g., 4500 = 4.5 × 10³).

5. Worked examples

Example 1. Is 0.272727… rational or irrational? It is non-terminating recurring, so it is rational (= 27/99 = 3/11).

Example 2. Simplify √12 + √27. = 2√3 + 3√3 = 5√3.

Example 3. Rationalise 1/(3 + √2). × (3 − √2)/(3 − √2) = (3 − √2)/(9 − 2) = (3 − √2)/7.

6. Common mistakes

Mistake: Calling a non-terminating recurring decimal irrational. Fix: Recurring decimals are rational.
Mistake: Adding unlike surds. Fix: Only like surds (same order and radicand) can be added.
Mistake: Wrong conjugate when rationalising. Fix: For (a + √b), multiply by (a − √b).

7. Practice (book-back style)

Classify 3.141592… (non-recurring) as rational or irrational.
Simplify √50 − √18.
Rationalise the denominator of 5/√5.
Express 0.000045 in scientific notation.
Write 7/8 in decimal form and classify it.

8. Answer key

Irrational (non-terminating, non-recurring).
= 5√2 − 3√2 = 2√2.
5/√5 × √5/√5 = 5√5/5 = √5.
= 4.5 × 10⁻⁵.
7/8 = 0.875 → terminating, rational.

9. Quick revision

Chapter 2 · rational/irrational, surds, scientific notation.
Rational = p/q; decimals terminating or recurring. Irrational = non-terminating non-recurring.
√a × √b = √(ab); only like surds add.
Rationalise using the conjugate; 1/(a + √b) × (a − √b)/(a − √b).
Scientific notation: m × 10ⁿ, 1 ≤ m < 10.

Key formulas & results

Everything you need to memorise, in one card. Screenshot this for revision.

Surd products

√a × √b = √(ab); √a ÷ √b = √(a/b)

For non-negative a, b.

Rationalising

1/(a + √b) × (a − √b)/(a − √b)

Multiply by the conjugate.

Scientific notation

m × 10ⁿ, 1 ≤ m < 10

n is an integer.

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Common mistakes & fixes

These are the exact errors that cost students marks in board exams. Read them once, save yourself the trouble.

WATCH OUT

✗ Calling a non-terminating recurring decimal irrational

✓ Recurring decimals are rational.

WATCH OUT

✗ Adding unlike surds

✓ Only like surds (same order and radicand) can be added.

WATCH OUT

✗ Wrong conjugate when rationalising

✓ For (a + √b), multiply by (a − √b).

NCERT exercises (with solutions)

Every NCERT exercise from this chapter — what it covers and how many questions to expect.

Multiple choice questions

MCQs on number types, surds and scientific notation.

Questions

Surd simplification

Simplify and operate on surds.

Questions

Rationalising problems

Rationalise denominators and classify numbers.

Questions

Practice problems

Try each one yourself before tapping "Show solution". Active recall > rereading.

Q1EASY· Concept

Classify 3.141592… (non-recurring) as rational or irrational.

▶ Show solution

Irrational (non-terminating, non-recurring).

Q2MEDIUM· Surds

Simplify √50 − √18.

▶ Show solution

5√2 − 3√2 = 2√2.

Q3MEDIUM· Rationalise

Rationalise the denominator of 5/√5.

▶ Show solution

5/√5 × √5/√5 = 5√5/5 = √5.

Q4EASY· Notation

Express 0.000045 in scientific notation.

▶ Show solution

4.5 × 10⁻⁵.

Q5MEDIUM· Rationalise

Rationalise 1/(3 + √2).

▶ Show solution

× (3 − √2)/(3 − √2) = (3 − √2)/(9 − 2) = (3 − √2)/7.

Q6EASY· Concept

Write 7/8 in decimal form and classify it.

▶ Show solution

0.875 — terminating, so rational.

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Chapter 2 of Samacheer Kalvi Class 9 Mathematics.
•Rational = p/q; decimals terminating or recurring.
•Irrational = non-terminating non-recurring (√2, π).
•√a × √b = √(ab); only like surds add.
•Rationalise with the conjugate; 1/(a + √b) × (a − √b)/(a − √b).
•Scientific notation: m × 10ⁿ, 1 ≤ m < 10.

Tamil Nadu (TNBSE) marks blueprint

Where the marks come from in this chapter — so you can plan your prep.

Typical chapter weightage: 6-9 marks across MCQ, surd and rationalising problems

Question type	Marks each	Typical count	What it tests
MCQ	1	1-2	Number classification
Surds	2-3	1-2	Simplification and operations
Rationalising	2-3	1	Conjugate method

Prep strategy

Classify numbers by decimal type
Simplify surds to like form before adding
Use the conjugate to rationalise
Practise scientific notation conversions

Where this shows up in the real world

This chapter isn't just an exam topic — it lives in the world around you.

Measurement

Irrational numbers like √2 appear in diagonals and circles.

Science notation

Scientific notation handles very large and small quantities.

Engineering

Surds occur in exact lengths and ratios.

Exam strategy

Battle-tested tips from teachers and toppers for this chapter.

Reduce surds to like form before adding
Always rationalise using the conjugate
Keep 1 ≤ m < 10 in scientific notation
Classify by the decimal pattern

Going beyond the textbook

For olympiad aspirants and curious learners — topics that build on this chapter.

Prove that √2 is irrational.
Show that the sum of a rational and an irrational number is irrational.

Where else this chapter is tested

CBSE board isn't the only one — other exams test this chapter too.

TN Class 9 Annual ExamHigh

Foundation / NTSE MathematicsMedium

School unit testsHigh

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Questions students ask

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If, in lowest terms, the denominator's only prime factors are 2 and/or 5, the decimal terminates; otherwise it is non-terminating recurring.

Removing the surd from the denominator gives a simpler, standard form that is easier to evaluate and compare.

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