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Page 1: Introduction to Number Systems

Mathematics begins with counting. Over time, we developed different types of numbers to solve real-world problems.

Natural numbers helped us count objects. Whole numbers included zero. Integers allowed negative values for debt or temperature below zero.

Rational numbers let us express fractions, and irrational numbers filled the gaps on the number line.

Together, rational and irrational numbers form the set of real numbers, which represent every point on the infinite number line.

This chapter explores all these types in depth, with proofs, examples, and exam-focused practice.

We will cover decimal expansions, rationalisation, operations on real numbers, and important theorems.

By the end, you will confidently handle any board exam question on Number Systems.

Let's begin the journey!

Page 2: Types of Numbers - Hierarchy

Natural Numbers (N): {1, 2, 3, 4, ...} – Used for counting.
Whole Numbers (W): {0, 1, 2, 3, ...} – Natural numbers including zero.
Integers (Z): {..., -3, -2, -1, 0, 1, 2, 3, ...} – Positive, negative, and zero.
Rational Numbers (Q): Numbers of the form p/q where p and q are integers, q ≠ 0.
Irrational Numbers: Numbers that cannot be expressed as p/q (e.g., √2, π).
Real Numbers (R): Union of rational and irrational numbers.

Every real number has a unique position on the number line.

Rational numbers have either terminating or repeating decimals.

Irrational numbers have non-terminating, non-repeating decimals.

Page 3: Rational Numbers in Detail

A number is rational if it can be written as p/q with gcd(p,q) = 1 and q ≠ 0.

Examples: 3/4, -5/7, 0.25 (= 1/4), 0.333... (= 1/3).

Equivalent rationals: 2/4 = 1/2 = 4/8 (multiply/divide numerator and denominator by same number).

Example: Express 0.375 as a fraction in lowest terms.
0.375 = 375/1000 = 3/8 (divide by 125).

Between any two rational numbers, there are infinitely many rationals.

Method: (a + b)/2 or multiply numerator and denominator to create gaps.

Page 4: Finding Rational Numbers Between Two Numbers

To find n rational numbers between a and b (a < b):

Method 1: Take average repeatedly.

Method 2: Write a = a×(n+1)/(n+1), b = b×(n+1)/(n+1), then take integers in between.

Example: Find 5 rationals between 2 and 3.
2 = 12/6, 3 = 18/6
So: 13/6, 14/6, 15/6, 16/6, 17/6 → 2.166, 2.333, 2.5, 2.666, 2.833

Practice: Find 4 rationals between 1/3 and 1/2.

Page 5: Decimal Expansion of Rational Numbers

For a rational p/q in lowest terms:

Terminating decimal if q has prime factors only 2 or 5 (or both).
Non-terminating repeating otherwise.

Maximum digits after decimal in terminating case = max power of 2 or 5 in q.

Example:
1/2 = 0.5 (terminating, q=2)
1/5 = 0.2 (terminating, q=5)
1/8 = 0.125 (terminating, q=2³)
1/6 = 0.1666... (repeating, q=2×3)
1/7 = 0.142857142857... (repeating cycle 142857)

Page 6: Irrational Numbers - Definition and Examples

Irrational numbers cannot be expressed as p/q for any integers p, q (q ≠ 0).

Their decimal expansion is infinite and non-repeating.

Common examples:

√2 ≈ 1.414213562...
√3 ≈ 1.732050807...
π ≈ 3.141592653...
e ≈ 2.718281828...
√5, √7, √11, etc. (non-perfect squares)

If n is not a perfect square, √n is irrational.

Page 7: Proof that √2 is Irrational (By Contradiction)

Assume √2 is rational, i.e., √2 = p/q where p, q are co-prime integers, q ≠ 0.

Then √2 × q = p

Square both sides: 2q² = p²

⇒ p² is even ⇒ p is even (since square of odd is odd).

Let p = 2k for some integer k.

Then 2q² = (2k)² = 4k²

⇒ q² = 2k²

⇒ q² even ⇒ q even.

So both p and q are even ⇒ they have common factor 2.

This contradicts our assumption that p and q are co-prime.

Hence, √2 is irrational.

Page 8: Proof that √3 is Irrational

Assume √3 = p/q (p, q co-prime, q ≠ 0).

Then 3q² = p²

p² divisible by 3 ⇒ p divisible by 3 (since 3 is prime).

Let p = 3k

Then 3q² = 9k² ⇒ q² = 3k²

q² divisible by 3 ⇒ q divisible by 3.

Both p and q divisible by 3 → contradiction.

Hence √3 is irrational.

General rule: If n is not a perfect square, √n is irrational.

Page 9: Representing Irrationals on Number Line

To represent √n on number line:

Draw a line segment AB of length n units.
Extend AB to C such that BC = 1 unit.
Find midpoint O of AC.
Draw semicircle with diameter AC and center O.
Draw perpendicular from B to the semicircle, intersecting at D.
BD = √n

Example: For √5, AB = 5, BC = 1, AC = 6 → BD = √5.

Page 10: Operations on Real Numbers

Real numbers are closed under addition, subtraction, multiplication, and division (except by zero).

Key results:

Rational ± Rational = Rational
Rational ± Irrational = Irrational
Irrational ± Irrational can be rational (e.g., √2 + (2 - √2) = 2)
Rational × Irrational = Irrational (if rational ≠ 0)

Example: Simplify √8 + √18
= 2√2 + 3√2 = 5√2 (irrational)

Page 11: Rationalising the Denominator

When denominator has irrational part, multiply numerator and denominator by conjugate.

Conjugate of (a + √b) is (a - √b).

(a + √b)(a - √b) = a² - b

Example 1: Rationalise 1/(√5 + √2)
Multiply by (√5 - √2):
= (√5 - √2)/(5 - 2) = (√5 - √2)/3

Example 2: 5/(3 + 2√2) = 5(3 - 2√2)/(9 - 8) = 5(3 - 2√2)

Page 12: Laws of Exponents for Real Numbers

For real numbers a, b > 0 and rational/irrational exponents:

a^m × a^n = a^(m+n)
a^m / a^n = a^(m-n)
(a^m)^n = a^(mn)
a^0 = 1
a^(-n) = 1/a^n
(ab)^m = a^m × b^m

Example: Simplify (8^(1/3) × 27^(1/3))³
= (2 × 3)³ = 6³ = 216

Page 13: Key Identities Summary

(√a + √b)(√a - √b) = a - b
(a + √b)(c + √d) = ac + √(ad) + √(bc) + √(bd)
Rationalisation formulas
Decimal expansion rules based on denominator factors

These identities are frequently tested in board exams.

Page 14: Practice Questions - Easy Level (1-10)

Is zero a rational number? Justify.
Express 0.999... as fraction.
Find three rational numbers between 1/4 and 1/3.
Which of these terminate: 1/3, 1/8, 1/7, 1/20?
Is √4 rational or irrational? Why?
Simplify √50 + √8.
Rationalise 1/(√3 + 1).
Is π rational? Why not?
Find irrational between 2 and 3.
Classify √16/9.

Page 15: Practice Questions - Medium Level (11-20)

Prove that √5 is irrational.
Find five rationals between 0.1 and 0.2.
Simplify (√5 + √3)^2.
Rationalise 7/(√7 - √5).
Show that 1/√2 is irrational.
Express 0.001001001... as fraction.
Simplify √98 / √2.
Is 0.1010010001... irrational? Why?
Represent √3 on number line (describe steps).
Simplify (3 + √2)(3 - √2).

Page 16: Practice Questions - Hard Level (21-30)

Prove that √2 + √3 is irrational.
Simplify 1/(√5 + √3 + √2).
Show that (√3 + √2)^2 + (√3 - √2)^2 = 10.
Find rational and irrational between √2 and √3.
Prove that 3 + 2√5 is irrational.
Simplify (16)^(1/4) × (81)^(1/4).
Is cube root of 2 irrational? How?
Rationalise denominator of 1/(2 + √3 + √5).
Show that 0.232332333... is irrational.
Prove that log₂3 is irrational.

Page 17: NCERT Exercise 1.1 Type Questions

Typical questions on rational/irrational classification, decimal types, and basic operations.

Focus on identifying terminating decimals and expressing repeating as fractions.

Common patterns: 0.142857 repeating (1/7), 0.333... (1/3), etc.

Remember: Pure repeating = divide by 9s, mixed repeating = formula.

Page 18: NCERT Exercise 1.2 Type Questions

Focus on representing numbers on number line and finding rationals between two numbers.

Geometric construction of √n is important for 2-3 mark questions.

Practice drawing accurate diagrams.

Page 19: NCERT Exercise 1.3 Type Questions

Heavy on rationalisation and simplification of expressions with surds.

Learn conjugate method thoroughly.

Common forms: 1/(a + b + √c), multiply by conjugate step by step.

Page 20: NCERT Exercise 1.4 Type Questions

Laws of exponents with real bases and rational exponents.

Simplify expressions like (a^m)^n × a^(p).

Negative exponents and fractional exponents are key.

Page 21: Important Theorems Summary

If √n is irrational, then k√n is irrational (k rational ≠ 0).
Rational + Irrational = Irrational.
Product of non-zero rational and irrational is irrational.
Decimal expansion rule for rational numbers.
Proof by contradiction for irrationality.

Page 22: Common Mistakes to Avoid

Thinking all decimals are rational.
Forgetting to check lowest terms in proofs.
Misapplying rationalisation (wrong conjugate).
Thinking √(a+b) = √a + √b (wrong!).
Confusing terminating vs repeating decimals.

Page 23: Previous Year Board Questions

Typical 1-mark: Classify number.

2-mark: Rationalise or simplify.

3-mark: Prove irrationality.

4-mark: Multiple steps with rationalisation.

5-mark: Construction + proof combined.

Page 24: Exam Strategy Tips

Start proofs with clear assumption.
Show all steps in rationalisation.
Draw neat number line diagrams.
Check decimal type by prime factors of denominator.
Revise all proofs before exam.

Page 25: Quick Revision Formula Sheet

Terminating decimal ⇔ denominator factors 2 and/or 5
(a + √b)(a - √b) = a² - b
a^m × a^n = a^(m+n)
Rational + Irrational = Irrational
√n irrational if n not perfect square

Master these 5 points → full marks guaranteed!

Page 26: Final Motivation

You've reached the end of this 27-page guide!

Number Systems is the foundation — master it now, and the rest of Class 9 Maths becomes easier.

Practice daily. Solve past papers. Believe in yourself.

Board Buddy is always here for you 🦖

You've got this!

Class 9 Mathematics