Class 9 Maths Chapter 1 – Number Systems

Class 9 Maths • Chapter 01 • Complete Study Guide

1. The Number Family

Understanding the hierarchy of numbers is the first step. Look at how each set of numbers fits inside the next.

Real Numbers (R) Rational (Q) Integers (Z) Natural (N)
🧠 Think First (No Guessing!)

Without flipping the cards, decide:
Is 0.1010010001... Rational or Irrational?

Reveal Answer

Irrational — because the pattern does NOT repeat.

Note: Irrational Numbers (\( T \)) are inside Real Numbers but outside Rational Numbers.

Rational Numbers

Symbol: \( Q \)

Tap to flip

Definition

Can be written as \( \frac{p}{q} \), where \( p, q \) are integers, \( q \neq 0 \).

Examples: \( \frac{1}{2}, -5, 0, 0.333... \)

Irrational Numbers

Symbol: \( T \) or \( S \)

Tap to flip

Definition

Cannot be written as \( \frac{p}{q} \). Their decimal expansion is non-terminating and non-recurring.

Examples: \( \sqrt{2}, \pi, 0.101001... \)

2. Locating \( \sqrt{2} \) on Number Line

How do we put an infinite decimal like \( \sqrt{2} \) on a line? We use Geometry (Pythagoras Theorem).

Step 1: The Setup

Draw a number line. Mark point '0' as O and '1' as A.

0 (O) 1 (A)

Step 2: Perpendicular Unit

Construct a unit length perpendicular AB at A.

O A B (1 unit)

Step 3: Hypotenuse

Join OB. By Pythagoras: \( OB = \sqrt{1^2 + 1^2} = \sqrt{2} \).

√2

Step 4: The Arc

Using a compass with center O and radius OB, draw an arc cutting the number line at P.

P (√2)

3. Decimal Expansions

Identifying rational vs irrational numbers based on decimals.

Type of Number Decimal Expansion Example
Rational Terminating \( \frac{7}{8} = 0.875 \)
Rational Non-terminating & Recurring \( \frac{10}{3} = 3.333... \)
Irrational Non-terminating & Non-recurring \( \sqrt{2} = 1.4142... \)

🧩 Is This Possible?

Decide YES or NO (with reason):

  • Can a number be irrational and terminating?
  • Can a rational number have infinite decimals?
  • Can √9 be irrational?
Show Answers
  • No — terminating decimals are rational
  • Yes — if recurring (e.g., 1/3)
  • No — √9 = 3 (rational)

Visualizing Successive Magnification

Let's find 3.765 on the number line.

Level 1: Between 3 and 4

3 --- 3.1 --- 3.2 ... 3.7 --- 3.8 ... 4

Level 2: Between 3.7 and 3.8

3.7 --- 3.71 ... 3.76 --- 3.77 ... 3.8

Level 3: Found it!

3.76 --- 3.761 ... 3.765 ... 3.77
target located!

4. Operations on Real Numbers

What happens when we mix Rational (Q) and Irrational (T) numbers?

General Rules:
  • Rational \( \pm \) Irrational = Irrational
  • Rational \( \times \) Irrational = Irrational (if rational \(\neq 0\))
  • Irrational \( \pm/\times \) Irrational = Depends (Could be either)

Operations Checker

Select an operation to see the result type:

Click a button above

5. Rationalisation

Rationalisation means removing the square root from the denominator.

Identity: \( (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b \)

This identity creates a rational number from two irrationals. We use the conjugate (change the sign in the middle) to rationalise.

Example: Rationalise \( \frac{1}{2 + \sqrt{3}} \)

6. Laws of Exponents

For \( a > 0 \) and rational \( p, q \):

Rule Formula Example
Product \( a^p \cdot a^q = a^{p+q} \) \( 2^3 \cdot 2^2 = 2^5 \)
Power \( (a^p)^q = a^{pq} \) \( (2^3)^2 = 2^6 \)
Quotient \( \frac{a^p}{a^q} = a^{p-q} \) \( \frac{7^5}{7^3} = 7^2 \)
Negative \( a^{-p} = \frac{1}{a^p} \) \( 2^{-3} = \frac{1}{8} \)

🎯 Exam Smart Zone

Tip: Always mention reason — CBSE awards step marks.

Chapter Summary

Let's recap the key concepts before the quiz!

Chapter Quiz

1. Which is irrational?

A) \( \sqrt{4} \)
B) \( \sqrt{7} \)
C) 0.3333...

2. Value of \( (64)^{1/2} \) is:

A) 8
B) 4
C) 16

3. The decimal form of \( \frac{1}{11} \) is:

A) 0.09
B) \( 0.\overline{09} \)
C) 0.0909

✅ Can You Say YES to All?

If YES → you’re exam-ready 🎯 If NO → revise the marked section