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.
🧠 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.
Step 2: Perpendicular Unit
Construct a unit length perpendicular AB at A.
Step 3: Hypotenuse
Join OB. By Pythagoras: \( OB = \sqrt{1^2 + 1^2} = \sqrt{2} \).
Step 4: The Arc
Using a compass with center O and radius OB, draw an arc cutting the number line at P.
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.
This identity creates a rational number from two irrationals. We use the conjugate (change the
sign in the middle) to rationalise.
Example: Rationalise \( \frac{1}{2 + \sqrt{3}} \)
Multiply num & den by \( 2 - \sqrt{3} \)
Denominator becomes \( (2)^2 - (\sqrt{3})^2
= 4 - 3 = 1 \)
Result: \( 2 - \sqrt{3} \)
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
- CBSE asks √2 construction to test geometry + number sense
- Decimal expansion questions test classification, not calculation
- Rationalisation checks identity usage
- Laws of exponents often appear as simplification steps
Tip: Always mention reason — CBSE awards step marks.
Chapter Summary
Let's recap the key concepts before the quiz!
-
Rational Numbers (\(Q\)): Can be written as \( \frac{p}{q} \). Decimals are either
terminating or recurring.
-
Irrational Numbers (\(T\)): Cannot be written as \( \frac{p}{q} \). Decimals are
non-terminating and non-recurring (e.g., \( \sqrt{2}, \pi \)).
-
Real Numbers (\(R\)): The collection of all Rational and Irrational numbers. Every
real number represents a unique point on the number line.
-
Rationalisation: To remove a root from the denominator of \( \frac{1}{\sqrt{a} +
\sqrt{b}} \), multiply by the conjugate \( \sqrt{a} - \sqrt{b} \).
-
Laws of Exponents: Remember: \( a^m \cdot a^n = a^{m+n} \), \( (a^m)^n = a^{mn} \),
and \( a^0 = 1 \).
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?
- I can classify any number instantly
- I know why √2 is irrational
- I can rationalise without memorising
- I understand exponent rules, not just apply them
If YES → you’re exam-ready 🎯
If NO → revise the marked section