Exercise 1.1 Class 9 Math Notes 2025 – Real Numbers

Need help with Exercise 1.1 Class 9 Maths (Unit 1: Real Numbers)? Our comprehensive guide provides everything you need to master real numbers! Get detailed explanations, step-by-step solutions, common mistake alerts, and real-world applications. Updated for the 2025 Punjab Board exams, this complete resource helps you understand rational and irrational numbers, master number line plotting, and score maximum marks in your exams.

1. Learn the rules of real numbers.
2. Tell the difference between rational (like fractions) and irrational numbers (like √2).
3. Plot numbers like √2 on a number line.
4. Avoid mistakes students often make and score higher in your exams!

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Common Mistakes to Avoid

1. Misclassifying Recurring Decimals

Mistake:
Thinking that repeating decimals are irrational numbers.
Example:
Believing that 0.373737… is irrational.

Correction:
Repeating decimals are rational numbers, because they can be written as fractions.
✅ 0.373737… = ³⁷⁄₉₉

➤ Tip: If a decimal repeats or terminates, it is rational. Only non-repeating, non-terminating decimals (like √2 or π) are irrational.

2. Incorrectly Placing Irrational Numbers on Number Lines

Mistake:
Placing irrational numbers like √2 or π incorrectly, as if they were exact fractions.

Correction:
Use approximations when placing irrational numbers.
✅ √2 ≈ 1.414, so it lies just past 1.4 on the number line.

➤ Tip: Even though irrational numbers can’t be expressed as exact decimals, they still have a precise location on the number line. Use approximations to plot them.

3. Confusing Additive and Multiplicative Identity

Mistake:
Mixing up the identities of 0 and 1 in arithmetic.

Correction:
Additive Identity:  a ＋ 0 ＝ a
Multiplicative Identity:  a × 1 ＝ a

➤ Tip:
 • Additive identity (0): “Adding nothing”
 • Multiplicative identity (1): “Multiplying by one — no change”

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Solved Problems

Problem 1: Is (2 − √2)(2 + √2) rational?

Solution: (2 − √2)(2 + √2) = 2² − (√2)² = 4 − 2 = 2 (Rational)

Why Students Get This Wrong:

• Forgetting the formula (a − b)(a + b) = a² − b²
• Incorrectly assuming (irrational) × (irrational) = irrational

Problem 2: Insert 2 Rational Numbers Between ⅓ and ¼

Step-by-Step: Convert to decimals: ⅓ ≈ 0.333, ¼ = 0.25

Average method: (0.333 + 0.25)/2 = 0.2915 → ⁷⁄₂₄

Second number: (0.2915 + 0.25)/2 = 0.27075 → ¹³⁄₄₈

Problem 3: Plot √2 on number line

Construction Steps:

1. Draw the base line: Mark points 0 and 1 on a horizontal number line
2. Create perpendicular: At point 1, draw a vertical line of exactly 1 unit upward
3. Form right triangle: Connect the top point to 0, creating a right triangle
4. Apply Pythagorean theorem: The hypotenuse length = √(1² + 1²) = √2
5. Transfer the length: Use a compass to mark this √2 distance from 0 on the number line
6. Mark the point: This gives you the exact location of √2 ≈ 1.414

👉 Exercise 1.2 Class 9 Math Solutions
👉 9 Class Math New Book PDF – Download & Overview

Key Definitions

1. Rational Number:
   A number that can be expressed as 𝑝/𝑞, where 𝑝 and 𝑞 are integers and 𝑞 ≠ 0.
   Examples: ³⁄₄, 0.6, 2.353535…
2. Irrational Number:
   A number that cannot be expressed as 𝑝/𝑞. It’s decimal form is Non-terminating, non-repeating.
   Examples: √2, √3, π, 𝑒.
3. Real Numbers:
   Includes all rational and irrational numbers.
4. Properties of Real Numbers:
   • Commutative: a + b = b + a | ab = ba
   • Associative: (a + b) + c = a + (b + c) | (ab)c = a(bc)
   • Distributive: a(b + c) = ab + ac
   • Additive Identity:  a+ 0 = a
   • Multiplicative Identity: a x 1 = a
   • Additive Inverse: a+ (-a) = 0
   • Multiplicative Inverse: 𝗮 × ¹⁄𝗮 = 1

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Important Formulas

1. Decimal to Fraction Conversion:

\(
x = 0.\overline{4}
\)
\(\text{Multiply Both Sides with 10}\)
\(
10x = 4.\overline{4}
\)
\(
\text{Now subtract the original } x \text{ from } 10x:
\)
\(
10x – x = 4.\overline{4} – 0.\overline{4} = 4
\)
\(
9x = 4
\)
\(
\text{Therefore, } x = \frac{4}{9}
\)

1. Sum/Product Rules:
   • Rational + Irrational = Irrational: 5 + √11
   • Rational × Irrational = Irrational: 2 × √3 = 2√3
2. Rationalizing Expressions:
   • Example: (2 − √2)(2 + √2) = 2² − (√2)² = 4 − 2 = 2

📐 Real Life Examples

Exercise 1.1 class 9 math is part of the new curriculum and is focused on the mathematical concepts that are closer to daily life applications. Here are few examples that give students an idea that math is not boring meaningless calculations rather math is deeply integrated in almost every aspect of our daily life.

1. In Mobile Games:
In mobile game development, mathematical values like √5 are critical for calculating diagonal movement speeds or designing curved trajectories. For instance, if a character moves 2 units horizontally and 1 unit vertically, the total distance traveled diagonally is √(2² + 1¹) = √5 units. Game engines use these calculations to ensure smooth animations, realistic physics, and accurate pathfinding in games like Clash of Clans or Subway Surfers.

2. While Shopping:
Shopkeepers use decimals like 0.75 or 1.99 when setting prices. They also use the distributive property to give discounts — for example, when something is “Buy 1, get 1 at 50% off”, they calculate the total price using math rules like:
a × (b + c) = a × b + a × c.

3. Engineering and Banking
Architects use √2 to calculate diagonal supports in square structures. For example, a 1m x 1m square room requires a diagonal beam of √2 meters to reinforce walls. While bankers rely on rational decimals for interest rates!

Video Lecture on Exercise 1.1 Class 9 Math

📺 Watch this helpful YouTube video explaining each question of Exercise 1.1:

https://www.youtube.com/watch?v=cdf-m1_bR3Y

Short Answer Questions

Q1: Is (2 − √2)(2 + √2) rational or irrational?
Yes, it’s rational! Using the formula (a − b)(a + b) = a² − b², we get:
2² − (√2)² = 4 − 2 = 2 → Rational

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Q2: How to represent √3 on a number line?
Use the Pythagorean theorem:

• Draw a right triangle with base 1 unit (from 0 to 1).
• Height = 1 unit ⇒ hypotenuse = √2
• Extend height 1 unit again from √2 ⇒ new hypotenuse = √3
• Point D represents √2 while point F represents √3.

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Q3: Why is 5 + √11 irrational?
The sum of a rational number (5) and an irrational number (√11) is always irrational.

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Q4: What’s the difference between associative and commutative properties?

• Commutative: Order doesn’t matter → a + b = b + a
• Associative: Grouping doesn’t matter → (a + b) + c = a + (b + c)

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Q5: How to convert 0.373737… into a fraction?
Let x = 0.373737…
Multiply both sides by 100: 100x = 37.373737…
Subtract: 100x − x = 37
So, x = ³⁷⁄₉₉

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Q6: Why is √7 irrational?
√7 cannot be expressed as p/q where p and q are integers.
Its decimal expansion is non-terminating and non-repeating.

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Q7: What’s the additive identity property?
Adding 0 to any number leaves it unchanged:
a + 0 = a
Example: 16 + 0 = 16

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Q8: How to insert rational numbers between ⅓ and ¼?
Use the average method:

• First rational number: (⅓ + ¼)/2 = ⁷⁄₂₄
• Second: (⁷⁄₂₄ + ¼)/2 = ¹³⁄₄₈