THE WORLD OF NUMBERS Class 9 Maths

 

📘 CHAPTER NOTES: THE WORLD OF NUMBERS

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🔹 1. Early Number Systems

• Ancient humans used marks and tallies to count.
• Example: Ishango Bone
  • Contains number patterns like 11, 13, 17, 19 (prime numbers).
  • Shows early understanding of doubling (multiplication).

Key Idea:

➡ Numbers are thousands of years old and evolved with human needs.

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🔹 2. Indian Contribution to Numbers

• Indus Valley used weights & measures for trade.
• Vedic texts used powers of 10 (up to very large numbers).
• Foundation of modern decimal system.

Important:

➡ India introduced the place value system and zero (0).

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🔹 3. Concept of Zero (Śhūnya)

• Introduced by Brahmagupta (628 CE).
• Defined:
  👉 $a - a = 0$

Rules:

• $a + 0 = a$
• $a - 0 = a$
• $a × 0 = 0$

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🔹 4. Integers (Z)

Include:

• Positive numbers → Fortunes (Dhana)
• Negative numbers → Debts (Ṛiṇa)
• Zero

Number Line:

... –3, –2, –1, 0, 1, 2, 3 ...

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🔹 5. Arithmetic of Integers

Rules:

• (+) + (+) = +
• (–) + (–) = –
• (+) × (–) = –
• (–) × (–) = +

Important Concept:

👉 Negative × Negative = Positive
(Deleting a debt increases wealth)

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🔹 6. Rational Numbers (ℚ)

Definition:

A number of the form:

$$\frac{p}{q}, \quad q \ne 0$$

Includes:

• Integers
• Fractions (positive & negative)

Properties:

• Closed under +, –, ×, ÷ (except divide by zero)

• Infinite representations:

  $$\frac{1}{2} = \frac{2}{4} = \frac{3}{6}$$

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🔹 7. Operations on Rational Numbers

Addition/Subtraction:

$$\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$$

Multiplication:

$$\frac{a}{b} × \frac{c}{d} = \frac{ac}{bd}$$

Division:

$$\frac{a}{b} ÷ \frac{c}{d} = \frac{a}{b} × \frac{d}{c}$$

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🔹 8. Number Line Representation

• Divide interval into equal parts.
• Move right → positive
• Move left → negative

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🔹 9. Absolute Value

$$|x| = \text{distance from 0}$$

Examples:

• |–5| = 5
• |3| = 3

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🔹 10. Density Property

➡ Between any two rational numbers, infinite rational numbers exist.

Example:
Between 1 and 2:

$$\frac{1 + 2}{2} = \frac{3}{2}$$

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🔹 11. Irrational Numbers

• Cannot be written as $\frac{p}{q}$
• Non-terminating, non-repeating decimals

Examples:

• $\sqrt{2}$
• π

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🔹 12. Proof of Irrationality (√2)

• Assume $\sqrt{2} = \frac{p}{q}$
• Leads to contradiction → not possible
  👉 Hence irrational

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🔹 13. Real Numbers (ℝ)

Combination of:

• Rational + Irrational numbers

➡ Forms complete number line

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🔹 14. Decimal Expansion

Rational Numbers:

1. Terminating → 0.25
2. Repeating → 0.333...

Condition:

Denominator must have only 2 and/or 5 as prime factors → terminating

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🔹 15. Converting Decimals to Fractions

Example:

Let $x = 0.333...$

$$10x = 3.333...$$ $$10x - x = 3$$ $$9x = 3 ⇒ x = \frac{1}{3}$$

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🔹 16. Cyclic Numbers

Example:

$$\frac{1}{7} = 0.142857...$$

Digits rotate when multiplied:

• 142857 × 2 = 285714

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🔹 17. Important Sets Summary

📝 TOPIC-WISE PRACTICE QUESTIONS

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🔸 1. Basic Concepts

1. What are prime numbers in the Ishango bone?
2. What is the importance of zero?

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🔸 2. Integers

1. Calculate:
   (i) –5 + 7
   (ii) –8 – 6
   (iii) (–4) × (–3)
2. Explain why (– × – = +).

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🔸 3. Rational Numbers

1. Write 5 as a rational number.
2. Find equivalent fractions of $\frac{3}{4}$.

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🔸 4. Operations

1. $\frac{2}{3} + \frac{5}{6}$
2. $\frac{7}{8} - \frac{1}{4}$
3. $\frac{3}{5} × \frac{10}{9}$

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🔸 5. Number Line

1. Represent $\frac{3}{4}$
2. Represent –$\frac{5}{2}$

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🔸 6. Absolute Value

1. Find:
   • |–9|
   • |5 – 8|

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🔸 7. Density Property

1. Find 3 rational numbers between:
   • 1 and 2
   • $\frac{1}{2}$ and $\frac{3}{4}$

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🔸 8. Irrational Numbers

1. Is √4 irrational? Why?
2. Prove √3 is irrational (try yourself).

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🔸 9. Decimals

1. Convert:
   • 0.25
   • 0.666...
2. Check if terminating:
   • $\frac{7}{20}$
   • $\frac{3}{14}$

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🔸 10. Higher Thinking

1. Why is division by zero not allowed?
2. Why are rational numbers dense?
3. Explain 0.999… = 1

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🔸 11. Word Problems

1. Temperature drops from 5°C to –10°C. Change?
2. A trader earns ₹500, loses ₹300 → final?

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🔸 12. Challenge Questions

1. Find 5 rational numbers between 2 and 3
2. Show:

$$\frac{a+b}{2}$$

lies between a and b