Quick Revision Notes: Class 9 Mathematics

Chapter 1: Number Systems (সংখ্যা पद्धति)

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### 1. Introduction to Numbers (संख्याओं का परिचय)

• Natural Numbers (प्राकृत संख्याएँ - N): Counting numbers starting from 1.
  • Formula/Set: N = {1, 2, 3, 4, ...}
• Whole Numbers (पूर्ण संख्याएँ - W): Natural numbers including zero.
  • Formula/Set: W = {0, 1, 2, 3, 4, ...}
• Integers (पूर्णांक - Z): All natural numbers, 0, and negatives of natural numbers.
  • Formula/Set: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
• Rational Numbers (परिमेय संख्याएँ - Q): Numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0.
  • Example: 2/3, -5/7, 0, 5 (since 5 = 5/1)
• Irrational Numbers (अपरिमेय संख्याएँ): Numbers that cannot be expressed in the form p/q (where p and q are integers and q ≠ 0). Their decimal expansion is non-terminating and non-recurring.
  • Example: √2, √3, π
• Real Numbers (वास्तविक संख्याएँ - R): The collection of all rational and irrational numbers.
  • Note: Every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number.

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### 2. Rational Numbers Between Two Numbers

• To find rational numbers between two given rational numbers $r$ and $s$ ($r < s$):
  • Method 1: Find the mean (r + s) / 2.
  • Method 2: Make the denominators equal (say $n$) and find equivalents by multiplying the numerator and denominator by (n + 1).

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### 3. Decimal Expansions of Real Numbers

The decimal expansion of a rational number is either terminating or non-terminating recurring (repeating).

1. Terminating (सांत): The remainder becomes zero after a finite number of steps.
   • Example: 3/8 = 0.375
2. Non-terminating Recurring (असांत आवर्ती): The remainder repeats after a certain stage.
   • Example: 1/3 = 0.333... = 0.¯3
3. Irrational Numbers: Their decimal expansions are non-terminating non-recurring (असांत अनावर्ती).
   • Example: 1.4142135... (√2)

Converting Non-terminating Recurring Decimals to p/q Form:

• Let x = 0.¯6 (Equation 1)
• Multiply by 10 (since 1 digit is repeating): 10x = 6.¯6 (Equation 2)
• Subtract (1) from (2): 10x - x = 6.¯6 - 0.¯6
• 9x = 6 => x = 6/9 = 2/3

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### 4. Operations on Real Numbers (वास्तविक संख्याओं पर संक्रियाएँ)

For positive real numbers $a$ and $b$:

1. √(ab) = √a * √b
2. √(a/b) = √a / √b
3. (√a + √b)(√a - √b) = a - b
4. (a + √b)(a - √b) = a^2 - b
5. (√a + √b)(√c + √d) = √ac + √ad + √bc + √bd
6. (√a + √b)^2 = a + 2√(ab) + b

Rationalization (हर का परिमेयकरण):

To rationalize the denominator of 1 / (√a + b), multiply the numerator and denominator by the conjugate (√a - b):

• 1 / (√a + b) * (√a - b) / (√a - b) = (√a - b) / (a - b^2)

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### 5. Laws of Exponents for Real Numbers (घातांक के नियम)

Let $a > 0$ be a real number and $m, n$ be rational numbers. Then:

1. a^m * a^n = a^(m + n) (Product Law)
2. (a^m)^n = a^(m * n) (Power of a Power Law)
3. a^m / a^n = a^(m - n) (Quotient Law)
4. a^m * b^m = (ab)^m (Power of a Product Law)
5. a^0 = 1 (Zero Exponent Law)
6. a^(-m) = 1 / a^m (Negative Exponent Law)
7. n√a = a^(1/n) (Radical Form to Exponent Form)