Quick Revision Notes: Class 7 Mathematics

Chapter: Integers (पूर्णांक)

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### Concept 1: What are Integers? (पूर्णांक क्या हैं?)

• Definition: Integers are a collection of numbers containing whole numbers and negative numbers.
• Set of Integers: ... , -3, -2, -1, 0, 1, 2, 3, ...
• Types of Integers:
  1. Positive Integers (धनात्मक पूर्णांक): 1, 2, 3, 4, ... (Greater than 0)
  2. Negative Integers (ऋणात्मक पूर्णांक): -1, -2, -3, -4, ... (Less than 0)
  3. Zero (शून्य): Neither positive nor negative. It is neutral.

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### Concept 2: Representation on Number Line (संख्या रेखा पर निरूपण)

• Numbers increase as we move to the right on a number line.
• Numbers decrease as we move to the left on a number line.
• Every positive integer is greater than every negative integer.
• Zero is less than every positive integer and greater than every negative integer.

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### Key Rules for Addition and Subtraction (जोड़ और घटाव के नियम)

1. Addition (जोड़):

• Same Signs: Add the absolute values and put the common sign.
  • (+) + (+) = (+) [Example: (+5) + (+3) = +8]
  • (-) + (-) = (-) [Example: (-5) + (-3) = -8]
• Opposite Signs: Subtract the smaller absolute value from the larger one and put the sign of the number with the larger absolute value.
  • (+) + (-) = Sign of the larger number. [Example: (+5) + (-3) = +2]
  • (-) + (+) = Sign of the larger number. [Example: (-5) + (+3) = -2]

2. Subtraction (घटाव):

• To subtract an integer, change the sign of the integer being subtracted and then add it.
  • a - (-b) = a + b
  • a - (+b) = a - b

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### Key Rules for Multiplication (गुणा के नियम)

When multiplying integers, multiply their absolute values and apply the following sign rules:

• (+) × (+) = (+) [Positive × Positive = Positive]
• (-) × (-) = (+) [Negative × Negative = Positive]
• (+) × (-) = (-) [Positive × Negative = Negative]
• (-) × (+) = (-) [Negative × Positive = Negative]

Shortcut:

• Even number of negative signs in multiplication $\rightarrow$ Result is Positive (+)
• Odd number of negative signs in multiplication $\rightarrow$ Result is Negative (-)

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### Key Rules for Division (भाग के नियम)

Division follows the same sign rules as multiplication:

• (+) ÷ (+) = (+)
• (-) ÷ (-) = (+)
• (+) ÷ (-) = (-)
• (-) ÷ (+) = (-)

Important Properties of Division:

• Division by zero (a ÷ 0) is not defined (परिभाषित नहीं है).
• 0 ÷ a = 0 (where a ≠ 0).

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### Properties of Integers (पूर्णांकों के गुण)

1. Closure Property (संवृत गुण):

   • Integers are closed under Addition, Subtraction, and Multiplication.
   • a + b, a - b, and a × b are always integers.
   • Integers are not always closed under Division. (3 ÷ 2 is not an integer).

2. Commutative Property (क्रमविनिमय गुण):

   • Addition: a + b = b + a (Valid)
   • Multiplication: a × b = b × a (Valid)
   • Note: Not valid for Subtraction and Division.

3. Associative Property (सहचर्य गुण):

   • Addition: (a + b) + c = a + (b + c) (Valid)
   • Multiplication: (a × b) × c = a × (b × c) (Valid)

4. Distributive Property of Multiplication over Addition (वितरण गुण):

   • a × (b + c) = (a × b) + (a × c)

5. Additive Identity (योज्य तत्समक):

   • 0 is the additive identity for integers.
   • a + 0 = 0 + a = a

6. Multiplicative Identity (गुणात्मक तत्समक):

   • 1 is the multiplicative identity for integers.
   • a × 1 = 1 × a = a

7. Additive Inverse (योज्य प्रतिलोम):

   • The additive inverse of an integer a is -a.
   • a + (-a) = 0