MP Board Class 8 Mathematics

Revision Notes & Formula Sheet

Rational Number (परिमेय संख्या): Any number that can be expressed in the form p / q, where p and q are integers and q ≠ 0.
- Examples: 2/3, -5/7, 0 (0/1), 4 (4/1).
- Note: 0 is a rational number because it can be written as 0/1.
- Division by zero (a / 0) is undefined.
Standard Form: A rational number p / q is in standard form if p and q have no common factors other than 1, and denominator q is positive.
- Example: -4/6 simplified to standard form is -2/3.

Property	Addition	Subtraction	Multiplication	Division
Closure	Yes	Yes	Yes	No (due to 0)
Commutative	Yes	No	Yes	No
Associative	Yes	No	Yes	No

1. Closure Property (संवृत गुणधर्म):
- Addition: If a and b are rational numbers, then (a + b) is also a rational number.
- Subtraction: (a - b) is also a rational number.
- Multiplication: (a * b) is also a rational number.
- Division: (a ÷ b) is NOT always a rational number (because a ÷ 0 is undefined).
2. Commutative Property (क्रमविनिमेय गुणधर्म):
- Addition: a + b = b + a
- Multiplication: a * b = b * a
- (Subtraction and Division are NOT commutative: a - b ≠ b - a and a ÷ b ≠ b ÷ a)
3. Associative Property (सहचर गुणधर्म):
- Addition: a + (b + c) = (a + b) + c
- Multiplication: a * (b * c) = (a * b) * c
- (Subtraction and Division are NOT associative)
4. Distributive Property (वितरण गुणधर्म):
- Over Addition: a * (b + c) = (a * b) + (a * c)
- Over Subtraction: a * (b - c) = (a * b) - (a * c)

Additive Identity (योज्य तत्समक):
- Zero (0) is the additive identity for rational numbers.
- Formula: a/b + 0 = 0 + a/b = a/b
Multiplicative Identity (गुणात्मक तत्समक):
- One (1) is the multiplicative identity for rational numbers.
- Formula: a/b * 1 = 1 * a/b = a/b
Additive Inverse / Negative of a Number (योज्य प्रतिलोम):
- The additive inverse of a/b is -a/b.
- Property: (a/b) + (-a/b) = 0
- Example: Additive inverse of 3/5 is -3/5; Additive inverse of -7/2 is 7/2.
Multiplicative Inverse / Reciprocal (गुणात्मक प्रतिलोम या व्युत्क्रम):
- The reciprocal of a non-zero rational number a/b is b/a.
- Property: (a/b) * (b/a) = 1
- Example: Reciprocal of 3/4 is 4/3; Reciprocal of -5 is -1/5.
- Important: Zero (0) has no reciprocal.

Addition:
- (a / c) + (b / c) = (a + b) / c (Same denominator)
- For different denominators, find the LCM of denominators first.
Subtraction:
- (a / c) - (b / c) = (a - b) / c
Multiplication:
- (a / b) * (c / d) = (a * c) / (b * d) = (Product of Numerators) / (Product of Denominators)
Division:
- (a / b) ÷ (c / d) = (a / b) * (d / c) = (a * d) / (b * c)

Finding Rational Numbers Between Two Numbers:
- Method 1 (Mean Method): The average (a + b) / 2 lies between a and b.
- Method 2 (Same Denominator Method):
  1. Make denominators equal by finding the LCM.
  2. Multiply numerator and denominator of both numbers by (n + 1) if you need n rational numbers between them.
Numbers equal to their own reciprocals: 1 and -1.
Rational number equal to its negative: 0.