Rational Numbers: Class 8 NCERT Math

Introduction to Rational Numbers

• An introduction to Class 8 NCERT Mathematics that focuses on rational numbers
• The goal is to make math seem less intimidating by performing regular study, practice, and video watching
• Numbers began as a way to measure and communicate needs or feelings in value and size
• It was hard for early humans to communicate amounts and temperatures without numbers

Evolution of Number Systems

• The first numbers were natural numbers like 1, 2, and 3 and it is also known as counting numbers.
• Natural numbers did not originally include the idea of zero.
• Whole numbers consists of all natural numbers including zero, which fulfilled the gap in the natural number system.
• Integers consist of positive and negative numbers as well as zero.
• Humans needed negative numbers to represent things such as debt or temperatures below zero as they evolved.
• Humans started dividing things which brought about fractions and decimals.

Definition and Components of Rational Numbers

• Rational numbers include whole numbers, natural numbers, integers, fractions, and decimals.
• Rational numbers can be written as P/Q, where P and Q are integers, but Q cannot be zero.
• Numbers like √2 that can't be accurately written as P/Q are called irrational numbers.
• Decimals like 2.1 can be rational numbers because they can be converted into fractions like 21/10.
• For a number to be classified as rational, the numerator and denominator must both be integers.

Properties of Rational Numbers: Closure Property

• Properties are characteristics of a particular thing
• Closure property says that performing mathematical operations on numbers of the same type (like integers) will result in a number that is also of the same type.
• The Closure Property confirms if math operations result in the same category you start with.
• Closure is applicable to intergers if adding two integers always results in an integer
• If adding results in non-like number, it wont be compatible in such case
• Natural Numbers:
  • Closed under addition, which always results in a natural number.
  • Not closed under subtraction and division.
  • Closed under multiplication.
• Whole Numbers:
  • Closed under addition and multiplication.
  • Not closed under subtraction and division.
• Integers:
  • Closed under addition, subtraction, and multiplication.
  • Not closed under division.
• Rational Numbers:
  • Closed under addition, subtraction, and multiplication.
  • Division generally is not closed, it has an exception when the denominator is zero, unless division by zero is not allowed.

Properties of Rational Numbers: Commutative Property

• Commutative property says that changing the order of numbers does not change the answer, such as A + B = B + A.
• Natural Numbers:
  • Addition is commutative.
  • Subtraction does not apply.
  • Multiplication is commutative.
  • Division does not apply.
• Whole Numbers:
  • Addition is commutative.
  • Subtraction does not apply.
  • Multiplication is commutative.
  • Division does not apply.
• Integers:
  • Addition is commutative.
  • Subtraction does not apply.
  • Multiplication is commutative.
  • Division does not apply.
• Rational Numbers:
  • Addition is commutative
  • Subtraction does not apply.
  • Multiplication is commutative
  • Division does not apply.

Properties of Rational Numbers: Associative Property

• Associative property is when re-grouping to link terms does not change the answer.
• A + (B + C) = (A + B) + C
• Natural Numbers:
  • Addition applicable.
  • Subtraction does not apply.
  • Multiplication applicable.
  • Division does not apply.
• Whole Numbers:
  • Addition applicable.
  • Subtraction does not apply.
  • Multiplication applicable.
  • Division does not apply.
• Integers:
  • Addition applicable.
  • Subtraction does not apply.
  • Multiplication applicable.
  • Division does not apply.
• Rational Numbers:
  • Addition applicable.
  • Subtraction does not apply.
  • Multiplication applicable.
  • Division does not apply.

Application of the Properties

• Commutative, Associative and Distributive properties are used to Simplify the expressions or equations.
• Computations are easier when numbers are re-arranged.
• Properties help in easier calculations..

Role of Zero and One

• Zero:
  • Adding zero to a number doesn't change it.
• Additive inverse:
  • A number to add to get a zero
  • Seven's additive inverse is -7
• One:
  • Multiplying one by a number doesn't change it.
• Multiplicative inverse:
  • Number multiplies to get one
  • Sevens multiplicative interverse is -1/7, also known as reciprocal.

Think, Discuss, and Write: Rational Numbers.

• Question: "If a property holds true for rational numbers, does that mean it will also be valid for integers, whole numbers, and others?"
• Answer:
  • "Not necessarily. While some properties like the associative and commutative properties may hold across different number systems, others like the closure property won't always be applicable."
  • "The validity depends on the characteristics/definition of the number system being considered."
  • "It isn't always guaranteed that what works for a broader number set (like rational numbers) will automatically apply to smaller subsets (like whole numbers)."
  • "A key factor is whether the specific properties align with all possible cases within the given number system; any exceptions will make the property invalid for that number system."

Distributive Property

• A(B + C) = AB + AC.
• Distributive property applicable for solving a number expression.

Textbook Examples with Solutions

• Simplification of numerical equations.
• Application of commutative, associative and Distributive property to solve questions in an easy way.
• "Negative of a negative" equals the original value.

Conclusion

• The material reviewed key concepts of rational numbers, their properties such as commutative, associative, and distributive laws, the roles of zero and one, additive and multiplicative inverses.
• It also covered textbook examples for a clear explanation of how these concepts can be practically applied.