Chapter -1 Rational Numbers
❖ INTRODUCTION:--)
Natural numbers – The numbers which are starting from 1 to infinity, know as Natural Numbers. These are also known as Counting Number. It is denoted by (N).
Examples – N{1,2,3,4,5,6,7,8,9…….so on}.
Whole numbers – The numbers which are starting from 0 to infinity, known as Whole Numbers. These are also known as Counting Numbers Including 0. It is denoted by (W).
Examples – W{0,1,2,3,4,5,6,7,8,9……so on}.
Integers – The numbers which include positive numbers and negative numbers including zero, known as Integers. It is denoted by (Z).
Examples – Z{-1,-2,-3,0,3,2,1}.
Rational numbers – The numbers which has 𝑝 𝑞 form where p and q are integers and q≠0, known as Rational Numbers. It is denoted by (Q).
Examples – Q {1/3 , 2/3 , 3/3 }
❖ PROPERTIES OF RATIONAL NUMBERS:--)
There are mainly 4 types of Rational Numbers
Associative Property – In this property, brackets are interchange of digits. So, the formed formula is like “(a+b)+c = a+(b+c). Here, digits will be only 3. We can use any operation (+,-,×,÷)
Commutative Property – In this property, digits are interchange. So, the formed formula is like “a+b+c = c+b+a. Here, digit will be at least 2. We can use any operation (+,-,×,÷)
Closure Property - In this property, we add/sub/multiply/divide and got a number. So, the formed formula is a+b = c. Here, digit will be at least 2. We can use any operation (+,-,×,÷)
Distributive Property – In this property, we multiply one digit one by one with two digits. So, the formed formula is a×(b+c). Here, digit will be 3. We can use multiply operation only.
❖ NUMBERS OPERATION CLOSED IN ASSOCIATIVE PROPERTY :--)
Whole numbers: These are closed under Addition and Multiplication.
Natural numbers: These are closed under Addition and Multiplication.
Integers: These are closed under Addition and Multiplication.
Rational Numbers: These are close under Addition and Multiplication.
❖ NUMBERS OPERATION CLOSED IN COMMUTATIVE PROPERTY :--)
Whole numbers: These are closed under Addition and Multiplication.
Natural numbers: These are closed under Addition and Multiplication.
Integers: These are closed under Addition and Multiplication.
Rational Numbers: These are closed under Addition and Multiplication.
❖ NUMBERS OPERATION CLOSED IN CLOSURE PROPERTY :--)
Whole numbers: These are closed under Addition and Multiplication.
Natural numbers: These are closed under Addition and Multiplication.
Integers: These are closed under Addition, subtraction and multiplication.
Rational numbers: These are closed under Addition, subtraction and multiplication.
❖ NUMBERS OPERATION CLOSED IN DISTRIBUTIVE PROPERTY :--)
Whole numbers: These are closed under Addition and Subtraction.
Natural numbers: These are closed under Addition and Subtraction.
Integers: These are closed under Addition and Subtraction.
Rational Numbers: These are close under Addition and Subtraction.
❖ THE ROLE OF 0 :--)
By seeing given below formulas we able to understand that if we will add or sub any number with 0 we got same number,
1. A+0 = 0+A → A
2. B + 0 = 0+B → B
By seeing given below formulas we able to understand that if we will multiply or divide any number with 0 we got 0,
1. A×0 = 0×A → 0
2. B÷0 = 0÷B → 0
❖ THE ROLE OF 1:--)
By seeing given below formulas we able to understand if we will add or sub 1 to any number then the number got increase itself 1 more,
1. A+1=B+1 → B = C (let’s suppose that by increasing 1 with A equal to B and B equal to C).
2. B–1=C-1 → A = B (let’s suppose that by decreasing 1 with B equal to A and C equal to B).
By seeing given below formulas we able to understand if we will multiply or divide 1 to any number then same number came,
1. A×1=B×1 → A=B
2. A÷1=B÷1 → A=B
❖ ADDITIVE INVERSE :--) In additive inverse, plus change into minus and minus into plus, known as Additive Inverse.
Ex-
1. (-2) → 2
2. 5 → (-5)
❖ MULTIPLICATIVE INVERSE :--) In multiplicative inverse we reciprocal the number means denominator come in place of numerator and numerator comes in place of denominator.
Ex-
1. 2 → 1 2
2. 3 4 → 4 3
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