In simple terms, Integers are natural numbers and their negatives. The set of Integers is denoted as ′Z′ or ′I’. Closure property applies to integers in the case of addition, subtraction, and multiplication but not division. Commutative Property Commutative property applies to integers in the case of addition and multiplication but not subtraction and division. Associative Property Associative property applies to integers in the case of addition and multiplication but not subtraction and division.

Chapter 1 Rational Numbers Class 8 Notes by MSVGo covers all the theory, properties and arithmetic operations involved in rational numbers. All the concepts are clearly explained down below.

Rational Numbers are numbers in the form of p/q such that q>0. It is denoted by “Q”. If the numerator and denominator are coprime and q>0 then the Rational Number is of the standard form.

A rational number can be defined as an integer that can be represented in the form of p/q where q is greater than 0. Any fraction can be defined as the rational number, where the denominator and numerator are integers, and the denominator is not equal to zero. A rational number can be represented by the letter “Q”.

The rational numbers include positive, negative numbers, and zero.

A rational number p/q is said to be in standard form if p and q are co-primes and q is always a positive integer.

Rational Numbers in standard form examples: 5/12, -10/3, 1/1, 0/1, etc.

i) Positive Rational Numbers: The sign of both the numerator and denominator are the same, i.e., either both are positive, or both are negative. Ex: 2/5, −7/−9

ii) Negative Rational Numbers: The sign of both the numerator and denominator are the same, i.e., if the numerator is negative, the denominator will be positive. Similarly, if the numerator is positive, the denominator is negative. Ex: 2/−3, −7/8

iii) Zero Rational Numbers: The numerator is always zero. Ex: 0/3, 0/8

i) Closure Property: The addition, subtraction, and multiplication operations result in closure Property, i.e., for any two rational numbers in these operations, the answer is always a rational number.

ii) Commutative Property: The various order of rational numbers in the operations like addition and multiplication results in the same answer. Ex: 2/3+4/8= 4/8+2/3, ...

iii) Associative Property: The grouping order does not matter in the operations like addition or multiplication, i.e., the place where we add the parenthesis does not change the answer. Ex: 8/9+(4/5+6/7) = (8/9+4/5) +6/7

iv) Distributive Property: The rational numbers are distributed in the following way:

a(b+c) =ab+ac

a(b−c) =ab−ac

General Properties: 

• A rational number can be a fraction or not, but vice versa is true.

• Rational numbers can be denoted on a number line.

• There is an ′n′ number of rational numbers between any two rational numbers.

 

Role of Zero: Also known as the Additive Identity

Whenever ′0′ is added to any rational number, the answer is the Rational number itself.

Ex: If ′a′ is any rational number, then a+0=0+a=a.

b + 0 = 0 + b = b, where b is an any integer.

c + 0 = 0 + c = c, c is a rational number.

Zero is called the additive identity for the addition of rational numbers.

 

Role of One: Also known as the Multiplicative Identity.

Whenever ′1′ is multiplied by any rational number, the answer is the Rational number itself.

Ex: If ′a′ is any rational number, then a×1=1×a=a

b × 1 = b, where b is an integer.

c × 1 = c, c is a rational number.

1 is the multiplicative identity for rational numbers.

Additive Inverse:

The Additive Inverse of any rational number is the same rational number with the opposite sign. The additive inverse of a/b is −a/b. Similarly, the additive inverse of −a/b is a/b, where a/b is the rational number. 

a + (-a) = 0; a is a whole number.

b +(-b) = 0; b is an integer.

(a/b) + (-a/b) = 0; a/b is a rational number.

Multiplicative Inverse: Also known as the Reciprocal.

The Multiplicative Inverse of any rational number is the inverse of the same rational number. The multiplicative inverse of a/b is b/a. Similarly, the multiplicative inverse of b/a is a/b, where a/b and b/a are any rational numbers.

Distributivity of Multiplication over Addition and Subtraction

For all rational numbers a, b, and c,

a (b + c) = ab + ac

a (b – c) = ab – ac

 

• We draw a line.

• We mark a point O on it and name it 0. Mark a point to the right of 0. Name it 1. The distance between these two points is called unit distance.

• Mark a point to the right of 1 at a unit distance and name it 2.

• Proceeding in this manner, we can mark points 3, 4, 5,

• Similarly, we can mark – 1, – 2, – 3, – 4, – 5, ……… to the left of 0. This line is called the number line.

• This line extends indefinitely on both sides.

The positive rational numbers are represented by points on the number line to the right of O whereas the negative rational numbers are represented by points on the number line to the left of O.

Any rational number can be represented on this line. The denominator of the rational number specifies the number of equal parts into which the first unit has been divided whereas the numerator specifies how many of these parts are to be taken into consideration.

 

We can find infinitely many rational numbers between any two given rational numbers. We can take the help of the idea of the mean for this purpose.

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Q1. What are Direct and Inverse Proportions for Class 8?

Answer. Direct and Inverse proportions are chapter 13 in Class 8 Maths. This chapter explains to the students to find the relation between any two quantities. It tells how one quantity changes if there is any variation with another quantity. This chapter is very important for a student as this chapter plays a vital role in our daily lives too. This chapter teaches the concepts of proportions through various types of relatable questions. One can score full marks in this chapter if understood well.

 

Q2. Mention the difference between direct and inverse proportions in Class 8?

Answer. Following are the differences between direct and inverse proportions:

In Direct proportion, the quantities vary directly from each other whereas, in Inverse proportions, the quantities vary inversely. In Direct proportion, both the quantities increase or decrease at the same time, whereas in Inverse proportion if one increases the other decreases. 

Q3. Mention the properties of Rational numbers

Answer. The list of properties of rational numbers can be given as follows:

• Closure

• Commutativity

• Associativity

• The role of zero (0)

• The role of 1

• Negative of a number

• Reciprocal

• Distributivity of multiplication over addition for rational numbers.