Class 8 mathematics chapter 1 key points

Introduction to Rational Numbers

Whole Numbers and Natural Numbers

Natural numbers are set of numbers starting from 1 counting up to infinity. The set of natural numbers is denoted as ′N′Whole numbers are set of numbers starting from 0 and going up to infinity. So basically they are natural numbers with the zero added to the set. The set of whole numbers is denoted as ′W′Closure Property Closure property is applicable for whole numbers in the case of addition and multiplication while it isn’t in the case for subtraction and division. This applies to natural numbers as well. Commutative Property Commutative property applies for whole numbers and natural numbers in the case of addition and multiplication but not in the case of subtraction and division. Associative Property Associative property applies for whole numbers and natural numbers in the case of addition and multiplication but not in the case of subtraction and division.

Integers

In simple terms Integers are natural numbers and their negatives. The set of Integers is denoted as ′Z′ or ′I′Closure Property 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 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.

Rational Numbers

A rational number is a number that can be represented as a fraction of two integers in the form of pq, where q must be non-zero. The set of rational numebrs is denoted as Q.

For example: −57 is a rational number where -5 and 7 are integers. Even 2 is a rational number since it can be written as 21 where 2 and 1 are integers.

Properties of Rational Numbers

Closure Property of Rational Numbers

For any two rational numbers a and ba∗b=c∈Q i.e. For two rational numbers say a and b the results of addition, subtraction and multiplication operations gives a rational number. Since the sum of two numbers ends up being a rational number, we can say that the closure propertyapplies to rational numbers in the case of addition.
For example : The sum of 23 +34 =(8+9)12 =1712 is also a rational number where 17 and 12 are integers. The difference between two rational numbers result in a rational number. Therefore, the closure property applies for rational numbers in the case of subtraction.
For example : The difference between 45 −34 =(16−15)20 =120 is also a rational number where 1 and 20 are integers. The multiplication of two rational numbers results in a rational number. Therefore we can say that the closure property applies to rational numbers in the case of multiplication as well.
For example : The product of 12 ×−45 =−410 =−25 which is also a rational number where -2 and 5 are integers. In the case with division of two rational numbers, we see that for a rational number a, a÷0 is not defined. Hence we can say that the closure property does not apply for rational numbers in the case of division.

Commutative Property of Rational Numbers

For any two rational numbers a and ba∗b=b∗a. i.e., Commutative property is one where in the result of an equation must remain the same despite the change in the order of operands. Given two rational numbers a and b, (a+b) is always going to be equal to (b+a). Therefore addition is commutative for rational numbers.
For example: 23 +43 = 43 + 23 ⇒67 = 67 Considering the difference between two rational numbers a and b, (a−b) is never the same as (b−a). Therefore subtraction is not commutative for rational numbers.
For example: 23 − 43 = −23 Whereas 43 − 23 = 23 When we consider the product of two rational numbers a and b, (a×b) is the same as (b×a). Therefore multiplication is commutative for rational numbers.
For example: 23 × 43 = 89 43 × 23 = 89 Considering the division of two numbers a and b, (a÷b) is different from (b÷a). Therefore division is not commutative for rational numbers.
For example: 2÷3=23 is definitely different from 3÷2=32

Associative Property of Rational Numbers

For any three rational numbers a,b and c, (a∗b)∗c=a∗(b∗c). i.e., Associative property is one where the result of an equation must remain the same despite a change in the order of operators. Given three rational numbers a,b and c, it can be said that : (a+b)+c = a+(b+c). Therefore addition is associative. (a−b)−c≠a−(b−c). Because (a-b)-c = a-b-c whereas a-(b-c) = a-b+c. Therefore we can say that subtraction is not associative. (a×b)×c=a×(b×c). Therefore multiplication is associative.(a÷b)÷c≠(a÷b)÷c. Therefore division is not associative.

Distributive Property of Rational Numbers

Given three rational numbers a,b and c, the distributivity of multiplication over addition and subtraction is respectively given as : a(b+c)=ab+aca(b−c)=ab−ac

Negatives and Reciprocals

Negation of a Number

For a rational number ab, ab + 0 = ab. i.e., when zero is added to any rational number the result is the same rational number. Here ‘0′ is known as additive identity for rational numbers. If (ab)+(−ab)=(−ab)+(ab)=0, then it can be said that the additive inverse or negative of a rational number ab is −ab. Also −ab is the additive inverse or negative of ab.
For example : The additive inverse of −218 is −(−218)=218

Reciprocal of a Number

For any rational number ab, ab×1=ab. i.e., When any rational numbers is multiplied by ‘1’ ,the result is same rational number. Therefore ‘1’  is called multiplicative identity for rational numbers. If ab×cd=1, then it can be said that the cd is reciprocal or the multiplicative inverse of a rational number ab. Also ab is reciprocal or the multiplicative inverse of a rational number cd For example : The reciprocal of 23 is 32 as 23×32=1

Comments

Post a Comment

Popular posts from this blog

Class 8 science chapter 1 very short questions with answers

Very Short Answer Questions Q. 1. If you are given a dry piece of land for cultivation what will you do before  sowing the seeds? [NCERT Exemplar] Ans: The field will be watered, tilled and ploughed before sowing seeds. Q. 2. During which months do farmers grow mustard in India? [NCERT Exemplar] Ans. October to March. Q. 3. Which activity of the farmer can promote growth of earthworms and  microbes in the field? [NCERT Exemplar] Ans. Loosening the soil/maintaining high moisture levels in soil. Q. 4. What are organic foods? [NCERT Exemplar] Ans. Crops cultivated without using any chemicals like fertilisers, pesticides,  weedicides, etc. are called organic foods. Q. 5. Why does a farmer rotate crops in a field? Ans. A farmer rotates crops as it maintains fertility of soil, gives better yields and  prevents crop diseases and pests. Q. 6. What do you mean by transplantation? Ans. The process of shifting seedlings from the nursery to the main field is called  transpl...
Read more

CLASS 8 HISTORY CHAPTER 1 SHORT QUESTIONS

Short Answer Questions Q. 1. How did James Mill view India? Ans. James Mill was not very positive towards India and considered Asian society as a whole at lower level of civilisation than Europe. He was well aware of religious intolerance, caste taboos and superstitious practices that dominated Indian social life. He suggested that the British should conquer all the territories of India to ensure the enlightenment and happiness of Indian people. For India was not capable of progress without the help of the British. He had a strong opinion that only British rule could civilise India. Q. 2. How did the British conquer India and establish their rule? Ans. The British conquered India in the following ways: (a) They conquered and overpowered local nawabs and rajas. (b) They gained control over economy and society. The collected revenue helped them to meet all their expenses. (c) They brought changes in rulers, tastes, customs and practices. Thus, they moulded everything in their favour. Q. ...
Read more