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What are the important points about rational numbers Class 8?
In the number system, rational numbers are numbers that can be expressed as a ratio of two integers. They can also be the quotient of the ratio if the rational number is an integer. If the rational number is represented by the ratio p/q, then q must be a non-zero integer.
What are the properties of a rational number Class 8 Chapter 1?
Some important formulas and properties covered in NCERT solutions for class 8 maths chapter 1 are given below: Closure Property: a + b, a - b, a × b results in a rational number. Commutative property: a + b = b + a; a × b = b × a. Associative property: a + (b + c) = (a + b) + c; a × (b × c) = (a × b) × c.
What I found challenging in chapter rational numbers?
Adding to the difficulty of learning rational number arithmetic are its complex relations to whole number arithmetic. For fractions, when denominators are equal, numerators are added or subtracted as if they were whole numbers, but the common denominator is maintained unchanged (3/5 + 4/5 = 7/5).
What is the role of zero in a rational number Class 8?
0 will be the additive identity for rational numbers. 1 will be the multiplicative identity for the rational numbers.
CBSE Class 8 Maths Notes Chapter 1 Rational Numbers
Here we have provided CBSE Class 8 Maths Notes Chapter 1 Rational Numbers for the ease of students so that they can prepare better in their exams.
Neha Tanna22 Aug, 2024
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CBSE Class 8 Maths Notes Chapter 1:
Chapter 1 of CBSE Class 8 Maths, titled Rational Numbers, introduces the concept of numbers that can be written in the form p/q
. The chapter begins by reviewing integers and fractions before expanding into rational numbers.
Students learn how to represent rational numbers on a number line, find equivalent rational numbers, and express them in their standard form. CBSE Class 8 Maths Notes Chapter 1 also explains the operations of addition, subtraction, multiplication, and division of rational numbers, along with their properties such as closure, commutative, associative, and distributive properties.
CBSE Class 8 Maths Notes Chapter 1 Overview
Chapter 1 of CBSE Class 8 Maths focuses on Rational Numbers, which are numbers that can be expressed in the form of p/q
, where
p
and
q
are integers. The chapter begins by revising the properties of whole numbers, integers, and fractions, then extends this understanding to rational numbers.
Key concepts covered include the representation of rational numbers on a number line, finding equivalent rational numbers, and the standard form of rational numbers. The chapter also delves into the
addition, subtraction, multiplication,
and
division
of rational numbers, emphasizing how to handle positive and negative values.
Students learn about the properties of rational numbers, such as
closure, commutative, associative,
and
distributive
properties under different operations. The concept of the
additive inverse
(negative of a rational number) and the
multiplicative inverse
(reciprocal) is also introduced.
CBSE Class 8 Maths Notes Chapter 1 PDF
Here we have provided CBSE Class 8 Maths Notes Chapter 1 Rational Numbers pdf for the ease of students. Students can download this pdf and access it without internet.
CBSE Class 8 Maths Notes Chapter 1 Rational Numbers
Here we have provided CBSE Class 8 Maths Notes Chapter 1 Rational Numbers -
Rational numbers are those that are intrinsically closed in many mathematical operations and are used in a wide range of mathematical applications, including addition, subtraction, and multiplication.
Introduction to Rational Numbers
We will study rational numbers, their characteristics, how to depict rational numbers on a number line, and how to use examples to obtain rational numbers between any two rational numbers in the chapter Rational Numbers for Class 8.
Whole Numbers and Natural Numbers
A collection of numbers called natural numbers begins at 1 and goes all the way up to infinity. ′N′ stands for the set of natural numbers.
N is the Set of Natural Numbers = {1,2,3,... }
Whole numbers are collections of numbers that extend from 0 to infinity. In essence, the zero has been introduced to the set of natural numbers. The letter "W" stands for the set of whole numbers.
The whole numbers (W) are {0,1,2,3,...}
Properties of Natural Numbers and Whole Numbers
Closure Property
Property Closure When adding and multiplying whole numbers, closure property holds true; however, when subtracting and dividing, it does not. This also holds true for natural numbers.
Commutative Property
Equivalency of Properties When it comes to addition and multiplication, the commutative characteristic holds true for whole numbers and natural numbers, but not when it comes to subtraction and division.
Associative Property
Relationship-Building Assemblage When it comes to addition and multiplication, associative property holds true for whole numbers and natural numbers, but not when it comes to subtraction and division.
Integers
Simply put, natural numbers and their negatives are known as integers. The symbols "Z" or "I" stand for the set of integers.
Z: The Set of Integer Numbers = {3, 2, 3, 0, 1, 2, 3,... }
Properties of Integers
Closure Property
- Property Closure Integers are subject to the closure property in addition, subtraction, and multiplication but not division.
Commutative
Property-
Equivalency of Properties Integers have the commutative property when it comes to addition and multiplication, but not subtraction or division.
Associative
Property
- Relationship-Building Assemblage When adding and multiplying numbers, the associative condition holds true; however, division and subtraction do not.
Rational Numbers
A rational number is one that has the form p/q, where q must not be 0, and may be expressed as a fraction of two integers. Q is the abbreviation for the set of rational numbers.
For instance, if -5 and 7 are integers, then -5/7 is a rational number. Given that it can be expressed as 2/1, where 2 and 1 are integers, even 2 is a rational number.
Properties of Rational Numbers
Closure Property of Rational Numbers
a∗b=c∈Q holds for any two rational numbers, a and b. The product of addition, subtraction, and multiplication of two rational integers, let's say a and b, yields a rational number. We can say that the closure property applies to rational numbers in the case of addition since the sum of two numbers is ultimately a rational number.
For instance, when 17 and 12 are integers, the addition of 2/3 + 3/4 = (8+9)/12 = 17/12 is likewise a rational number. A rational number is the difference of two rational numbers. Consequently, in the case of subtraction, the closure property holds true for rational numbers.
For instance, where 1 and 20 are integers, the difference between 4/5 – 3/4 = (16–15)/20 = 1/20 is likewise a rational number. A rational number is produced when two rational numbers are multiplied. As a result, we may state that the closure property also holds true for rational numbers when they are multiplied.
As an illustration, consider the rational number (-2/5) that results from multiplying 1/2 by (-4/5) = (-4/10) where -2 and 5 are integers. When two rational numbers are divided, we can observe that a rational number a·0 is not defined for that number a. Therefore, in the case of division, we can say that the closure property does not hold true for rational numbers.
Commutative Property of Rational Numbers
A∗b=b∗a for any two rational numbers, a and b. To put it another way, a commutative property requires that an equation's solution hold true even if the operands' order changes. When two rational numbers, a and b, are given, (a+b) will always equal (b+a). Thus, for rational numbers, addition is commutative.
For example: 2/3 + 4/3 = 4/3 + 2/3
⇒ 6/3 = 6/3
⇒ 2 = 2
When comparing two rational numbers, a and b, it can be observed that (a−b) is never equal to (b−a). Consequently, for rational numbers, subtraction is not commutative.
For example: 2/3 – 4/3 = -2/3
but 4/3 – 2/3 = 2/3
The product of two rational numbers, a and b, is equal to (b×a) when considering (a×b). For rational numbers, multiplication is hence commutative.
For example: 2/3 × 4/3 = 8/9
and
4/3 × 2/3 = 8/9
When two numbers, a and b, are divided, (a÷b) differs from (b÷a). Thus, for rational numbers, division is not commutative.
For example: 2/3 is definitely different from 3/2.
Associative Property of Rational Numbers
(a∗b)∗c=a∗(b∗c) for any three rational numbers, a, b, and c. That is to say, an associative property requires that an equation's solution hold true even if the operators' order changes.
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
.
Example: 1/2 + (1/4 + 2/3) = (1/2 + 1/4) + 2/3
⇒ 17/12 = 17/12
Also,
1/2 x (1/4 x 2/3) = (1/2 x 1/4) x 2/3
⇒ 2/24 = 2/24
⇒1/12 = 1/12
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+ac
a(b−c)=ab−ac
Example: 1/2 x (1/2 + 1/4) = (1/2 x 1/2) + (1/2 x 1/4)
3/8 = 3/8
Negatives and Reciprocals
Negation of a Number
In the case of a rational number a/b, a/b+0 = a/b, meaning that any rational number multiplied by zero yields the same rational number. In this context, "0" denotes the additive identity for rational integers.
If (a/b) + (-a/b) = (-a/b) + (a/b) = 0, then it can be said that the
additive inverse
or
negative
of a rational number ab is −ab. Also, (−a/b) is the
additive inverse
or
negative
of a/b.
For example: The additive inverse of -21/8 is -(-21/8) = 21/8
Reciprocal of a Number
Any rational number multiplied by "1" yields the same rational number, hence for any given rational number a/b, a/b × 1 = a/b. Therefore, for rational numbers, '1' is referred to as the multiplicative identity.
It can be stated that c/d is the multiplicative inverse of a rational integer a/b, or reciprocal, if a/b × c/d = 1. Furthermore, a/b is the multiplicative inverse of a rational integer c/d, or reciprocal.
For example: The reciprocal of 2/3 is 3/2 as 2/3 × 3/2 = 1.
Representing on a Number Line
Representation of Rational Numbers on the Number Line
To represent a given rational number a/n on the number line, where a and n are integers:
Step 1: Split the distance in 'n' parts, the distance between two consecutive numbers.
For instance, we split the space between 0 and 1, 1 and 2, etc. into three pieces if we are given the rational number 2/3.
Step 2: indicate the rational numbers with labels until the number you need to indicate is included in the range.
Rational Numbers between Two Rational Numbers
Unlike whole numbers and natural numbers, the number of rational numbers that separate any two given rational numbers is not certain.
For instance, there are precisely seven natural numbers between 2 and 10, but there are an unlimited amount of possible numbers between 2 and 8/10.
Technique 1 Make sure the denominators of two rational numbers are the same. Any rational integer in between can be chosen once there is a common denominator.
Example: Find the rational numbers between ½ and ⅔.
LCM of denominators (2 and 3) = 6
½ × (3/3) = 3/6
⅔ × (2/2) = 4/6
We cannot find any number between 3 and 4. So will multiply each rational number 3/6 and 4/6 by 10/10.
3/6 × 10/10 = 30/60
4/6 × 10/10 = 40/60
Hence, the rational numbers between 1/2 and 2/3 are 31/60, 32/60, 33/60, 34/60, 35/60, 36/60, 37/60, 38/60 and 39/60.
Method 2
By computing the mean or midpoint of two rational numbers, we can always discover a rational number that falls between them.
Example: Find the rational numbers between 3 and 4.
Mean of 3 and 4 = (3+4)/2 = 7/2
Mean of 3 and 7/2 = 13/4
Hence, the two rational numbers between 3 and 4 are 7/2 and 13/4.
Benefits of CBSE Class 8 Maths Notes Chapter 1
The benefits of studying Chapter 1 of CBSE Class 8 Maths, Rational Numbers, include:
Strong Conceptual Foundation
: Understanding rational numbers forms the basis for many advanced mathematical topics like algebra, geometry, and data handling in future classes.
Improved Arithmetic Skills
: The chapter focuses on the addition, subtraction, multiplication, and division of rational numbers, enhancing basic arithmetic skills that are applicable in real-life situations.
Understanding Number Properties
: Students learn important mathematical properties such as
closure, commutative, associative,
and
distributive
, which helps them solve problems more efficiently.
Problem-Solving Skills
: The chapter includes exercises that challenge students to think logically and apply concepts, improving their analytical and problem-solving abilities.
Preparation for Higher Studies
: Mastery of rational numbers prepares students for more complex mathematical topics like polynomials, linear equations, and functions in higher grades.
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