What are Rational Numbers?
Do you ever come across numbers expressed in fractional forms and wonder why haven’t they expressed as other whole numbers? What is its significance? To answer these questions Vedantu has brought this write-up for you. On this page not this question will be answered, you will also learn about other properties associated with Rational Numbers.
In Mathematics, Rational Numbers are those numbers that can be expressed in the form of a/b where both ‘a’ and ‘b’ are integers, and b is not equal to 0. To be specific, Rational Numbers are integers that can be represented on the number line. For understanding the properties of Rational Numbers, we will consider the general properties of integers, including commutative, associative, and closure properties. So, let us go through these properties of Rational Numbers one by one.
Closure Property
According to the Closure Property, for two Rational Numbers, say, for example - 'a' and 'b,' the results of addition, subtraction, and multiplication operations shall always give another Rational Number. Therefore, we can say that the Rational Numbers are closed under the Mathematical operations of addition, subtraction, and multiplication.
Addition of Rational Numbers Under the Closure Property
According to the closure property, the result of the addition of two Rational Numbers, say, for example, 'a' and 'b' is also a Rational Number, that is, a + b is also a Rational Number. Let us try to understand the concept of the addition of Rational Numbers under the closure property with the help of an example.
We have two numbers, 1/2 and 3/4.
Let us assume a = 1/2 and b = 3/4.
We will now perform the Mathematical operation of addition on these two numbers.
a + b = 1/2 + 3/4 = (1*2 + 3*1)/4 = 5/4, which is also a Rational Number.
Hence, it is evident that Rational Numbers are closed under the Mathematical operation of addition.
Subtraction of Rational Numbers Under the Closure Property
According to the closure property, the result of the subtraction of two Rational Numbers, say, for example, 'a' and 'b' is also a Rational Number, that is, a - b is also a Rational Number. Let us try to understand the concept of subtraction of Rational Numbers under the closure property with the help of an example.
We have two numbers, 1/2 and 3/4.
Let us assume a = 1/2 and b = 3/4.
We will now perform the Mathematical operation of subtraction on these two numbers.
a - b = 1/2 - 3/4 = (1*2 - 3*1)/4 = -1/4, which is also a Rational Number.
Hence, it is evident that Rational Numbers are closed under the Mathematical operation of subtraction.
Multiplication of Rational Numbers Under the Closure Property
According to the closure property, the result of the multiplication of two Rational Numbers, say, for example, 'a' and 'b' is also a Rational Number, that is, a * b is also a Rational Number. Let us try to understand the concept of multiplication of Rational Numbers under the closure property with the help of an example.
We have two numbers, 1/2 and 3/4.
Let us assume a = 1/2 and b = 3/4.
We will now perform the Mathematical operation of multiplication on these two numbers.
a * b = 1/2 * 3/4 = 3/8, which is also a Rational Number.
Hence, it is evident that Rational Numbers are closed under the Mathematical operation of multiplication.
Why is the Mathematical Operation of Division Not Under the Closure Property?
The reason why the Mathematical operation of division is not under the closure property is that division by zero isn't defined. However, we can say that except '0,' all numbers are closed under the Mathematical operation of division. Let’s consider an example.
We have two numbers, 1/2 and 3/4.
Let us assume a = 1/2 and b = 3/4.
We will now perform the Mathematical operation of division on these two numbers.
a ÷ b = 1/2 ÷ 3/4 = 1/2 * 4/3 = 2/3, which is also a Rational Number.
Hence, it is evident that all Rational Numbers except ‘0’ are closed under the Mathematical operation of division.
Commutative Law
According to the Commutative Law, for Rational Numbers, the Mathematical operations of addition and multiplication are commutative.
Commutative Law of Addition
According to the commutative law of addition, for two Rational Numbers, say, 'a' and 'b': a + b = b + a. Let's consider an example to understand the commutative law of addition.
We have two numbers, 2/5 and 7/6.
Let us assume a = 2/5 and b = 7/6.
LHS
a + b = 2/5 + 7/6 = (2*6 + 7*5)/30 = (12 + 35)/30 = 47/30
RHS
b + a = 7/6 + 2/5 = (7*5 + 2*6)/30 = (35 + 12)/30 = 47/30
LHS = RHS
Hence, the Mathematical operation of addition is commutative for Rational Numbers.
Commutative Law of Multiplication
According to the commutative law of multiplication, for two Rational Numbers, say, 'a' and 'b': a * b = b * a. Let's consider an example to understand the commutative law of multiplication.
We have two numbers, 2/5 and 7/6.
Let us assume a = 2/5 and b = 7/6.
LHS
a * b = 2/5 * 7/6 = 14/30 = 7/15
RHS
b * a = 7/6 * 2/5 = 14/30 = 7/15
LHS = RHS
Hence, the Mathematical operation of multiplication is commutative for Rational Numbers.
NOTE – The Mathematical operations of subtraction and division are not commutative for Rational Numbers as a – b ≠ b – a, and a ÷ b ≠ b ÷ a.
Associative Law
Rational Numbers follow the associative property for the Mathematical operations of addition and multiplication. Let us say that we have three numbers, 'a,' 'b,' and 'c,' for addition, the associative law for Rational Numbers states that: a + (b + c) = (a + b) + c, and for multiplication, the associative law for Rational Numbers states that: a*(b*c) = (a*b)*c.
For example – we have three numbers, 5, -6, and 2/3.
Let us see how the associate law works on the addition of Rational Numbers.
LHS
5 + (-6 + 2/3) = -1/3
RHS
(5 – 6) + 2/3 = -1/3
LHS = RHS
Hence verified
NOTE – The Mathematical operations of subtraction and division are not associative for Rational Numbers.
Importance of Rational Numbers and Their Properties
The topic of Rational Numbers and Their Properties forms a very fundamental part of Mathematics. Operations related to Rational Numbers hold a lot of significance in calculations in the future. These calculations come in handy not just in Math but also in other subjects such as Physics and Chemistry.
Rational Numbers and Their Properties will help you in solving various Mathematics and Quant problems in entrance exams conducted for government jobs and higher studies in the future. GRE exams also require you to know this topic thoroughly.
Conclusion
After reading this write-up on Rational Numbers you will be in a position to define Rational Numbers and state their various properties related to their operations. To make sure you get hands-on experience each property has been explained with associated examples. After studying this topic from Vedantu’s website you will be able to solve problems involving Rational Numbers not just in your upcoming Math exams but anywhere in the future.
FAQs on Rational Numbers and Their Properties
1. What topics have been covered in Rational Numbers and Their Properties?
Vedantu has tried to cover Rational Numbers and Their Properties in a very holistic way. To achieve this following topics have been covered:
Definition of Rational Numbers
Closure property of Rational Numbers
Adding Rational Numbers
Multiplication of Rational Numbers
Division of Rational Numbers and related limitations
Commutative laws related to Rational Numbers
For addition
For multiplication
Associative law related to Rational Numbers
All the properties are explained with suitable examples so that it becomes easier for you to understand each of them. You can cover the topic of Rational Numbers in greater detail from Vedantu’s website.
2. What are Rational Numbers in Rational Numbers and Their Properties?
Very often people tend to define Rational Numbers as simply as fractions. It provides only a limited picture. Rational Numbers in Rational Numbers and Their Properties are not just fractions rather they are much more than that. They can be defined as numbers, either integer or whole, that can be expressed in the form of a/b, such that b is never equal to 0. But not every number can be written in this format. For a number to be rational it must satisfy the following conditions:
The denominator is an integer > 0
Numerator and denominator must have only 1 as a common divisor
3. I want to master Rational Numbers and Their Properties, how many problems should I solve?
Any topic of Mathematics can be mastered with practice. The same goes for Rational Numbers and Their Properties. There is no fixed number of sums that can help you master this topic. It varies from learner to learner. It is suggested you first go through the content given above. This will help you understand the definition and properties of Rational Numbers. Also, solve all the problems given above. After this pick up any standard book for Mathematics and solve all the problems related to it. You can also refer to RD Sharma’s books on Mathematics. Start with solving basic and simpler problems then proceed towards more complex ones.
4. Will Rational Numbers and Their Properties be useful in higher standards?
Yes!. The topic of Rational Numbers and Their Properties will be very useful in higher standards. Properties learnt here will come in handy when solving more complex problems in future not just in Mathematics but also in other subjects. It suggested you solve enough problems so that you get a very good command over the properties such as commutative and associative properties. These properties will also help you when you will appear for entrance exams like MBA in the future. You will face a high number of instances in higher standards where you will deal with Rational Numbers and their calculations, thus study this topic thoroughly.
5. How much time should I dedicate to learning Rational Numbers and Their Properties?
There is no fixed number of hours you need to dedicate towards learning Rational Numbers and Their Properties. Go through the text multiple times and understand the basic concept properly. After this, start solving sums related to it from various books of Math such as NCERTs, RD Sharma, etc. This whole exercise would take about 5-6 hours on average. It can differ from student to student. Keep practicing such questions whenever you come across them and keep polishing your Mathematical skills. You can refer to Vedantu’s website for many other such topics of Mathematics.