Rational Numbers and Their Properties


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 the 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.