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Chapter 1 – Rational Numbers

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

You might have come across several applications of rational numbers in your daily life without being aware of their importance and value in mathematics. In simpler words, rational numbers are any numbers that are involved in various mathematical applications, for example, addition, division, subtraction, and multiplication. You will also learn about different kinds of properties of rational numbers.

  • They are the set numbers starting from 1 and ending at infinity.
  • This set is denoted by  ‘N’.
AdditionSubtractionMultiplicationDivision
Closure propertyAppliesDoesn’t applyAppliesDoesn’t apply
Commutative propertyAppliesDoesn’t applyAppliesDoesn’t apply
Associative propertyAppliesDoesn’t applyAppliesDoesn’t apply
  • They are the set of numbers starting from 0 and ending at infinity.
  • The set is denoted by a ‘W’.
AdditionSubtractionMultiplicationDivision
Closure propertyAppliesDoesn’t applyAppliesDoesn’t apply
Commutative propertyAppliesDoesn’t applyAppliesDoesn’t apply
Associative propertyAppliesDoesn’t applyAppliesDoesn’t apply
  • They are a set of natural numbers with the addition of their negatives. 
  • The set is denoted by a ‘Z’.
AdditionSubtractionMultiplicationDivision
Closure propertyAppliesAppliesAppliesDoesn’t apply
Commutative propertyAppliesDoesn’t applyAppliesDoesn’t apply
Associative propertyAppliesDoesn’t applyAppliesDoesn’t apply

In this section, a proper introduction to rational numbers will be given to you. Rational numbers can be represented in a p/q form, where both p and q are integers and q is unequal to zero. 

This set of numbers is denoted by ‘Q’.

For example- -6/8,  2/5, -5,and 4 

Properties of Rational Numbers 

Below are some critical properties of rational numbers: 

 

1. Closure Property

For any two random rational numbers, say ‘a’ and ‘b’, a∗b=c∈Q.

  • * is any binary operation (multiplication, addition, subtraction, and division) 
  • C is the product of the binary operation applied between a and b.
  • All three numbers should belong to the set of rational numbers

 

  • Addition

Closure property pertains to rational numbers under addition.

For example:

3/5+ 1/2= (6+5)/10 = 11/10

  • Subtraction

Closure property pertains to rational numbers under subtraction.

For example: 

4/2- 1/2 = (4-1)/2 = 3/2 

  • Multiplication

Closure property pertains to rational numbers under multiplication.

For example:

4/7 x -2/5 = -8/35

  • Division

Closure property does not pertain to rational numbers under addition.

For rational number s, s÷0= not defined.

 

2. Commutative Property

For any two random rational numbers, say ‘a’ and ‘b’, a∗b= b*a. Therefore, the result of the equation should be constant regardless of the order of the operands.

  • Addition

Commutative property pertains to rational numbers under addition because for any two rational numbers a and b-

(a+b) = (b+a)

  • Subtraction

The commutative property does not pertain to rational numbers under subtraction because for any two rational numbers a and b-

(a-b) is unequal to (b-a)

  • Multiplication

Commutative property pertains to rational numbers under multiplication because for any two rational numbers a and b-

(a x b) = (b x a)

  • Division

The commutative property does not pertain to rational numbers under addition because for any two rational numbers a and b-

(a÷b) is unequal to (b÷a)

 

3. Associative Property

For any three random rational numbers, say ‘a’, ‘b’ and ‘c’, (a∗b)∗c=a∗(b∗c). Therefore, the result of the equation should be constant regardless of the order of the operands.

  • Addition

Associative property pertains to rational numbers under addition because for any three rational numbers a, b and c-

(a+b)+c = a+(b+c)

  • Subtraction

The associative property does not pertain to rational numbers under subtraction because for any two rational numbers a, b and c-

(a−b)−c≠a−(b−c) because (a-b)-c = a-b-c whereas a-(b-c) = a-b+c.

  • Multiplication

Associative property pertains to rational numbers under multiplication because for any three rational numbers a, b and c-

(a×b)×c=a×(b×c)

  • Division

The associative property does not pertain to rational numbers under addition because for any three rational numbers a, b and c- 

(a÷b)÷c≠(a÷b)÷c

 

4. Distributive Property

For any three rational numbers a, b and c- 

a(b+c)=ab+aca(b−c)=ab−ac

When zero is added to a random rational number, the result remains constant. Mathematically it is represented as- 

For a rational number p/q, p/q+ 0 = p/q

Zero, in this case, is referred to as additive identity.

If (p/q)+(−p/q)=(−p/q)+(p/q)=0 then the additive inverse or the negative of a rational number pq is -pq.

When one is multiplied to a random rational number, the result remains constant. Mathematically it is represented as- 

For a rational number p/q, p/q x 1 = p/q

1, in this case, is referred to as multiplicative identity.

If p/q x r/s = 1 then r/s is the multiplicative inverse of p/q. Also, p/q is the reciprocal of the multiplicative inverse r/s.

Representation of rational numbers on the number line can be divided into two steps. 

Step 1: Equally divide the distance between the two consecutive integers in the ‘n’ number of parts.

Step 2: label the rational numbers on the number line until it contains the number you have to mark.

The number of rational numbers between rational numbers is indefinite. 

How to find out rational numbers between rational numbers:

Method 1

Ensure that the two given natural numbers have the same denominator. Once that is settled, you can pick out any rational number that lies between them.

Method 2

We can invariably find a rational number between two rational numbers by mathematically calculating their midpoint or mean.

To sum up

We hope that you understood the nuances of rational numbers and all the properties associated with them.

  1. What is a rational number in maths? 

Rational numbers are any numbers that are involved in various mathematical applications, for example, addition, division and etc.

  1. What are five examples of rational numbers?

-7, 5/6, -7/9, 3, 9.

  1. Is zero a rational number?

Yes, zero is a rational number as it can be written as p/q, where p= 0 and q= non zero.

  1. Is 5 a rational number?

Yes. 5 can be written as 5/1.

  1. How can you identify a rational number?

Any number that can be written as a fraction, where the denominator is unequal to zero.

  1. Is 2/3  a rational number?

Yes. 2/3 is written in the p/q format and 3 unequal to zero.

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