The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:
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.
| Addition | Subtraction | Multiplication | Division | |
|---|---|---|---|---|
| Closure property | Applies | Doesn’t apply | Applies | Doesn’t apply |
| Commutative property | Applies | Doesn’t apply | Applies | Doesn’t apply |
| Associative property | Applies | Doesn’t apply | Applies | Doesn’t apply |
| Addition | Subtraction | Multiplication | Division | |
|---|---|---|---|---|
| Closure property | Applies | Doesn’t apply | Applies | Doesn’t apply |
| Commutative property | Applies | Doesn’t apply | Applies | Doesn’t apply |
| Associative property | Applies | Doesn’t apply | Applies | Doesn’t apply |
| Addition | Subtraction | Multiplication | Division | |
|---|---|---|---|---|
| Closure property | Applies | Applies | Applies | Doesn’t apply |
| Commutative property | Applies | Doesn’t apply | Applies | Doesn’t apply |
| Associative property | Applies | Doesn’t apply | Applies | Doesn’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.
Closure property pertains to rational numbers under addition.
For example:
3/5+ 1/2= (6+5)/10 = 11/10
Closure property pertains to rational numbers under subtraction.
For example:
4/2- 1/2 = (4-1)/2 = 3/2
Closure property pertains to rational numbers under multiplication.
For example:
4/7 x -2/5 = -8/35
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.
Commutative property pertains to rational numbers under addition because for any two rational numbers a and b-
(a+b) = (b+a)
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)
Commutative property pertains to rational numbers under multiplication because for any two rational numbers a and b-
(a x b) = (b x a)
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.
Associative property pertains to rational numbers under addition because for any three rational numbers a, b and c-
(a+b)+c = a+(b+c)
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.
Associative property pertains to rational numbers under multiplication because for any three rational numbers a, b and c-
(a×b)×c=a×(b×c)
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.
Rational numbers are any numbers that are involved in various mathematical applications, for example, addition, division and etc.
-7, 5/6, -7/9, 3, 9.
Yes, zero is a rational number as it can be written as p/q, where p= 0 and q= non zero.
Yes. 5 can be written as 5/1.
Any number that can be written as a fraction, where the denominator is unequal to zero.
Yes. 2/3 is written in the p/q format and 3 unequal to zero.
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