Rational numbers can be expressed as ratios of integers in the form p/q where q is not equal to 0. They have four main properties: commutativity, associativity, distributivity, and closure. The four main operations on rational numbers are addition, subtraction, multiplication, and division. To add or subtract rational numbers, they are first converted to have a common denominator before applying the numeric operation. To multiply and divide rational numbers, the numerators and denominators are multiplied or divided separately.

1. rational
numbers
By Somu Rajesh

2. Rational numbers
 Rational numbers are numbers which can be expressed in the form of p/q A
rational number is a number that can be express as the ratio of two integers. A
number that cannot be expressed that way is irrational. Where q is not equal to
0
E.g. ; p/q = -89/47
Where q not equal to zero and the rational fraction is to be written in its
standard form

3. Properties of rational numbers
There are four properties of rational numbers that are
 Commutativity property
 Associative property
 Distributive property
 Closure property

4. Distributive property
To “distribute” means to divide something or give a share or part of something.
According to the distributive property, multiplying the sum of two or more
addends by a number will give the same result as multiplying each addend
individually by the number and then adding the products together.
E.g.; 3/2*5/6+3/2*-2/6 = 3/2[5/6+-2/6]
3/2[5/6-2/6] = 3/2[3/6]
3/2[5/6-2/6] = 3/4

5. Types of distributive properties
There are two types of distributive properties that are
Distributive property of multiplication over addition
Distributive property of multiplication over subtraction

6. Distributive property of multiplication
over addition
The distributive property of multiplication over addition can be used when
you multiply a number by a sum. For example, suppose you want to multiply 3 by
the sum of 10 + 2. ... According to this property, you can add the numbers and
then multiply by 3. 3(10 + 2) = 3(12) = 36.
E.g.; 3/2*5/6+3/2*-2/6 = 3/2[5/6+-2/6]
3/2[5/6-2/6] = 3/2[3/6]
3/2[5/6-2/6] = 3/4

7. Distributive property of multiplication
over subtraction
The distributive property of multiplication over subtraction. For example, . is
like the distributive property of multiplication over addition. You
can subtract the numbers and then multiply, or you can multiply and
then subtract as shown below. This is called “distributing the multiplier.”
7/10(9/10 – 2/10) = 7/10(9/10) – 7/10(2/10)
7/10(9/10- 2/10) = 7/10*7/10
7/10(9/10 – 2/10) = 49/100

8. Commutative property
The word "commutative" comes from "commute" or "move around", so the
Commutative Property is the one that refers to moving stuff around. For
addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2. For
multiplication, the rule is "ab = ba"; in numbers, this means 2×3 = 3×2.
E.g. ; 2/3*-2/9 = -2/9*2/3
-4/27 = -4/27

9. Types of commutative properties
There are two types of commutative properties that are
 Commutative property over addition
 Commutative property over multiplication

10. Commutative property of addition
The sum of two rational numbers is always a rational number. If a/b and c/d are
any two rational numbers, then (a/b + c/d) is also a rational
number. Commutative property of addition of rational numbers: Two rational
numbers
e g ; 2/3 +4/3 = 4/3 +2/3
6/3 = 6/3
2 = 2

11. Commutativity property of
multiplication
The commutative property is a math rule that says that the order in which
we multiply numbers does not change the product.
2/3 * 4/3 = 4/3 * 2/3
8/3 = 8/3

12. Associative property
associative property states that you can add or multiply regardless of how the
numbers are grouped. By 'grouped' we mean 'how you use parenthesis'. In other
words, if you are adding or multiplying it does not matter where you put the
parenthesis.
E g ; [2/3 + 4/3] + 5/3 = 2/3 +[4/3 +5/3]
6/3 + 5/3 = 2/3 + 9/3
11/3 = 11/3

13. Types of Associative property
There are two types of associative properties that are
 Associativity property of addition
 Associativity property of multiplication

14. Associativity property of addition
According to the associative property of addition, the sum of three or more
numbers remains the same regardless of how the numbers are grouped. Here's an
example of how the sum does NOT change irrespective of how the addends are
grouped
[2/3+1/3]+3/3=2/3+[1/3+3/3]
3/3+3/3=2/3+4/3
6/3=6/3
2=2

15. Associativity property of multiplication
According to the associative property of multiplication, the product of three or
more numbers remains the same regardless of how the numbers are grouped.
Here's an example of how the product does not change irrespective of how the
factors are grouped.
[2/3*1/3]*3/3=2/3*[1/3*3/3]
2/9*3/3=2/3*3/9
6/27=6/27

16. Closure property of rational numbers
 The closure property states that for any two rational numbers a and b, a ÷ b
is also a rational number. The result is a rational number. ... So rational
numbers are not closed under division. But if we exclude 0, then all
the rational numbers are closed under division.

17. closure property of addition
 The set of whole numbers is closed under addition if the addition of any two
elements from the set produces another element in the set. ... If the addition of
any two elements in the set of whole numbers produces another element in the
set, then the set of whole numbers is closed under addition.

18. Closure property of subtraction
 Closure property under addition and multiplication is a closed operation,
where as under subtraction

19. Closure property of division
 Closure property under addition and multiplication is a closed operation,
where as under subtraction and division its not a closed operation

20. Operations of rational numbers
There are four main operations in rational numbers that are
 Addition
 Subtraction
 Multiplication
 division

21. Addition of rational numbers
 Addition of Rational Numbers
• When Given Numbers have same Denominator: In this case, we define (a/b +
c/b) = (a + c)/b.
• When Denominators of Given Numbers are Unequal: In this case we take the
(least common multiple) LCM of their denominators and express each of the
given numbers with this LCM as the common denominator.
3/10+4/5
3+8/10
11/10

22. Subtraction of rational numbers
 What is the rule for subtracting rational numbers?
 When subtracting rational numbers we follow the rules for
subtracting integers. CHANGE the sign of the second number to the opposite,
positive becomes negative, negative becomes positive.
If a/b and c/d are two rational numbers, then subtracting c/d from a/b means
adding additive inverse (negative) of c/d to a/b. ... The subtracting of c/d from
a/b is written as a/b - c/d.
If a/b and c/d are two rational numbers,
then subtracting c/d from a/b means adding additive
inverse (negative) of c/d to a/b. ... The subtracting of c/d
from a/b is written as a/b - c/d.

23. Multiplication of rational numbers
Rational numbers are numbers that can be written as the fraction of two integers.
To multiply rational numbers together, you multiply the tops and bottoms
separately to get your answer
2/5*4/5*2/3
16/75

24. Division of rational numbers
 To divide rational numbers, you turn the division problem into
a multiplication problem by flipping the second rational number. Then you go
ahead and multiply the tops and bottoms together to get your answer. If you
can simplify your problem before multiplication, you can go ahead and do so
to make your problem easier.

25. Inserting rational numbers between two
rational numbers
 The total number of these rational numbers is same as the number of
integers between -40 and 70, i.e., 70 - (-40) - 1 = 70 + 40 - 1 = 110 - 1 = 109.
Similarly, by re-writing -4/7 and 2/7 as -400/700 and 200/700, we
can insert 700 - (-400) - 1 = 700 + 400 - 1 = 1100 - 1 = 1099 rational numbers
between -4/7 and 2/7.

26. Video
Maths Rational Numbers part 1 (Introduction) CBSE Class 8 ...

27. More information
Rational Numbers Class 8 Notes and Important Questions

28. Questions
 What is a rational number
 What are properties rational numbers
 What are the operations of rational numbers

29. Thank you
By Rajesh and
Sai Samarth