CBSE Class 8CBSE
Share
Notifications

View all notifications

Properties of Rational Numbers - Closure

Login
Create free account


      Forgot password?

notes

1. Whole numbers - 

Operation  Numbers Remarks
Addition  0 + 5 = 5,
a whole number
4 + 7 = 11 
In general, a + b is a whole number for any two whole numbers a and b.
Whole numbers are closed under addition .
Subtraction 5 – 7 = – 2,
which is not a whole number.
Whole numbers are not closed under subtraction.
Multiplication 0 × 3 = 0,
a whole number
3 × 7 = 21 .
Is it a whole number? .
In general, if a and b are any two whole numbers, their product ab is a whole number.
Whole numbers are closed under multiplication.
Division 5 ÷ 8 = `5/8` which is not a whole number. Whole numbers are not closed under division.

2 . Integers : 

Operation  Numbers  Remarks
Addition – 6 + 5 = – 1,
an integer  Is – 7 + (–5) an integer? Is 8 + 5 an integer?  
In general, a + b is an integer for any two integers a and b.
Integers are closed under addition.
Subtraction 7 – 5 = 2, an integer Is 5 – 7 an integer?
– 6 – 8 = – 14,
an integer
– 6 – (– 8) = 2,
an integer Is 8 – (– 6) an integer?
In general, for any two integers a and b, a – b is again an integer. Check if b – a is also an integer.
Integers are closed under subtraction.
Multiplication  5 × 8 = 40,
an integer Is – 5 × 8 an integer?
– 5 × (– 8) = 40, an integer In general, for any two integers a and b, a × b is also an integer.
Integers are closed under multiplication.
Division 5 ÷ 8 = `5/8` which is not an integer. Integers are not closed under division.

3. Rational numbers : 

A number which can be written in the form `p/q`, where p and q are integers and q ≠ 0 is called a rational number. 
For example, `-2/3,6/7,9/-5` are all rational numbers.

1) How to add two rational numbers. lets check it.
for example - `3/8 + (-5)/7 =( 21 + (-40))/56= -19/56` - it is a rational number.
We say that rational numbers are closed under addition. That is, for any two rational numbers a and b, a + b is also a rational number. 

2) Will the difference of two rational numbers be again a rational number?  We have 
for example - `(-5)/7 - 2/3 =(-5 xx 3 - 2xx 7)/21 = (-29)/21`  -> It is rational number.
We say that rational numbers are closed under subtraction. That is, for any two rational numbers a and b, a – b is also a rational number. 

3) Let us now see the product of two rational numbers.
for example- `-2/3 xx 4/5 = -8/15; 3/7 xx 2/5 = 6/35`
(both the products are rational numbers)
We say that rational numbers are closed under multiplication. That is, for any two rational numbers a and b,  a × b is also a rational number. 

4) We note that `-5/3 ÷2/5 = -25/6 ` -> (it is rational number)
We find that for any rational number a, a ÷ 0 is not defined. So rational numbers are not closed under division.

Shaalaa.com | Closure Property of Rational Number

Shaalaa.com


Next video


Shaalaa.com


Closure Property of Rational Number [00:09:13]
S
Series 1: playing of 1
0%


S
View in app×