Check sibling questions

Let’s check closure for rational numbers

Operation Commutative Closed or not
Addition 2/5 + 4/5 = 6/5
6/5 is a rational number

Also,
(−3)/5 + 0 = (−3)/5
(−3)/5 is a rational number

So, rational numbers are closed under addition
So, if we add any two numbers,
we get a rational number

So, it is closed
Subtraction 2/5 – 4/5 = (2 − 4)/5 = (−2)/5
(−2)/5 is a rational number

Also,
(−3)/5 – 0 = (−3)/5
(−3)/5 is a rational number

So, rational numbers are closed under subtraction
So, if we subtract any two numbers,
we get a rational number

So, it is closed
Multiplication 2/5 × 4/5 = (2 × 4)/(5 × 5) = 8/25
8/25 is a rational number

Also,
(−3)/5 × 0 = 0
0 is a rational number

So, rational numbers are closed under multiplication
So, if we multiply any two numbers,
we get a rational number

So, it is closed
Division 2/5 ÷ 4/5 = 2/5  × 5/4 = 2/4 = 1/2
1/2 is a rational number

Also,
(−3)/5 ÷  0 = (−3)/5 × 1/0
 1/0 is not defined
∴ (−3)/5 × 1/0 is also not defined
So, it is not a rational number

So, rational numbers are not closed under division
So, if we divide any two numbers,
we do not get a rational number

So, it is not closed

To summarize

Numbers

Closed under

Addition

Subtraction

Multiplication

Division

Natural numbers

Yes

No

Yes

No

Whole numbers

Yes

No

Yes

No

Integers

Yes

Yes

Yes

No

Rational Numbers

Yes

Yes

Yes

No

 

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.