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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