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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 |
