No, integers are not closed under division.
Understanding Closure in Mathematics
In mathematics, a set is said to be "closed" under an operation if performing that operation on elements of the set always results in another element within the same set.
Why Integers Fail the Closure Test for Division
The reference states that when one integer is divided by another integer, the result is not always an integer.
- Example: 4 and 9 are both integers. However, 4 ÷ 9 = 4/9. 4/9 is a fraction and not an integer.
- Another Example: 10 and 3 are integers. Dividing 10 by 3 (10 ÷ 3) results in 3.333..., which is not an integer.
Because division can result in non-integer values (rational numbers, decimals, etc.) when starting with integers, integers are not closed under division.
Closure Property
| Operation | Closed under Integers? |
|---|---|
| Addition | Yes |
| Subtraction | Yes |
| Multiplication | Yes |
| Division | No |
