The closure property, when applied to division, reveals that the result of dividing two numbers within a specific set is not always a number that belongs to that same set. Specifically, for whole numbers, division is not closed.
Understanding Closure in Mathematics
In mathematics, a set of numbers is considered closed under a particular operation (like addition, subtraction, multiplication, or division) if performing that operation on any two numbers within the set always produces a result that is also within the same set.
Closure Property and Whole Numbers Under Division
According to the closure property, the result of the division of two whole numbers is not always a whole number. This means that whole numbers are not closed under division. The property states that a ÷ b is not always a whole number when 'a' and 'b' are whole numbers (with b ≠ 0).
Here are examples illustrating this point:
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Example 1: Result is a Whole Number
When you divide 14 by 7, both 14 and 7 are whole numbers. The result, 2, is also a whole number.
14 ÷ 7 = 2 (Whole number) -
Example 2: Result is Not a Whole Number
When you divide 7 by 14, both 7 and 14 are whole numbers. However, the result is a fraction, which is not a whole number.
7 ÷ 14 = ½ (Not a whole number)
This second example clearly demonstrates why the set of whole numbers is not closed under division. While some divisions of whole numbers yield whole numbers, others do not.
Illustrating with a Table
We can summarize the examples:
| Operation | Numbers Used | Result | Is Result a Whole Number? |
|---|---|---|---|
| 14 ÷ 7 | Whole | 2 | Yes |
| 7 ÷ 14 | Whole | ½ | No |
This lack of closure for division within the set of whole numbers necessitates the expansion to larger sets of numbers, such as rational numbers (which include fractions), to ensure that division (except by zero) always yields a result within the set.
