Yes, integers are closed under multiplication, meaning the product of any two integers is always another integer. Here are some examples.
Understanding Closure Under Multiplication
The closure property, in general, states that performing an operation on elements within a set will result in an element that is also within that same set. In the case of integers and multiplication, this means that when you multiply two integers, the result will always be an integer.
Examples Demonstrating Closure
Here are several examples demonstrating that integers are closed under multiplication:
| Integer 1 (a) | Integer 2 (b) | Product (a * b) | Result is an Integer? |
|---|---|---|---|
| 2 | 3 | 6 | Yes |
| -5 | 4 | -20 | Yes |
| 0 | 10 | 0 | Yes |
| -7 | -1 | 7 | Yes |
| 1 | 1 | 1 | Yes |
| -1 | -1 | 1 | Yes |
| 1 | -1 | -1 | Yes |
| 12 | 5 | 60 | Yes |
| -3 | -8 | 24 | Yes |
| 15 | 0 | 0 | Yes |
As you can see from the table above, no matter what two integers we choose, their product is always another integer.
Why This Matters
The closure property is fundamental in mathematics and helps us understand the behavior of different number systems under various operations. The fact that integers are closed under multiplication is a crucial property used in algebra and other mathematical fields. If this were not true, many mathematical operations and proofs would become significantly more complex.
In conclusion, the examples clearly demonstrate that integers are indeed closed under multiplication. The result of multiplying any two integers will always be another integer.
