Yes, an integer multiplied by an integer always results in an integer.
Understanding the Closure Property
The concept of integers and their behavior under multiplication is neatly explained by the closure property. According to this property, when you perform certain operations on integers, the result remains within the set of integers. The specific closure property that concerns us here states:
"When you add, subtract, or multiply two integers together, the result will always be an integer."
This means that no matter what two integers you choose, their product will always be an integer. Let's look at some examples.
Multiplication Examples
- Positive integers: 5 * 7 = 35 (35 is an integer)
- Negative integers: -3 * -4 = 12 (12 is an integer)
- Mixed signs: -2 * 6 = -12 (-12 is an integer)
- Zero: 0 * 9 = 0 (0 is an integer)
Key Takeaway
The closure property for integers, with regards to multiplication, ensures that the product of any two integers will always be a member of the set of integers. This is a fundamental principle in mathematics and provides a basis for numerous calculations and proofs. It also means that we don't have to worry about ending up with a non-integer when multiplying integers.
| Operation | Closure Property? | Result Example |
|---|---|---|
| Addition | Yes | 5 + 3 = 8 |
| Subtraction | Yes | 7 - 2 = 5 |
| Multiplication | Yes | 4 * -6 = -24 |
| Division | No | 5 / 2 = 2.5 (not an integer) |
