Dividing integers is the inverse, or opposite, operation of multiplying integers, and they share similar rules. As stated in the reference, dividing integers is the opposite operation of multiplication.
The Inverse Relationship
Multiplication and division are fundamentally linked. Just as addition and subtraction undo each other, multiplication and division are inverse operations. This means that one operation can "undo" the other.
- Example: If 3 * 4 = 12, then 12 / 4 = 3.
Rules for Signs
The rules for determining the sign (positive or negative) of the result are the same for both multiplication and division:
| Operation | Positive * Positive | Positive * Negative | Negative * Positive | Negative * Negative |
|---|---|---|---|---|
| Multiplication | Positive (+) | Negative (-) | Negative (-) | Positive (+) |
| Division | Positive (+) | Negative (-) | Negative (-) | Positive (+) |
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Rule 1: Dividing a positive integer by a positive integer yields a positive result (e.g., 10 / 2 = 5).
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Rule 2: Dividing a positive integer by a negative integer yields a negative result (e.g., 10 / -2 = -5).
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Rule 3: Dividing a negative integer by a positive integer yields a negative result (e.g., -10 / 2 = -5).
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Rule 4: Dividing a negative integer by a negative integer yields a positive result (e.g., -10 / -2 = 5).
Important Consideration: The Quotient
While the rules for signs are the same, a key difference is that the quotient (the result of division) is not always an integer.
- Example: 7 / 2 = 3.5 (which is not an integer).
This contrasts with multiplying integers, which always results in another integer. The reference notes, "it is not always necessary that the quotient will always be an integer."
Summary
- Division is the opposite of multiplication.
- Sign rules are identical for both operations.
- Division doesn't always result in an integer, unlike multiplication.
