The rules for subtracting integers involve transforming the subtraction problem into an addition problem using the additive inverse. Here's a breakdown:
Steps for Subtracting Integers
The core principle is that subtracting an integer is the same as adding its opposite.
- Keep the first integer just as it is. (Reference)
- Change subtraction to addition. Since subtraction is addition of the opposite, this step is crucial. (Reference)
- Change the sign of the second integer to the opposite sign. This means if it's positive, make it negative; if it's negative, make it positive. (Reference) This gives you the additive inverse.
- Add the two integers. Follow the rules for adding integers (same signs add and keep the sign; different signs subtract and take the sign of the larger absolute value). (Reference)
Here's a table summarizing the process:
| Step | Description | Example: 5 - 3 | Example: 5 - (-3) |
|---|---|---|---|
| 1 | Keep the first integer. | 5 | 5 |
| 2 | Change subtraction to addition. | 5 + | 5 + |
| 3 | Change the sign of the second integer. | 5 + (-3) | 5 + (+3) |
| 4 | Add the integers. | 5 + (-3) = 2 | 5 + 3 = 8 |
Examples and Explanations
Let's illustrate with more examples:
-
Example 1: -7 - 2
- Keep -7.
- Change subtraction to addition: -7 +
- Change the sign of 2 to -2: -7 + (-2)
- Add: -7 + (-2) = -9
-
Example 2: -4 - (-6)
- Keep -4.
- Change subtraction to addition: -4 +
- Change the sign of -6 to +6: -4 + (+6)
- Add: -4 + 6 = 2
-
Example 3: 3 - 8
- Keep 3.
- Change subtraction to addition: 3 +
- Change the sign of 8 to -8: 3 + (-8)
- Add: 3 + (-8) = -5
Key Insight
The reason this works is rooted in the definition of subtraction. Subtracting a number is the same as adding its additive inverse. The additive inverse of a number a is the number that, when added to a, results in zero. For example, the additive inverse of 5 is -5 because 5 + (-5) = 0.
By converting subtraction problems into addition problems using additive inverses, we can apply consistent rules for addition, simplifying the process and reducing errors.
