The additive inverse of an integer is the number that, when added to the integer, results in a sum of zero.
In simpler terms, the additive inverse is obtained by changing the sign of the integer. For example, if you have the integer 'n', its additive inverse is '-n', and vice-versa. This relationship is fundamental to understanding operations with integers and solving algebraic equations.
Understanding Additive Inverses
- Definition: The additive inverse of a number is the value that, when added to the original number, results in a sum of zero.
- Property: For any integer a, there exists an integer -a such that a + (-a) = 0.
Examples of Additive Inverses
| Integer (a) | Additive Inverse (-a) | Sum (a + (-a)) |
|---|---|---|
| 5 | -5 | 0 |
| -3 | 3 | 0 |
| 0 | 0 | 0 |
| 12 | -12 | 0 |
| -100 | 100 | 0 |
Importance of Additive Inverses
Additive inverses play a crucial role in various mathematical concepts:
- Solving Equations: Additive inverses are used to isolate variables in algebraic equations. For instance, to solve x + 5 = 0, you add the additive inverse of 5 (which is -5) to both sides of the equation.
- Number Systems: They are essential for defining the properties of integer sets and other number systems.
- Subtraction: Subtraction can be defined as adding the additive inverse. For example, a - b is the same as a + (-b).
In summary, the additive inverse of an integer is the number that, when added to the integer, gives a result of zero, effectively "canceling out" the original number.
