The opposite of 3 is -3.
Understanding Opposite Numbers
In mathematics, the "opposite" of a number refers to its additive inverse. This term is often preferred as it precisely describes the relationship between two numbers that, when added together, result in a sum of zero.
What is an Additive Inverse?
The additive inverse of any given number is the number that, when added to the original number, produces zero. For any real number, say n, its additive inverse is -n. This relationship can be expressed as:
- n + (-n) = 0
For example:
- For the number 3, its additive inverse is -3, because 3 + (-3) = 0.
- For the number -5, its additive inverse is 5, because -5 + 5 = 0.
This means that if a number A is the opposite of number B, then number B is also the opposite of number A.
Visualizing Opposites on a Number Line
Opposite numbers are characterized by their position on a number line. They are always:
- Equidistant from zero.
- Located on opposite sides of zero.
For instance, 3 is three units to the right of zero, while its opposite, -3, is three units to the left of zero. They are essentially mirror images of each other with respect to the origin (zero).
Key Properties of Opposite Numbers
Understanding these properties helps in various mathematical operations:
- Sum to Zero: The most fundamental property is that any number added to its opposite will always equal zero.
- Example: 15 + (-15) = 0
- Equal Absolute Value: A number and its opposite always have the same absolute value. The absolute value signifies the distance of a number from zero, regardless of direction.
- Example: The absolute value of 8 is 8 (|8| = 8), and the absolute value of -8 is also 8 (|-8| = 8).
Examples of Numbers and Their Additive Inverses
The following table illustrates more examples of numbers and their opposites:
| Number | Opposite (Additive Inverse) | Resulting Sum |
|---|---|---|
| 3 | -3 | 0 |
| -7 | 7 | 0 |
| 0.25 | -0.25 | 0 |
| -2/3 | 2/3 | 0 |
| 0 | 0 | 0 |
Grasping the concept of additive inverses is crucial for fundamental operations in arithmetic and algebra, including solving equations and understanding number system properties.
