When working with algebra reciprocal numbers are common, and when working in algebra inverse numbers are too. Get help with both here.
Every number has an opposite. In fact, every number has two opposites: the additive inverse and the reciprocal—or multiplicative inverse. Don't be intimidated by these technical-sounding names, though. Finding a number's opposites is actually pretty straightforward.
The first type of opposite is the one you might be most familiar with: positive numbers and negative numbers. For example, the opposite of 4 is -4, or negative four. On a number line, 4 and -4 are both the same distance from 0, but they're on opposite sides.
This type of opposite is also called the additive inverse. Inverse is just another word for opposite, and additive refers to the fact that when you add these opposite numbers together, they always equal 0.
-4 + 4 = 0
In this case, -4 + 4 equals 0. So does -20 + 20 and -x + x. In fact, any number you can come up with has an additive inverse. No matter how large or small a number is, adding it and its inverse will equal 0 every time.
If you've never worked with positive and negative numbers, you might want to review our lesson on negative numbers.
The main time you'll use the additive inverse in algebra is when you cancel out numbers in an expression. (If you're not familiar with cancelling out, check out our lesson on simplifying expressions.) When you cancel out a number, you're eliminating it from one side of an equation by performing an inverse action on that number on both sides of the equation. In this expression, we're cancelling out -8 by adding its opposite: 8.
|+ 8||+ 8|
Using the additive inverse works for cancelling out because a number added to its inverse always equals 0.