### Opposite numbers

Every number has actually 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. Recognize a number's opposites is in reality pretty straightforward.

You are watching: Find the reciprocal or multiplicative inverse

The very first type of opposite is the one you might be most familiar with: positive numbers and also negative numbers. For example, the opposite of 4 is -4, or negative four. Top top a number line, 4 and also -4 space both the same distance indigenous 0, however they're on the opposite sides. This form of the contrary is likewise called the additive inverse. Inverse is just an additional word for opposite, and also additive refers to the truth that once you add this opposite numbers together, they constantly equal 0.

-4 + 4 = 0

In this case, -4 + 4 equals 0. Therefore does -20 + 20 and -x + x. In fact, any type of number you deserve to come increase with has an additive inverse. No issue how big or tiny a number is, including it and its inverse will equal 0 every time.

If you've never functioned with confident and negative numbers, you might want to evaluation our lesson on negative numbers.

To discover the additive inverse:For positive numbers or variables, choose 5 or x: include a negative sign (-) to the left of the number: 5 → -5.
 x → -x 3y → -3y
For negative numbers or variables, like -5 or -x: Remove the an adverse sign: -10 → 10.
 -y → y -6x → 6x

The key time you'll use the additive train station in algebra is as soon as you cancel out number in an expression. (If you're not acquainted with cancelling out, inspect out our lesson on simple expressions.) once you cancel the end a number, you're eliminating it from one next of an equation by performing one inverse action on that number top top both sides of the equation. In this expression, we're cancelling out -8 by adding its opposite: 8.

 x - 8 = 12 + 8 + 8

Using the additive inverse works for cancelling out since a number added to its station always equals 0.

### Reciprocals and also the multiplicative inverse

The second type of opposite number has to do through multiplication and also division. It's dubbed the multiplicative inverse, yet it's much more commonly called a reciprocal.

To know the reciprocal, girlfriend must an initial understand the every entirety number can be created as a fraction equal to that number split by 1. Because that example, 6 can also be composed as 6/1.

 6 = 6 1

Variables can be composed this method too. For instance, x = x/1.

 x = x 1

The reciprocal the a number is this portion flipped upside down. In other words, the reciprocal has the original fraction's bottom number—or denominator—on top and also the top number—or numerator—on the bottom. So the reciprocal of 6 is 1/6 due to the fact that 6 = 6/1 and 1/6 is the inverse that 6/1. Below, you can see an ext reciprocals. Notice that the reciprocal of a number that's already a fraction is simply a flipped fraction.

 5y → 1 5y
 18 → 1 18
 3 → 4 4 3

And because reciprocal method opposite, the reciprocal of a reciprocal portion is a whole number.

 1 → 7 7
 1 → 2 2
 1 → 25 25

From spring at this tables, you could have currently noticed a simpler way to determine the reciprocal of a whole number: just write a fraction with 1 ~ above top and the original number top top the bottom.

Decimal numbers have actually reciprocals too! To find the mutual of a decimal number, change it to a fraction, then flip the fraction. Not sure just how to convert a decimal number to a fraction? examine out our lesson on converting percentages, decimals, and fractions.

Using reciprocals

If you've ever multiplied and divided fractions, the reciprocal can seem acquainted to you. (If not, girlfriend can constantly check the end our great on multiplying and also dividing fractions.) when you multiply 2 fractions, you multiply straight across. The numerators gain multiplied, and the denominators obtain multiplied.

 4 ⋅ 2 = 8 5 3 15

However, once you divide by a portion you upper and lower reversal the fraction over therefore the molecule is on the bottom and the denominator is top top top. In other words, you use the reciprocal. You usage the opposite number because multiplication and department are also opposites.

See more: What Does 1 Gram Look Like On A Digital Scale, How To Read A Gram Scale In The Easiest Way

 4 ÷ 2 = 4 ⋅ 3 = 12 5 3 5 2 10

### Practice!

Use the an abilities you just learned to fix these problems. After ~ you've addressed both set of problems, you deserve to scroll down to watch the answers.