Opposite numbers

Every number has an opposite. In fact, every number has actually two opposites: the additive inverse and also the reciprocal—or multiplicative inverse. Don't be intimidated by this technical-sounding names, though. Recognize a number's opposites is actually pretty straightforward.

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The additive inverse

The first type of the opposite is the one you could be most acquainted with: positive numbers and negative numbers. For example, the contrary of 4 is -4, or negative four. ~ above a number line, 4 and also -4 room both the exact same distance indigenous 0, yet they're on the contrary sides.

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This form of opposite is likewise called the additive inverse. Inverse is just an additional word because that opposite, and additive describes the fact that once you add this opposite number together, they always equal 0.

-4 + 4 = 0

In this case, -4 + 4 equates to 0. Therefore does -20 + 20 and also -x + x. In fact, any number you can come increase with has actually an additive inverse. No issue how large or small a number is, adding it and also its inverse will equal 0 every time.

If you've never worked with confident and an unfavorable numbers, you could want to evaluation our great on negative numbers.

To find the additive inverse:For hopeful numbers or variables, favor 5 or x: include a an unfavorable sign (-) to the left that the number: 5 → -5.
x-x
3y-3y
For an adverse numbers or variables, favor -5 or -x: Remove the negative sign: -10 → 10.
-yy
-6x6x
Using the additive inverse

The main time you'll usage the additive station in algebra is once you cancel out numbers in one expression. (If you're not familiar with cancelling out, examine out our lesson on simplifying expressions.) as soon as you cancel out a number, you're eliminating it from one next of one equation by performing one inverse action on the number top top both political parties of the equation. In this expression, we're cancelling out -8 by including its opposite: 8.

x- 8=12
+ 8+ 8

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

Reciprocals and also the multiplicative inverse

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

To understand the reciprocal, friend must very first understand the every whole number can be written as a fraction equal to that number divided by 1. For example, 6 can likewise be composed as 6/1.

6=6
1

Variables deserve to be composed this method too. Because that instance, x = x/1.

x=x
1

The reciprocal the a number is this portion flipped upside down. In various other words, the reciprocal has the original fraction's bottom number—or denominator—on top and also the peak number—or numerator—on the bottom. So the reciprocal of 6 is 1/6 because 6 = 6/1 and 1/6 is the inverse of 6/1.

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Below, you deserve to see more reciprocals. Notice that the reciprocal of a number that's currently a fraction is just a flipped fraction.

5y1
5y
181
18
34
43

And because reciprocal way opposite, the reciprocal of a reciprocal fraction is a whole number.

17
7
12
2
125
25

From looking at this tables, you might have already noticed a simpler means to determine the mutual of a totality number: just write a fraction with 1 on top and the original number on the bottom.

Decimal numbers have reciprocals too! To find the reciprocal of a decimal number, readjust it to a fraction, climate flip the fraction. No sure just how to convert a decimal number come a fraction? examine out ours lesson on converting percentages, decimals, and fractions.

Using reciprocals

If you've ever before multiplied and divided fractions, the reciprocal might seem acquainted to you. (If not, friend can constantly check out our great on multiplying and dividing fractions.) when you multiply 2 fractions, you multiply straight across. The numerators acquire multiplied, and the denominators acquire multiplied.

42 =8
5315

However, once you divide by a fraction you flip the portion over so the molecule is top top the bottom and also the denominator is top top top. In various other words, you use the reciprocal. You usage the opposite number because multiplication and department are additionally opposites.

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4÷2=43=12
535210

Practice!

Use the skills you just learned to deal with these problems. After you've resolved both set of problems, you have the right to scroll down to check out the answers.