Opposite numbers

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

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The additive inverse

The very first type of opposite is the one you can be most acquainted with: positive numbers and negative numbers. Because that example, the opposite of 4 is -4, or negative four. ~ above a number line, 4 and also -4 are both the same distance native 0, but they're on the contrary sides.


This type of opposite is additionally called the additive inverse. Inverse is just an additional word for opposite, and additive describes the fact that once you add this opposite numbers together, they constantly equal 0.

-4 + 4 = 0

In this case, -4 + 4 equates to 0. So does -20 + 20 and -x + x. In fact, any number you deserve to come up with has an additive inverse. No issue how large or tiny a number is, including it and also its inverse will certainly equal 0 every time.

If you've never worked with optimistic and negative numbers, you can want to evaluation our lesson on an unfavorable numbers.

To uncover the additive inverse:For confident numbers or variables, like 5 or x: add a an unfavorable sign (-) come the left the the number: 5 → -5.
For an adverse numbers or variables, favor -5 or -x: Remove the an adverse sign: -10 → 10.
Using the additive inverse

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

x- 8=12
+ 8+ 8

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

Reciprocals and also the multiplicative inverse

The second form of opposite number needs to do v multiplication and division. It's called the multiplicative inverse, yet it's more commonly referred to as a reciprocal.

To understand the reciprocal, you must first understand that every totality number can be created as a fraction equal to that number divided by 1. For example, 6 can additionally be composed as 6/1.


Variables can be created this way too. Because that instance, x = x/1.


The reciprocal the a number is this portion flipped upside down. In various other words, the reciprocal has actually the initial fraction's bottom number—or denominator—on top and also the optimal number—or numerator—on the bottom. So the mutual of 6 is 1/6 due to the fact that 6 = 6/1 and 1/6 is the inverse the 6/1.


Below, you deserve to see an ext reciprocals. Notice that the mutual of a number that's currently a portion is just a flipped fraction.


And due to the fact that reciprocal means opposite, the reciprocal of a reciprocal portion is a whole number.


From looking at this tables, you can have currently noticed a simpler way to identify the reciprocal of a totality number: simply write a portion with 1 top top top and also the initial number ~ above the bottom.

Decimal numbers have actually reciprocals too! To discover the reciprocal of a decimal number, change it come a fraction, then flip the fraction. No sure how to convert a decimal number come a fraction? examine out ours lesson on convert percentages, decimals, and fractions.

Using reciprocals

If you've ever multiplied and divided fractions, the reciprocal could seem acquainted to you. (If not, you can constantly check out our great on multiplying and dividing fractions.) as soon as you multiply 2 fractions, friend multiply directly across. The numerators gain multiplied, and the denominators obtain multiplied.

42 =8

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

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Use the skills you simply learned to solve these problems. After you've addressed both to adjust of problems, you have the right to scroll under to see the answers.