### 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.

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**include a an unfavorable sign (-) to the left that the number: 5 → -5.

*x*:x | → | -x |

3y | → | -3y |

**For an adverse numbers or variables, favor -5 or**Remove the negative sign: -10 → 10.

*-x*:-y | → | y |

-6x | → | 6x |

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.

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

5y | → | 1 |

5y |

18 | → | 1 |

18 |

3 | → | 4 |

4 | 3 |

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

1 | → | 7 |

7 |

1 | → | 2 |

2 |

1 | → | 25 |

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 reciprocalsIf 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.

4 | ⋅ | 2 | = | 8 |

5 | 3 | 15 |

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 | = | 4 | ⋅ | 3 | = | 12 | ||||

5 | 3 | 5 | 2 | 10 |

### 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.