### 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.

You are watching: The reciprocal of 1/6

### 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**add a an unfavorable sign (-) come the left the the number: 5 → -5.

*x*:x | → | -x |

3y | → | -3y |

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

*-x*:-y | → | y |

-6x | → | 6x |

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.

6 | = | 6 |

1 |

Variables can be created this way 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 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.

5y | → | 1 |

5y |

18 | → | 1 |

18 |

3 | → | 4 |

4 | 3 |

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

1 | → | 7 |

7 |

1 | → | 2 |

2 |

1 | → | 25 |

25 |

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 reciprocalsIf 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.

4 | ⋅ | 2 | = | 8 |

5 | 3 | 15 |

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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4 | ÷ | 2 | = | 4 | ⋅ | 3 | = | 12 | ||||

5 | 3 | 5 | 2 | 10 |

### Practice!

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.