### 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 additive 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**include a negative sign (-) to the left of the number: 5 → -5.

*x*:x | → | -x |

3y | → | -3y |

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

*-x*:-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 reciprocalsIf 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.