When working with algebra reciprocal numbers are common, and when working in algebra inverse numbers are too. Get help with both here.

The second type of opposite number has to do with **multiplication** and **division**. It's called the **multiplicative inverse**, but it's more commonly called a **reciprocal**.

To understand the reciprocal, you must first understand that every whole number can be written as a **fraction** equal to that number divided by** 1**. For example, 6 can also be written as 6/1.

6 | = | 6 |

1 |

Variables can be written this way too. For instance, x = x/1.

x | = | x |

1 |

The **reciprocal** of a number is this fraction flipped upside down. In other words, the reciprocal has the original fraction's bottom number—or **denominator**—on top and the top 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 can see more reciprocals. Notice that the reciprocal of a number that's already a fraction is just a flipped fraction.

5y | → | 1 |

5y |

18 | → | 1 |

18 |

3 | → | 4 |

4 | 3 |

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

1 | → | 7 |

7 |

1 | → | 2 |

2 |

1 | → | 25 |

25 |

From looking at these tables, you might have already noticed a simpler way to determine the reciprocal of a whole 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, change it to a fraction, then flip the fraction. Not sure how to convert a decimal number to a fraction? Check out our lesson on converting percentages, decimals, and fractions.

If you've ever **multiplied **and **divided fractions**, the reciprocal might seem familiar to you. (If not, you can always check out our lesson on multiplying and dividing fractions.) When you multiply two fractions, you multiply straight across. The numerators get multiplied, and the denominators get multiplied.

4 | ⋅ | 2 | = | 8 |

5 | 3 | 15 |

However, when you **divide **by a fraction you flip the fraction over so the numerator is on the bottom and the denominator is on top. In other words, you use the **reciprocal**. You use the **opposite** number because multiplication and division are also opposites.

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

5 | 3 | 5 | 2 | 10 |