# Reciprocal In Algebra

Turn it upside down!

## Reciprocal of a Number

To get the reciprocal of a **number**, we divide 1 by the number:

### Examples:

Number | Reciprocal | As a Decimal |
---|---|---|

2 | ^{1}/_{2} |
= 0.5 |

8 | ^{1}/_{8} |
= 0.125 |

1,000 | ^{1}/_{1,000} |
= 0.001 |

## Reciprocal of a Variable

Likewise, the reciprocal of a variable "x" is "1/x".

And the reciprocal of something more complicated like "x/y" is "y/x".

In other words **turn it upside down**.

### Example: What is the Reciprocal of x/(x−1) ?

Answer: take \frac{x}{(x−1)} and flip it upside down: \frac{(x−1)}{x}

### More Examples:

Expression | Reciprocal |
---|---|

2x | \frac{1}{2x} |

\frac{3}{x} | \frac{x}{3} |

\frac{(2x−3)}{(x+5)} | \frac{(x+5)}{(2x−3)} |

## Flipping a Flip

When we take the **reciprocal of a reciprocal** we end up back where we started!

### Example:

The reciprocal of \frac{ax}{y} is \frac{y}{ax}

The reciprocal of \frac{y}{ax} is \frac{ax}{y} (back again)

### Example: What is:

\frac{1}{^{1}/_{w}}

Answer: w

Why? because the reciprocal of ^{1}/_{w} is ^{w}/_{1}
which is just **w**

Or with numbers: What is \frac{1}{½} ? Divide 1 into halves, and the answer is **2**

## Notation

The reciprocal of "x" can be shown as:

\frac{1}{x} |
or |
x^{-1} (see exponents) |