# Inverse

Inverse means the **opposite in effect**. The reverse of.

It is a **general idea** in mathematics and has many meanings. Here are a few.

## The Inverse of Adding is Subtracting

Adding moves us one way, subtracting moves us the opposite way.

Example: **20 + 9 = 29** can be reversed by** 29 − 9 = 20** (back to where we started)

And the other way around:

Example: **15 − 3 = 12** can be reversed by** 12 + 3 = 15** (back to where we started)

#### Additive Inverse

The **additive inverse** is what we add to a number to **get zero**.

### Example: The additive inverse of −5 is +5, because −5 + 5 = 0.

Another example: the additive inverse of +7 is −7.

## The Inverse of Multiplying is Dividing

Multiplying can be "undone" by dividing.

Example: **5 × 9 = 45** can be reversed by **45 / 9 = 5**

It works the other way around too, dividing can be undone by multiplying.

Example: **10 / 2 = 5** can be reversed by **5 × 2 = 10**

#### Multiplicative Inverse

The **multiplicative inverse** is what we multiply a number by to get 1.

It is the reciprocal of a number.

### Example: The multiplicative inverse of 5 is \frac{1}{5}, because 5 × \frac{1}{5} = 1

#### But Not With 0

We can't divide by 0, so don't try!

### Example: **5 × 0 = 0** cannot be reversed by **0/0 = ???**

## Inverse of a Function

Doing a function and then its inverse will give us back the original value:

When the function f turns the apple into a banana,

Then the **inverse** function f^{-1} turns the banana back to the apple

Here we have the function **f(x) = 2x+3**, written as a flow diagram:

The **Inverse Function** goes the other way:

So the inverse of: 2x+3 is: (y−3)/2

Read Inverse of a Function to find out more.

## Inverse Sine, Cosine and Tangent

### Example: the sine function

The **sine** function sin takes angle θ and gives the ratio \frac{opposite}{hypotenuse }

The **inverse sine** function sin^{-1} takes the ratio \frac{opposite}{hypotenuse } and gives angle θ

A similar idea applies to cosine, tangent and other trig functions.

Read Inverse Sine, Cosine, Tangent to find out more.

## The Inverse of an Exponent is a Logarithm

Read logarithms to find out more, but basically:

The logarithm tells us what the exponent is!