# Zero

Zero shows that there is no amount.

Example: **6 − 6 = 0** (the difference between six and six is zero)

It is also used as a "placeholder" so we can write a numeral properly.

Example: **502** (five hundred and two) could be mistaken for **52** (fifty two) without the zero in the tens place.

## Zero is a very special number ...

It is halfway between −1 and +1 on the Number Line:

Zero is neither negative nor positive. But it is an even number.

## The Idea

The idea of **zero**, though natural to us now, was not natural to early humans ... if there is nothing to count, how can we count it?

**Example: you can count dogs, but you can't count an empty space:**

Two Dogs | Zero Dogs? Zero Cats? |
---|

An empty patch of grass is just an empty patch of grass!

## Zero as a Placeholder

But about 3,000 years ago people needed to tell the difference between numbers like **4** and **40.** Without the zero they look the same!

So zero is now used as a "placeholder": it shows "there is no number at this place", like this:

502 | This means 5 hundreds, |

## The Value of Zero

Then people started thinking of zero as an actual **number**.

### Example:

*"I had 3 oranges, then I ate the 3 oranges, now I have zero oranges...!"*

## Additive Identity

And zero has a special property: when we add it to a number we get that number back, unchanged

### Example:

7 + 0 = 7

Adding 0 to 7 gives the answer 7

Also 0 + 7 = 7

This makes it the **Additive Identity**, which is just a special way of saying "add 0 and we get the **identical** (same) number we started with".

## Special Properties

Here are some of zero's properties:

Property | Example |
---|---|

a + 0 = a | 4 + 0 = 4 |

a − 0 = a | 4 − 0 = 4 |

a × 0 = 0 | 6 × 0 = 0 |

0 / a = 0 | 0/3 = 0 |

a / 0 = undefined (dividing by zero is undefined) | 7/0 = undefined |

0^{a} = 0 (a is positive) |
0^{4} = 0 |

0^{0} = indeterminate |
0^{0} = indeterminate |

0^{a} = undefined (a is negative) |
0^{-2} = undefined |

0! = 1 ("!" is the factorial function) | 0! = 1 |