# 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  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, no tens, and 2 ones

## 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...!"

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

### Example:

7 + 0 = 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
0a = 0 (a is positive) 04 = 0
00 = indeterminate 00 = indeterminate
0a = undefined (a is negative) 0-2 = undefined
0! = 1 ("!" is the factorial function) 0! = 1