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, 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...!"
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 |
| 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 |