THE DIGIT ZERO

0 may refer to:
-a digit 0
-the integer between 1 and -1
-null
-0 (years) in some calendar  

Some of the properties of 0~
1. it is neither positive or negative

2. it is an even number: by definition, any number can divided by 2 is an even number, since 0 /2 = 0, then 0 is an even number

3. it is a natural number

4. additional property : any number added to 0 equal to itself

example:
1 + 0 = 1
2 + 3 + 0 = 2 + 3 = 5
a + b + 0 = a + b

5. additive inverse property : if the sum of two numbers is zero, then they are additive inverse or opposite of each other

example:
since -3 + 3 = 0, 3 is the additive inverse of -3

6. multiplication property : any number times zero is equal to zero

example:
4 x 0 = 0
53 x 0 = 0
p x 0 = 0
rq x 0 = 0

7. any number divided by 0 is undefined, it is neither zero, infinity or negative infinity

8. 0 divided by any number is equal to 0

9. 0 to the power of any number is equal to 0, but 0 to the power of 0 is equal to 1

example:

• 1⋅03=1⋅0⋅0⋅0=0
• 1⋅02=1⋅0⋅0=0
• 1⋅01=1⋅0=0
• 1⋅00=1=1

10. 0 factorial is equal to 1: 0! = 1

11.0 divided by 0 = indeterminate

if we do multiplication as follows:
 4 x 9 = 36,
the we know that
36 / 9 = 4 or 36 / 4 = 9

if we do the same with a number 0:
7 x 0 = 0
so what we gain from 0 / 7 or 0 / 0?


on the other hands, the associative rule: (a+b)+c=a+(b+c) will not always work, for example:

1+(infinity - infinity) = 1+0=1
(1+infinity) - infinity = infinity - infinity = 0