A lot of great & detailed answers above so I will just add my small note:
Zero has no value. It is more-or-less, "not allowed" to divide a number by 0. If it were possible, a lot of math rules would not work. If you try to divide by zero--in reality, you aren't actually trying to divide by anything. Just remember, dividing by zero is not actually dividing. Sure...you can pretend to divide by zero---you're just not really doing it! :)
Zero can be divided any number. But any number can not be divided by zero.
Example and explanation:
0/1 = 0. 0/6 = 0. 0/any number = 0. By these simple examples we conclude that zero can be divided by any number other than zero and the result is zero.
Now we take 5*1 = 5 implies 5/5 = 1.
9*2 = 18. So 18/9 = 2.
9*0 = 0, 5*0 = 0, 13*0 = 0. So any number * 0 = 0......(1)
Therefore ny number*0 = 0, implies that from 0/0 = any number. So 0/0 is an indeterminate number. There is no fixed answer to the form 0/0.
From (1) logically we get:
There is no number * zero = a number than zero.
Therefore, no number = a number other than zero/zero.
Or x/0 = no number, where x is any number.
Thus we can not divide a number by zero.
Now we go by a graph to see what happens to a continuous graph of 1/x as x approaches zero from positive side and negative side.
If you consider f(x) = 1/x as continuous function, then look at the values of 1/x as we go towards zero from positive side. It goes on increasing positively without any bound.
Now consider f(x) = 1/x .The value of 1/x is negative when we x< 0. We notice that when x approaches zero from negative side , we see that the value of x becomes lager and larger negatively without any bound.
Plot the graph and see whether we can decide anything any particular for f(x) = 1/x when x= 0. 1/x remains undecided. The graph has the greatest jump or discontinuity at x = 0. The same principle holds for for the value of any number/0 is undefined in mathematics.
There is nothing left to divide