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X/0 will be possible only if X=0
Division is a continous addition/substraction based on zero

Zero divides Zero

Independent thought by Kapil Krishnan

  • Statement: Zero divided by zero is the set of all numbers including real and complex.


Proof
When you divide X by Y, we need to do either add or subtract X, Y times upto zero. The result is the total count of your addition or subtraction. The count may be a real number or a complex number.

Example 1, we know that 6/3 is 2.
Operation selected = Subtraction. So the result will be positive number
Count=1, 6 - 3 = 3. Here, the result is not zero and it is required to try again now
Count=2, 3 - 3 = 0. Now the result is zero.
i.e., we can subtract the number 3 two times from 6.

Example 2, we know that -6/3= -2
Operation selected = Addition. so the result will be negative number
Count=1, -6 + 3 = -3. Here, the result is not zero and it is required to try again now
Count=2, -3 + 3 = 0. Now the result is zero.
i.e., we can add the number 3, negative two time from -6.

Example 3, We know that 0/X=0, where X is any number
Here, the dividend is already zero, so it is not required to perform any addition/subtraction X times to make it zero. In this case, count=0 only. i.e., result is also zero.

Note: In all the above examples, you can make the dividend as zero only once if you go for either addition/subtraction of the divisor.

Now, let us consider the famous doubt of great Mathematician Sreenivas Ramanujan when he was a school student.

Background: If you divide X banana to X people then everyone will get one banana.
Ramanujan' question for the above statement: If 0 banana divided to 0 people then everyone will get one banana each?

I would say that Ramanujan's statement is perfectly right.

Example 4, 0/0 = {..,0,..} where dividend is already zero, so one of the result is zero.
Now operation selected = Subtraction. So the result will be a positive number
Count=1, 0 - 0 = 0. So we got one more result, that is +1
Count=2, 0 - 0 = 0. So got one more result, that is +2

You can see that all the real positive numbers will be the result of 0/0. If we performed addition operation then you can find that all the negative numbers will be one of the result and same will be applicable to complex numbers also.

If X/Y=Z then Y x Z = X. For example 0.23 is one of the solution of 0/0. So 0/0=0.23 and also 0 x 0.23 = 0. This will be true for all numbers.

Now we can consider another example X/0. What do you think about it?
X - 0 = X, Count=1
X - 0 = X, Count=2 ...
Also X + 0 = X, Count=1
X + 0 = X, Count=2 ...

You can't make the dividend as zero and of course infinity is not the right answer. That is division by zero is not possible except for zero. Basically division is an operation based on zero. That is how to make the dividend as zero by adding/subtracting divisor times. You can do 'n' number of addition or subtraction to get the dividend as zero, but remember that at a time you can choose either subtraction or addition only.