### Why 0÷0 doesn't have a value?

In this short post we're going to take a look at why 0/0 is really undefined.

### # Overview

It's an old mathematical issue, which has been debated many times over the centuries with mostly the same result: 0/0 does not have a defined value. But now, we're going to take a look at why this is true.

### # Illustration

To illustrate this, let's consider the following sum:

\sum_{k=0}^(n)b^k = b^0 + b^1 + b^2 + ... + b^n

Deriving a closed form to this sum, we get:

\sum_{k=0}^(n)b^k = (b^(n+1) - 1) / (b-1)

For example, when b=3 and n=4, we have:

3^0 + 3^1 + 3^2 + 3^3 + 3^4 = (3^(4+1) - 1) / (3-1)

All good so far. However, if we set b=1, we have a special case:

(1^(n+1) - 1) / (1-1) = 0/0

We know that 1^k=1 for any k>=0, therefore:

\sum_{k=0}^(n)1^k = n+1

but when b=1, our closed-form evaluates to 0/0 for any value of n. From this we can conclude that 0/0 does not have a certain value.

Taking this example a little bit further, we can also show that 0^0 = 1:

\sum_{k=0}^(n)0^k = (0^(n+1) - 1) / (0-1) = (-1) / (-1) = 1

We can agree that 0^k=0 for any k > 0, but because the initial k in the sum is 0, we have to evaluate 0^0 to 1 in order to have the same result as that of the closed-form, which does not require this assumption.

### # Conclusion

In this post we concluded the followings: 0/0 is undefined and 0^0 = 1.