Resolved Question

If you divide by zero, and zero is nothing, then you are dividing by nothing. Isn't this the same thing as not dividing at all. Then wouldn't you get what you started with. Example 4 divided by 0 = 4?

Let's go back to the fundamentals:

In any mathematical field we have a unity and zero. The unity is defined as the multiplicative identity (multiply it and there will be no change), and the zero is the additive identity (add a number with it, and nothing will change).

It is easy to identify that in our real number system; 1 is the unity, and 0 is the zero (this may not be the case in other fields). Now, division is merely defined as MULTIPLICATION WITH THE NUMBER'S MULTIPLICATIVE INVERSE.

So not dividing at all, is the same as multiplying with the inverse of 'not at all', which is logically equivalent to "not multiplying at all". And by definition that is numerically equivalent to multiplying with the "unity = 1", and has NOTHING to do with the zero element of the real number system, a.k.a 0.

And division of zero is undefinied, since there is, by definition, no multiplicative inverse to the zero element in the real number system. And we can obviously not multiply with something undefined.
(infinity is not part of the real number system. And 1/0 =/= infinity. But lim{x-->0} 1/x = infinity... We could extend the real line, but anything (except zero) multiplied with +/- Infinity = +/- infinity which is not a number anyhow)

In simple wording. Not dividing == not multiplying = multiplying with 1.