Let's settle this shit once and for all

[Magilla]

So what they're saying is you go to a supermarket, right? And you buy a cake and you decide you don't want to cut up the cake, right?

But that's exactly the trouble.  If you don't cut the cake, you aren't performing a division.   In order for your analogy to work, you would have to divide the cake into zero pieces.  But you aren't dividing anything.  1/0 is undefined, 1 not divided is 1.
Boru

To extend . . .
For the sake of argument, let's assume that when cutting, you cut a cake into equal portions, (not possible in practice).

What do we mean when we divide a cake into two?
You halve it right?
Thus, you have two pieces of cake and two people get an equal share. If they trot off with their portions, there is no cake left behind on the cutting platter.
Two people get 1/2 of the cake each, and nothing else remains for the sharing.

• Thus 1/2 means two parts of equal size, and nothing left over, (no crumbs for the poor).
  

If you cut the cake into three, then all of the above using threes, and . . .
Three people get 1/3 of the cake each, and nothing else remains for the sharing.

• Thus 1/3 means three parts of equal size, and nothing left over, (no crumbs for the poor).
  

It is therefore logical that the following must pertain:-

• 1/0 means zero parts of equal size, and nothing left over, (no crumbs for the poor).
  

So how many people get cake, and what is left behind on the platter, for the poor, (it should be nothing, right) ???

You have a cake - 1

You do NOT cut this cake into pieces - 1/0

What are you left with after NOT cutting the cake? The goddamn cake, right?  

Then why the hell do
Math lords call this - 0/1= Undefined

The above analysis changes the rules, and is invalid.

What is the result of 1/(0.5) (ie. 1 ÷ 1/2)? The answer is two. But we've gone into the twilight zone. In real terms with a real cake, what is going on???
We cut a cake a half a time, get two cakes, and none left over !!!

Try it for 1/(0.000001) = 1000000 We cut the cake one one-hundred thousandth of a time, and get pieces of size a million !!!

As the divisor gets smaller, (closer to zero), in real terms, the result gets more ludicrous. The series of steps as we approach a smaller and smaller divisor, is that we divide the cake by something akin to a lesser proportion, and portions achieved approach infinity in size.

Thus at the limit, as the divisor approaches zero, getting smaller and smaller, 1/0 means we divide the cake proportionally so tiny it vanishes, yet each portion produced is infinite in size.
It simply doesn't compute. There isn't a piece of graph paper possible to display the results in full . . . as the divisor plotted on the x-axis gets smaller, so the vertical portion size vertical axis vanishes into the nether regions of the universe and beyond.  In the limit, as the divisor reaches zero, the outcome is labelled UNDEFINED.

1/0 is undefined - it has no sensible result.