Are you into mathematics? Do you have any cake?

[Kernel Sohcahtoa]

What's so "trololololol" about my question?

According to you guys it's absolutely impossible to divide a cake with no one but perfectly possible to add absolutely nothing to a cake.
You can't just "add" nothing to a cake, that would defeat the purpose of the addition operation in the first place, no addition took place there. So why is it okay to add nothing to a cake but not okay to divide a cake among no one? Seems like a perfectly reasonable question to me.

If I divide my cake with nobody how much cake will each person have? Zero. So 1/0=0?

Hello, sir.  I have noticed that in your posts 1/0 was equated to 1.  In order to understand why this is not correct, it may be useful to understand how the graph of 1/x looks and behaves




Now, notice that once we get to the values in between 0 and 1, the values of Y start shooting up.  For example, 1 divided by x=1/2 yields y=2, 1 divided by x=1/4 yields y=4, 1 divided by x=1/1,000 yields y=1,000, 1 divided by x=1/1,000,000,000 yields y=1,000,000,000 and so on. Hence, the closer x gets to zero (zero being the limit of this function in this case), then this will cause the y value to get closer to infinity.  Hence, 1/0 would result in infinity, which is not defined.

* Also notice that we can observe the same thing in between the values of -1 and 0: as x approaches zero from the left side, then y will approach negative infinity

Now, the operation of 1+0 is valid, because zero is the identity element of addition: this means that if we add 0 to any real number, then we will get that same real number.  For example, 1+0=1;100+0=100; and so on. This may be better illustrated by seeing a graph of 1+x

*in terms of division and multiplication, the identity element is 1 (100/1=100, 2/1=2, 1/1=1 and so on).  Hence, when you are not sharing the cake with your friends, then that implies that you get the whole cake, which is equivalent to dividing by the identity element of 1, or 1/1=1.

Here's a graph of y=1+x, which we can equivalently call y=x+1:



Notice that for any x value in the domain (all of the possible values that can be inputted into x of our function x+1) of real numbers, y will have a value in the range (all of the outputs that our produced from inputting x into our function) of real numbers (in fact, each x will be mapped to exactly one y value). Hence, each x value will produce a y value that is defined.  For example, 1+ x=0 yields y=1.

Hence, adding by zero is valid, because zero is the identity element of addition: it will produce that same value that we are adding zero to.  However, dividing by zero is invalid, because that would result in infinity and infinity is not defined. 

I hope this has been useful in answering your inquiry, sir.  Live long and prosper.

Edit

P.S. Regarding y=1/x and the cake problem, we would need to restrict our domain to the natural numbers or positive integers {1,2,3,4,5,6,7,.......} in order for your cake cutting division problem to make sense.