Let's settle this shit once and for all

[Kernel Sohcahtoa]

You don't have a cake - 0

You cut this virtually non existent cake into two pieces - 0/2

What are you left with? NOTHING

Math lords call this - 0/2=0

Ok, so that's in the clear. Moving on :

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

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? You DON'T cut up the cake. Well guess what? You're screwed cos POOF! You open your bag and your cake is gone. All you're left with is a note that reads "error LOL"

I'm nowhere close to being gifted at math (I enjoy it and that's all), but IMO, in order to understand math, students will ultimately have to put the concepts in their own words and internalize them in a manner that best suits their learning styles.  Now, with that said, I'd like to challenge you to continue with your reasoning in bold.  Lets assume that the statement is that "if one person has a cake and doesn't divide it into any pieces, then he or she has 1/1=1 cake (lets say that this is true).  Now, lets assume to the contrary that this is not true.  In fact, suppose that 1/0 is the correct way to divide a cake, so that one person can have a whole cake to himself or herself.

Now, moving forward from this reasoning along with your reasoning in the op, see if you can generate some kind of contradiction.  For example, what happens when you divide 1 cake by 1/2 or 1 cake by 1/4 or 1 cake by 1/100.  As we divide by numbers that get closer and closer to zero, what happens to the number of cakes that we have?  If we were to divide by zero, then how many cakes would we have?