Quandaries
and Queries 

Ok, our teacher gave us this riddle, and I cannot for the life of me figure it out. He said that there are three problems with the following proof:
I already know that you cannot divide both sides by x because we do not know what it is. But I can't see anything else wrong with this proof. Can you help me? 

Well it is really a very nice problem. I think you should focus on the line
what does this mean and when is it true? It certainly is true if x is an integer like 5: 5^{2} = 5+5+5+5+5; but what happens if x is 1/2 for example? Is (1/2)^{2} = 1/2 + 1/2 + ... + 1/2 when 1/2 is added 1/2 times? Doesn't make much sense does it? We need to be precise in writing x^{2} = x+x+x+...+x+x (x is added x times) and state when it actually holds? Is it just for x an integer? Positive integers? Good luck, 

