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:

f(x) = x2
f(x) = x*x
x2 = x*x
x2 = x+x+x+...+x+x
(dervative of x squared) d/dx(x2)= d/dx(x+x+x+...+x+x) (dervative of x added x times)
2x=1+1+1+...+1 (assume 1 is added x times)
And if you divide both sides by x:

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

x2 = x+x+x+...+x+x (x is added x times)

what does this mean and when is it true? It certainly is true if x is an integer like 5: 52 = 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 x2 = 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,


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