760-word version:
The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.

 

- Georg Cantor

 

1=0.999… I could not accept the equation as it was. "There should be an infinitesimal difference between the two numbers," I thought. "How can two distinct numbers be the same?" Explanations were: (1) "1/3=0.333… Multiply each number by three. Then, you will get 1=0.999…" But they did not explain why 1/3=0.333…. I even grew doubtful of the existence of the number 1/3, for I never could successfully cut an apple into three exactly identical pieces...

 

501-word version:
1=0.999… I could not accept the equation as it was. "There should be an infinitesimal difference between the two numbers," I thought. "How can two distinct numbers be the same?" Explanations were: (1) "1/3=0.333…. Multiply each number by three. Then, you will get 1=0.999…." But they did not explain why 1/3=0.333…. I even grew doubtful of the existence of the number 1/3, for I never could successfully cut an apple into three exactly identical pieces...

 

487-word version:
1=0.999… I could not accept the equation as it was. "There should be an infinitesimal difference between the two numbers," I thought. "How can two distinct numbers be the same?" Explanations were: (1) "1/3=0.333…. Multiply each number by three. Then, you will get 1=0.999…." But they did not explain why 1/3=0.333…. I even grew doubtful of the existence of the number 1/3, for I never could successfully cut an apple into three exactly identical pieces...

 

348-word version:
1=0.999… I could not accept the equation as it was. "There should be an infinitesimal difference between the two numbers," I thought. Explanation was "x=0.9+0.09+0.009+…. 10x=9+0.9+0.09…. So, 9x=9. x=1." I, however, could not accept this approach, for it did not clearly solve the problem: x=1+10+100+…...