John Allen Paulos responds to the question, “What insight or idea has thrilled or excited you?” in The Atlantic:
I could mention my first introduction to Godel’s theorem about the essential incompleteness of mathematics; or my first encounter with the Banach-Tarski theorem in topology showing that a sphere the size of a pea can be decomposed into a finite number of pieces and put back together to get a sphere the size of a basketball; or Russell’s paradox about the set of all sets that do not belong to themselves; or any number of counterintuitive results in probability theory. All of these mathematical ideas excited me in high school and college, but I will concentrate instead on the thrill I felt in elementary school when I saw that the power of simple arithmetic was sufficient to vanquish a bully, my fifth-grade teacher. It still evokes the same emotions in me that it did decades ago.
I was about 10 years old and enthralled with baseball. I loved playing the game and aspired to be a major league shortstop. (My father played in college and professionally in the minor leagues.) I also became obsessed with baseball statistics and noted that a relief pitcher for the then Milwaukee Braves had an earned run average (ERA) of 135. (The arithmetic details are less important than the psychology of the story, but as I dimly recall, the pitcher had allowed the opposing team to score five runs and had got only one batter out. Getting one batter out is equivalent to pitching one-third of an inning, one-27th of a complete nine-inning game––and allowing five runs in one-27th of a game translates into an ERA of 5/(1/27) or 135.)
Impressed by this extraordinarily bad ERA, I mentioned it diffidently to my teacher during a class discussion of sports. He looked pained and annoyed and sarcastically asked me to explain the fact to my class. Being quite shy, I did so with a quavering voice, a shaking hand, and a reddened face. (A strikeout in self-confidence.) When I finished, he almost bellowed that I was confused and wrong and that I should sit down.