### Thought for the Day: Nine Lines of Gemara, Three Powerful Lessons in Thinking, Learning, and Teaching

There is a standard line of reasoning in logic known as

*reductio ad absurdum*. It is usually employed when a direct proof would be difficult/impossible and it works as follows: You want to prove that some proposition is true; say, for example, that the there is no smallest, positive, non-zero, rational number. The first step is to consider the opposite; in this case, suppose that there*is*, in fact, a smallest, positive, non-zero, rational number; call it*. Next, we formulate some logical implications of our supposition; in our case, divide that smallest, positive, non-zero, rational number by two; call that***x***. We have posited that***y****is the smallest, positive, non-zero, rational number, which implies that any other positive, non-zero, rational number -- including***x***y**-- is bigger than*x*. So by assuming that there is, in fact, a smallest, positive, non-zero, rational number, we have shown by logical inference that would mean there is a number that is both half the size*and*larger than th…