Suppose we start the growth process at eight o’clock in the morning and have carefully calculated that there are just sufficient nutrients to ensure that the container is completely filled with bacteria by exactly twelve noon. Here’s the question: at what time between eight a.m. and noon is the container half full? Those who get the wrong answer typically suggest a time over halfway between 8:00 and 12:00, such as 10:30 or 11:15. The correct answer, which comes as a surprise to some, is 11:59, just one minute before noon. I’m sure you’ve got it: since the population doubles every minute, it
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