![]() Jordan tells you your guess is high, so you know the answer is somewhere on the number line between 0 and 1 2. You try starting exactly in the middle and guess 1 2. You have to guess a number somewhere along the number line from 0 to 1. What if fractions are allowed? Suppose Jordan picks a number between 0 and 1, for example 3 22. However, during that class, you realize that you have been assuming Jordan will always pick an integer. You march off to your second class victoriously, confident that you will be prepared for Jordan’s next challenge. But then you realize: there is nothing special about −100 and 100! If you start with a number between −10, you know you will eventually guess the correct number even if it takes a few more guesses. If Jordan’s number is −32, and you have already figured out that −33 is too low and −31 is too high, then you know the answer is −32. You guess, and by going higher and lower you get closer and closer to the target. ![]() You decide to take the bait, and you quickly discover that this does not change the game much. ![]() ![]() Feeling pleased that you are getting closer, you ask, “How about 75?” “You got it!” Jordan replies, and you march triumphantly off to your first class of the day.īut after class, you again run into Jordan, who has apparently been thinking about ways to stump you: why stick to positive numbers? What if you also allow negative numbers? “Now I am thinking of a number between negative 100 and 100,” Jordan says gleefully. On Monday morning, your friend Jordan walks up to you and says, “I’m thinking of a number between 1 and 100.” Being a good sport, you play along and guess 43. What makes fractions so special? We explore how we can recognize the decimal representation of fractions and how fractions can be used to approximate any real number as closely as we wish. Centuries later, while we regularly use numbers that cannot be written as fractions, those numbers that can be written as fractions remain powerful tools. Legend has it that the first person in ancient Greece who discovered that there are numbers that cannot be written as fractions was thrown overboard from a ship. ![]()
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