At the beginning of the year, I always like to teach a lesson that will hopefully excite and inspire my students and also impress them with how much their new math teacher knows. One problem I use to do this is the old .999 =1 problem. I love this problem, because it serves as a perfect example of how mathematics is more complex and philosophical than students think. If you aren’t familiar with the proof, or you want a refresher, check it out here:
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I start by writing .999 and 1 on the board, and ask the class which symbol belongs in between them, <, >, or =. I let the class discuss this for 5 minutes or so, then I make them vote. After voting, I like to have a discussion in the class about why people chose what they did. If anyone said .999 is greater than 1, I ask them to justify their reasoning. Usually it becomes clear quickly that this is not a realistic answer choice. The real fun comes when we debate less than or equal. Obviously less than is the most common choice, but when pressed to answer why, students usually have difficulty justifying their choice. This process of thinking about why they believe something and having to justify it in class is the most valuable part of the lesson, don’t miss it. The students who said they are equal usually rest on some explanation based on rounding.
Finally, when I have tortured them enough and everyone is dying to hear the answer, I show them several different proofs for the problem. You may want to vary your approach based on the level of your students. A simple proof is to ask students what ⅓ x3 is. If they multiply by hand, they will get 1, if they divide 13, then multiply by 3, they will get .999.
Another simple demonstration is to investigate the pattern of numbers divided by 9. Start with 1/9, and you get .1111. 2/9 is .2222, now when you get to 8/9 you get .8888, so shouldn’t 9/9 be .9999? Of course it is actually 1, however .999 fits this pattern, and is actually also correct.
Finally, you may want to include a true algebraic proof. I have included a basic algebraic proof and a more advanced one based on infinite geometric series below.
I have also had the pleasure of teaching classes of exceptionally bright competition math students. For students at a high level, this problem may be too easy for them. There will likely be several students who already know the answer. For them, consider breaking the class into groups and challenging them to come up with as many unique proofs as they can for this problem.