Sharrod asked me this, more or less, during the first week of school. Only a few days ago did I have time in class to answer the question. There is a long version of this story and a short one. The long version is not very exciting, but the short one ... well?
I saw the proof for the following claim and that sealed the deal for me. 0.999999999... = 1 Notice, that is not an approximation sign. That is an EQUAL sign. This can't be true, right? I mean, isn't 0.999999999... infinitesimally less than than 1? Well, if something is infinitesimal, it is really small. Almost zero. The problem is that it is not zero. So, clearly the left side is less than the right side (albeit infinitesimally so). Right? Maybe you know something about limits from Calculus or Pre-calculus. Maybe, you say, "Okay, the decimal expansion on the left side is an infinite series and as the number of terms approaches infinity, the infinitesimal amount that is less than 1 approaches zero and the left side equals the right side." Sure. Only I saw a proof that didn't require Real Analysis, just high school algebra. That is what did it for me. Here it is. Proof: Let x = 0.999999999... Then 10x = 9.999999999.... (multiplication by 10, moves the decimal) Subtracting x from both sides, we have: 10x - x = 9.999999999... - 0.999999999... Simplifying, we have: 9x = 9 Division by 9 on both sides results in the following: x = 1 But we already stated that x = 0.999999999... Hence, 0.999999999... = 1. QED Wild.
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AuthorI teach Geometry and Geometry support at Maynard H. Jackson High School. ArchivesCategories |