5/7/2023 0 Comments Infinitesimals 0.999![]() ….the best explanations describe both the essence of the paradox ( that there is a difference between the notation and the thing being notated) as well as its details ( what decimal notation actually means). Uri Zarfaty makes a good remark on what a good explanation should be like: can’t be smaller than 1, or there would be a number of the form 0.99….99 bigger than it, violating #3. Any number smaller than 1 has a number of the form 0.99…99 bigger than it (this follows from the fact that any positive number has a number of the form 0.00.001 smaller than it). 0.999… is bigger than any number of the form 0.99…99.įrom #1 and #2, 0.999… is either 1 or a number smaller than 1.But even without knowing what it means, you can conclude it must be 1 from a few plausible assumptions: Maybe you are not so sure what the expression 0.999… means. and 1 that stubbornly persists in the face of the arguments presented before it.Īs usual, Quora serves up a ton of good answers that I’m reproducing below in full, mostly for future reference.īefore I go on, here’s a highly-upvoted plausibility argument by Michael Hochster: in the first place) isn’t enough there’s a feeling that there should be a non-zero yet non-measurable distance between 0.999…. = 0 (assuming the student doesn’t feel that there’s a “lingering 1” at the end of 0.000…. Sometimes even the assertion that “two numbers are equal if the difference between them is zero” combined with the fact that 1- 0.999…. (It’s probably a good teacher’s password though, as is the next argument.) The standard “proof” (technically correct but also completely unenlightening if you’re not convinced in the first place) uses the representation “1/3 = 0.333….” and the fact that 1 is three times 1/3 students report that it seems more like arithmetic sleight-of-hand, in the manner of the “proof” that 0 = 1. Asserting that it is “true by definition” is completely unilluminating I personally feel that it shouldn’t even count as an answer in the first place if the question is genuine and the intention is to convince. Quora’s answer wiki, “ Why is 0.999… equal to 1?“Ĭonvincing beginning students of math that 0.999… = 1 turns out to be an interestingly challenging and rewarding exercise in pedagogy. ![]() Two different labels are given to the same number.” “In asking “how can two things that are unequal be equal”, you confuse the map with the territory.
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