"0.99999999... = 1 ?"의 두 판 사이의 차이

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==노트==
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===말뭉치===
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# One might argue that when multiplying by ten, the right hand side shifts to the left a decimal place, leaving a terminating zero at the end of the infinite string of 9's ( ).<ref name="ref_81c60dff">[https://math.wikia.org/wiki/Proof:The_Decimal_0.999..._is_Equivalent_to_1 Proof:The Decimal 0.999... is Equivalent to 1]</ref>
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# And, therefore, when subtracting (with a one after the last of infinite nines).<ref name="ref_81c60dff" />
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# It is impossible to find a "next higher" number that is both larger to a given value but couldnt have been smaller (see argument from averages).<ref name="ref_81c60dff" />
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# One might argue that after the infinitely many zeros, there is going to be a 1 ( ).<ref name="ref_81c60dff" />
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# The sequence S = (0.9, 0.99, 0.999, … ) is bounded above by 1.0, so the least upper bound is not greater than 1.0.<ref name="ref_daa472c2">[https://thatsmaths.com/2019/01/10/really-0-999999-is-equal-to-1-surreally-this-is-not-so/ Really, 0.999999… is equal to 1. Surreally, this is not so!]</ref>
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# If we consider the surreal number 0.999… (see earlier post), we may assume that there are ω 9s after the decimal.<ref name="ref_daa472c2" />
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# Return to the Lessons Index | Do the Lessons in Order | Print-friendly page How Can 0.999... = 1?<ref name="ref_d594d900">[https://www.purplemath.com/modules/howcan1.htm How Can 0.999... = 1?]</ref>
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# Whatever the reason, many students (me included) have, at one time or another, felt uncomfortable with this equality.<ref name="ref_d594d900" />
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# The ellipsis (the "dot, dot, dot" after the last 9 ) means "goes on forever in like manner".<ref name="ref_d594d900" />
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# Yes, at any given stop, at any given stage of the expansion, for any given finite number of 9 s, there will be a difference between 0.999...9 and 1 .<ref name="ref_d594d900" />
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# In mathematics, 0.999... (also written as 0.9, in repeating decimal notation) denotes the repeating decimal consisting of infinitely many 9s after the decimal point.<ref name="ref_acb2a2a7">[https://en.wikipedia.org/wiki/0.999... Wikipedia]</ref>
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# In other words, "0.999..." and "1" represent the same number.<ref name="ref_acb2a2a7" />
 +
# The technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined.<ref name="ref_acb2a2a7" />
 +
# More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all base representations.<ref name="ref_acb2a2a7" />
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# So when we learn math in elementary school we are told: Every real number can be written as a decimal expansion (maybe infinite) and every possible decimal expansion is a real number.<ref name="ref_48c663f0">[https://math.stackexchange.com/questions/11/is-it-true-that-0-999999999-dots-1 Is it true that $0.999999999\dots=1$?]</ref>
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# we take it on faith that: Every real number can be written as a decimal expansion (maybe infinite) and every possible decimal expansion is a real number.<ref name="ref_48c663f0" />
 +
# (So it can be all the elements less than 1/2 but it can't be all the elements less than or equal to 1/2).<ref name="ref_48c663f0" />
 +
# Notice sometimes the cut will occur at a rational number (all the rationals less than 1/2), but sometimes it will occur at points "between" the rational numbers.<ref name="ref_48c663f0" />
 +
# 다행히도, 임의의 무한소수에 대해 수열의 극한값은 존재하고, 그 극한값은 이 수열의 상한(supremum), 풀어 쓰면 '모든 자연수에 대해보다 크거나 같은 숫자의 집합에서 가장 작은 수' 와 같다.이를 증명하기는 어렵지 않다.<ref name="ref_36083143">[https://namu.wiki/w/0.999%E2%80%A6%3D1 0.999…=1]</ref>
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# IF the two numbers 0.99999... and 1 were NOT equal, you would have a number to represent the difference (the GAP or Distance between them) - and it should not be 0.<ref name="ref_36a36aab">[http://mathcentral.uregina.ca/QQ/database/QQ.09.01/catherine1.html 0.999999=1?]</ref>
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# Of course it is a bit obscure since we are dealing with 'infinite processes' (sometimes called limits) whenever we work with decimal numbers which do no terminate.<ref name="ref_36a36aab" />
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# If you stop the expansion of 9s at any finite point, the fraction you have (like .99999 = 99999/100000) is never equal to 1.<ref name="ref_cfd5cdc1">[https://www.relativelyinteresting.com/does-0-99999-really-equal-1/ Does 0.99999… really equal 1?]</ref>
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# But each time you add a 9, the margin error is smaller (with each 9, the error is actually ten times smaller).<ref name="ref_cfd5cdc1" />
 +
# You can show (using calculus or other summations) that with a large enough number of 9s in the expansion, you can get arbitrarily close to 1.<ref name="ref_cfd5cdc1" />
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# To prove it to yourself that 0.9999… = 1, consider that if they weren’t equal, there would be a number E that is greater than zero such that E = (1 — 0.9999…).<ref name="ref_b8db7dcd">[https://www.tcg.com/blog/why-099999-1-proof-and-limits/ Why 0.99999… = 1, proof, and limits]</ref>
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# Since we’re playing this game, you counter and make E smaller, say 10^(-10), and I turn around and say “make D = 11” (because 1 — 0.99999999999 < 10^(-10) ).<ref name="ref_b8db7dcd" />
 +
# Specifically, if E > 10^(-X) for some positive integer X, then setting D = X will do it.<ref name="ref_b8db7dcd" />
 +
# There is no E that is greater than zero such that E = (1 — 0.9999…).<ref name="ref_b8db7dcd" />
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# Well, it’s “true” in the same way numbers must be there (positive) or not there (zero) — it’s true because we’ve implicitly excluded other possibilities.<ref name="ref_e1d14936">[https://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/ A Friendly Chat About Whether 0.999… = 1 – BetterExplained]</ref>
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# The idea behind limits is to find some point at which “a – b” becomes zero (less than any number); that is, we can’t tell the “number to test” and our “result” as different.<ref name="ref_e1d14936" />
 +
# It’s still tough to compare items when they take such different forms (like an infinite series).<ref name="ref_e1d14936" />
 +
# Then if you ask for 1, and I give you 0.99, the difference is 0.01 (one hundredth) and you don’t know you’ve been tricked!<ref name="ref_e1d14936" />
 +
# Student: Are you (looks at Vadim) saying that you (looks at Teacher) made up the definition of a limit just in order to make 0.999…=1?<ref name="ref_8fbbe88f">[https://blogs.helsinki.fi/kulikov/2011/07/04/0-999/ 0.999999….. = 1?]</ref>
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# But by the least upper bound axiom of real numbers, if a set of real numbers has a upper bound, then it has a least upper bound (that is a real number).<ref name="ref_7912a8ac">[https://artofproblemsolving.com/wiki/index.php/0.999... Art of Problem Solving]</ref>
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# That must mean that one-third of it ("point 3 repeating") is also irrational.<ref name="ref_c122e3b9">[https://www.hltv.org/forums/threads/2070372/09999-1 Forum thread: 0.9999... = 1]</ref>
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# And whenever you have an infinite number of parts of something (an infinite number of places in a number, in this case) you might well expect something amazing to happen.<ref name="ref_e8240b85">[https://www.uhigh.ilstu.edu/math/thompson/.999999%20repeating1.htm Ask Dr. Math]</ref>
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# Said another way, if your number x = 0.999999999... is such that the difference "x-1" is smaller than any positive number (however small), then we consider x - 1 = 0, so that x = 1.<ref name="ref_e8240b85" />
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# 실제로 아이들을 가르쳐본 경험상 중학생까지는 일단 표기부터 다르니까 0.999...<ref name="ref_8f26c17c">[http://m.blog.naver.com/lovetaehong/130076182331 0.999999... = 1 에 대한 쉬운 이야기]</ref>
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# 대략적인 원리를 살펴보자면 고등학교 방법으로는 lim(x->1) x+1 를 구해보면 2 라는 것을 알 수 있잖아?<ref name="ref_8f26c17c" />
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# 작은 범위(δ값으로 제한을 두는)의 x값에서 x+1과 2의 차이가 임의의 양수(ε>0)보다 항상 작음을 증명함으로서 lim(x->1) x+1 = 2 임을 보이는 방법이야!<ref name="ref_8f26c17c" />
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# ε-δ 방법이 있고 극한값이 이런 개념이 있지'라고 곧바로 이해가 된다면 그 사람은 이미 이해하고 있는 사람이었겠지...아마 0.999...<ref name="ref_8f26c17c" />
  
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===소스===
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<references />
  
 
==많이 나오는 질문과 답변==
 
==많이 나오는 질문과 답변==

2021년 2월 12일 (금) 08:52 판

개요

0.9999… = 1 이다.



메모


노트

말뭉치

  1. One might argue that when multiplying by ten, the right hand side shifts to the left a decimal place, leaving a terminating zero at the end of the infinite string of 9's ( ).[1]
  2. And, therefore, when subtracting (with a one after the last of infinite nines).[1]
  3. It is impossible to find a "next higher" number that is both larger to a given value but couldnt have been smaller (see argument from averages).[1]
  4. One might argue that after the infinitely many zeros, there is going to be a 1 ( ).[1]
  5. The sequence S = (0.9, 0.99, 0.999, … ) is bounded above by 1.0, so the least upper bound is not greater than 1.0.[2]
  6. If we consider the surreal number 0.999… (see earlier post), we may assume that there are ω 9s after the decimal.[2]
  7. Return to the Lessons Index | Do the Lessons in Order | Print-friendly page How Can 0.999... = 1?[3]
  8. Whatever the reason, many students (me included) have, at one time or another, felt uncomfortable with this equality.[3]
  9. The ellipsis (the "dot, dot, dot" after the last 9 ) means "goes on forever in like manner".[3]
  10. Yes, at any given stop, at any given stage of the expansion, for any given finite number of 9 s, there will be a difference between 0.999...9 and 1 .[3]
  11. In mathematics, 0.999... (also written as 0.9, in repeating decimal notation) denotes the repeating decimal consisting of infinitely many 9s after the decimal point.[4]
  12. In other words, "0.999..." and "1" represent the same number.[4]
  13. The technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999... is commonly defined.[4]
  14. More generally, every nonzero terminating decimal has two equal representations (for example, 8.32 and 8.31999...), which is a property of all base representations.[4]
  15. So when we learn math in elementary school we are told: Every real number can be written as a decimal expansion (maybe infinite) and every possible decimal expansion is a real number.[5]
  16. we take it on faith that: Every real number can be written as a decimal expansion (maybe infinite) and every possible decimal expansion is a real number.[5]
  17. (So it can be all the elements less than 1/2 but it can't be all the elements less than or equal to 1/2).[5]
  18. Notice sometimes the cut will occur at a rational number (all the rationals less than 1/2), but sometimes it will occur at points "between" the rational numbers.[5]
  19. 다행히도, 임의의 무한소수에 대해 수열의 극한값은 존재하고, 그 극한값은 이 수열의 상한(supremum), 풀어 쓰면 '모든 자연수에 대해보다 크거나 같은 숫자의 집합에서 가장 작은 수' 와 같다.이를 증명하기는 어렵지 않다.[6]
  20. IF the two numbers 0.99999... and 1 were NOT equal, you would have a number to represent the difference (the GAP or Distance between them) - and it should not be 0.[7]
  21. Of course it is a bit obscure since we are dealing with 'infinite processes' (sometimes called limits) whenever we work with decimal numbers which do no terminate.[7]
  22. If you stop the expansion of 9s at any finite point, the fraction you have (like .99999 = 99999/100000) is never equal to 1.[8]
  23. But each time you add a 9, the margin error is smaller (with each 9, the error is actually ten times smaller).[8]
  24. You can show (using calculus or other summations) that with a large enough number of 9s in the expansion, you can get arbitrarily close to 1.[8]
  25. To prove it to yourself that 0.9999… = 1, consider that if they weren’t equal, there would be a number E that is greater than zero such that E = (1 — 0.9999…).[9]
  26. Since we’re playing this game, you counter and make E smaller, say 10^(-10), and I turn around and say “make D = 11” (because 1 — 0.99999999999 < 10^(-10) ).[9]
  27. Specifically, if E > 10^(-X) for some positive integer X, then setting D = X will do it.[9]
  28. There is no E that is greater than zero such that E = (1 — 0.9999…).[9]
  29. Well, it’s “true” in the same way numbers must be there (positive) or not there (zero) — it’s true because we’ve implicitly excluded other possibilities.[10]
  30. The idea behind limits is to find some point at which “a – b” becomes zero (less than any number); that is, we can’t tell the “number to test” and our “result” as different.[10]
  31. It’s still tough to compare items when they take such different forms (like an infinite series).[10]
  32. Then if you ask for 1, and I give you 0.99, the difference is 0.01 (one hundredth) and you don’t know you’ve been tricked![10]
  33. Student: Are you (looks at Vadim) saying that you (looks at Teacher) made up the definition of a limit just in order to make 0.999…=1?[11]
  34. But by the least upper bound axiom of real numbers, if a set of real numbers has a upper bound, then it has a least upper bound (that is a real number).[12]
  35. That must mean that one-third of it ("point 3 repeating") is also irrational.[13]
  36. And whenever you have an infinite number of parts of something (an infinite number of places in a number, in this case) you might well expect something amazing to happen.[14]
  37. Said another way, if your number x = 0.999999999... is such that the difference "x-1" is smaller than any positive number (however small), then we consider x - 1 = 0, so that x = 1.[14]
  38. 실제로 아이들을 가르쳐본 경험상 중학생까지는 일단 표기부터 다르니까 0.999...[15]
  39. 대략적인 원리를 살펴보자면 고등학교 방법으로는 lim(x->1) x+1 를 구해보면 2 라는 것을 알 수 있잖아?[15]
  40. 작은 범위(δ값으로 제한을 두는)의 x값에서 x+1과 2의 차이가 임의의 양수(ε>0)보다 항상 작음을 증명함으로서 lim(x->1) x+1 = 2 임을 보이는 방법이야![15]
  41. ε-δ 방법이 있고 극한값이 이런 개념이 있지'라고 곧바로 이해가 된다면 그 사람은 이미 이해하고 있는 사람이었겠지...아마 0.999...[15]

소스

  1. ↑ 1.0 1.1 1.2 1.3 Proof:The Decimal 0.999... is Equivalent to 1
  2. ↑ 2.0 2.1 Really, 0.999999… is equal to 1. Surreally, this is not so!
  3. ↑ 3.0 3.1 3.2 3.3 How Can 0.999... = 1?
  4. ↑ 4.0 4.1 4.2 4.3 Wikipedia
  5. ↑ 5.0 5.1 5.2 5.3 Is it true that $0.999999999\dots=1$?
  6. ↑ 0.999…=1
  7. ↑ 7.0 7.1 0.999999=1?
  8. ↑ 8.0 8.1 8.2 Does 0.99999… really equal 1?
  9. ↑ 9.0 9.1 9.2 9.3 Why 0.99999… = 1, proof, and limits
  10. ↑ 10.0 10.1 10.2 10.3 A Friendly Chat About Whether 0.999… = 1 – BetterExplained
  11. ↑ 0.999999….. = 1?
  12. ↑ Art of Problem Solving
  13. ↑ Forum thread: 0.9999... = 1
  14. ↑ 14.0 14.1 Ask Dr. Math
  15. ↑ 15.0 15.1 15.2 15.3 0.999999... = 1 에 대한 쉬운 이야기

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