# "0.99999999... = 1 ?"의 두 판 사이의 차이

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(사용자 2명의 중간 판 17개는 보이지 않습니다) | |||

1번째 줄: | 1번째 줄: | ||

− | + | ==개요== | |

0.9999… = 1 이다. | 0.9999… = 1 이다. | ||

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− | + | ==메모== | |

− | + | * http://sprott.physics.wisc.edu/pickover/pc/9999.html | |

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− | + | ==노트== | |

+ | ===말뭉치=== | ||

+ | # 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> | ||

+ | # And, therefore, when subtracting (with a one after the last of infinite nines).<ref name="ref_81c60dff" /> | ||

+ | # 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" /> | ||

+ | # One might argue that after the infinitely many zeros, there is going to be a 1 ( ).<ref name="ref_81c60dff" /> | ||

+ | # 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> | ||

+ | # 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" /> | ||

+ | # 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> | ||

+ | # Whatever the reason, many students (me included) have, at one time or another, felt uncomfortable with this equality.<ref name="ref_d594d900" /> | ||

+ | # The ellipsis (the "dot, dot, dot" after the last 9 ) means "goes on forever in like manner".<ref name="ref_d594d900" /> | ||

+ | # 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" /> | ||

+ | # 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> | ||

+ | # 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" /> | ||

+ | # 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> | ||

+ | # 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> | ||

+ | # 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> | ||

+ | # 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" /> | ||

+ | # 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> | ||

+ | # 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" /> | ||

+ | # 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> | ||

+ | # 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" /> | ||

+ | # 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> | ||

+ | # 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> | ||

+ | # 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> | ||

+ | # 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> | ||

+ | # 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> | ||

+ | # 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" /> | ||

+ | # If we do this, we'll get a 9.999990 (slide the decimal in the 0.999999 right 1 places, you'll get 9.999990).<ref name="ref_718b3275">[http://www.webmath.com/cgi-bin/fraction.cgi?decimal=.999999&op=to_fraction&back=dec2fract.html Webmath.com: Doing math with fractions]</ref> | ||

+ | # If you can warranted that you always do the best on your job level, your boss will automatically think that you must get (absolutely get) new level on your job.<ref name="ref_6495662e">[https://brilliant.org/discussions/thread/explain-why-0999-1/ Explain why 0.999... = 1 ?]</ref> | ||

+ | # The standard algebra proof (which, if you modify it a little, works to convert any repeating decimal into a fraction) runs something like this.<ref name="ref_a97b282a">[https://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html Polymathematics: No, I'm Sorry, It Does.]</ref> | ||

+ | # Those are my first two demonstrations that our fact is true (the last one is at the end).<ref name="ref_a97b282a" /> | ||

+ | # The word "geometric" means that each term of the series is the identical multiple (in this case 1/10) of the previous term.<ref name="ref_a97b282a" /> | ||

+ | # You’ll see the same kind of thing in all languages that support your hardware’s floating-point arithmetic (although some languages may not display the difference by default, or in all output modes).<ref name="ref_a8b68441">[https://python.readthedocs.io/en/v2.7.2/tutorial/floatingpoint.html 14. Floating Point Arithmetic: Issues and Limitations — Python 2.7.2 documentation]</ref> | ||

+ | # The documentation for the built-in round() function says that it rounds to the nearest value, rounding ties away from zero.<ref name="ref_a8b68441" /> | ||

+ | # Since the decimal fraction 2.675 is exactly halfway between 2.67 and 2.68, you might expect the result here to be (a binary approximation to) 2.68.<ref name="ref_a8b68441" /> | ||

+ | # For fine control over how a float is displayed see the str.format() method’s format specifiers in Format String Syntax.<ref name="ref_a8b68441" /> | ||

+ | # If you had a hat with an infinite amount of slips of paper consisting of an infinite amount of numbers, the probability of pulling any random number is zero (1/infinity = 0).<ref name="ref_1579dacc">[https://www.neogaf.com/threads/0-9999-1-true-or-false-and-why.385028/ 0.9999 = 1, true or false and why?]</ref> | ||

+ | # In fact, the probability of pulling any real subset of numbers is zero (i.e. 10000/infinity = 0).<ref name="ref_1579dacc" /> | ||

− | + | ===소스=== | |

+ | <references /> | ||

− | + | ==많이 나오는 질문과 답변== | |

− | + | 많이 나오는 질문과 답변이 아래 링크에 정리되어 있다. | |

− | + | * http://gall.dcinside.com/list.php?id=mathematics&no=28665&page=1&bbs= | |

− | + | ** 디씨인사이드 수학 갤러리, [http://gallog.dcinside.com/tmxkskgkwk ε-δ]작성 | |

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## 2021년 2월 12일 (금) 08:55 기준 최신판

## 개요

0.9999… = 1 이다.

## 메모

## 노트

### 말뭉치

- 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]} - And, therefore, when subtracting (with a one after the last of infinite nines).
^{[1]} - 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]} - One might argue that after the infinitely many zeros, there is going to be a 1 ( ).
^{[1]} - 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]} - If we consider the surreal number 0.999… (see earlier post), we may assume that there are ω 9s after the decimal.
^{[2]} - Return to the Lessons Index | Do the Lessons in Order | Print-friendly page How Can 0.999... = 1?
^{[3]} - Whatever the reason, many students (me included) have, at one time or another, felt uncomfortable with this equality.
^{[3]} - The ellipsis (the "dot, dot, dot" after the last 9 ) means "goes on forever in like manner".
^{[3]} - 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]} - 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]} - In other words, "0.999..." and "1" represent the same number.
^{[4]} - 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]} - 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]} - 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]} - 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]} - (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]} - 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]} - 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]} - 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]} - 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]} - 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]} - But each time you add a 9, the margin error is smaller (with each 9, the error is actually ten times smaller).
^{[8]} - 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]} - 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]} - 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]} - Specifically, if E > 10^(-X) for some positive integer X, then setting D = X will do it.
^{[9]} - There is no E that is greater than zero such that E = (1 — 0.9999…).
^{[9]} - 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]} - 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]} - It’s still tough to compare items when they take such different forms (like an infinite series).
^{[10]} - 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]} - 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]} - 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]} - That must mean that one-third of it ("point 3 repeating") is also irrational.
^{[13]} - 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]} - 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]} - If we do this, we'll get a 9.999990 (slide the decimal in the 0.999999 right 1 places, you'll get 9.999990).
^{[15]} - If you can warranted that you always do the best on your job level, your boss will automatically think that you must get (absolutely get) new level on your job.
^{[16]} - The standard algebra proof (which, if you modify it a little, works to convert any repeating decimal into a fraction) runs something like this.
^{[17]} - Those are my first two demonstrations that our fact is true (the last one is at the end).
^{[17]} - The word "geometric" means that each term of the series is the identical multiple (in this case 1/10) of the previous term.
^{[17]} - You’ll see the same kind of thing in all languages that support your hardware’s floating-point arithmetic (although some languages may not display the difference by default, or in all output modes).
^{[18]} - The documentation for the built-in round() function says that it rounds to the nearest value, rounding ties away from zero.
^{[18]} - Since the decimal fraction 2.675 is exactly halfway between 2.67 and 2.68, you might expect the result here to be (a binary approximation to) 2.68.
^{[18]} - For fine control over how a float is displayed see the str.format() method’s format specifiers in Format String Syntax.
^{[18]} - If you had a hat with an infinite amount of slips of paper consisting of an infinite amount of numbers, the probability of pulling any random number is zero (1/infinity = 0).
^{[19]} - In fact, the probability of pulling any real subset of numbers is zero (i.e. 10000/infinity = 0).
^{[19]}

### 소스

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Proof:The Decimal 0.999... is Equivalent to 1 - ↑
^{2.0}^{2.1}Really, 0.999999… is equal to 1. Surreally, this is not so! - ↑
^{3.0}^{3.1}^{3.2}^{3.3}How Can 0.999... = 1? - ↑
^{4.0}^{4.1}^{4.2}^{4.3}Wikipedia - ↑
^{5.0}^{5.1}^{5.2}^{5.3}Is it true that $0.999999999\dots=1$? - ↑ 0.999…=1
- ↑
^{7.0}^{7.1}0.999999=1? - ↑
^{8.0}^{8.1}^{8.2}Does 0.99999… really equal 1? - ↑
^{9.0}^{9.1}^{9.2}^{9.3}Why 0.99999… = 1, proof, and limits - ↑
^{10.0}^{10.1}^{10.2}^{10.3}A Friendly Chat About Whether 0.999… = 1 – BetterExplained - ↑ 0.999999….. = 1?
- ↑ Art of Problem Solving
- ↑ Forum thread: 0.9999... = 1
- ↑
^{14.0}^{14.1}Ask Dr. Math - ↑ Webmath.com: Doing math with fractions
- ↑ Explain why 0.999... = 1 ?
- ↑
^{17.0}^{17.1}^{17.2}Polymathematics: No, I'm Sorry, It Does. - ↑
^{18.0}^{18.1}^{18.2}^{18.3}14. Floating Point Arithmetic: Issues and Limitations — Python 2.7.2 documentation - ↑
^{19.0}^{19.1}0.9999 = 1, true or false and why?

## 많이 나오는 질문과 답변

많이 나오는 질문과 답변이 아래 링크에 정리되어 있다.

- http://gall.dcinside.com/list.php?id=mathematics&no=28665&page=1&bbs=
- 디씨인사이드 수학 갤러리, ε-δ작성