0.99999999... = 1 ?

개요

0.9999… = 1 이다.

메모

• http://sprott.physics.wisc.edu/pickover/pc/9999.html

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 ( ).
2. And, therefore, when subtracting (with a one after the last of infinite nines).
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).
4. One might argue that after the infinitely many zeros, there is going to be a 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.
6. If we consider the surreal number 0.999… (see earlier post), we may assume that there are ω 9s after the decimal.
7. Return to the Lessons Index | Do the Lessons in Order | Print-friendly page How Can 0.999... = 1?
8. Whatever the reason, many students (me included) have, at one time or another, felt uncomfortable with this equality.
9. The ellipsis (the "dot, dot, dot" after the last 9 ) means "goes on forever in like manner".
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 .
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.
12. In other words, "0.999..." and "1" represent the same number.
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.
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.
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.
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.
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).
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.
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.
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.
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.
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.
23. But each time you add a 9, the margin error is smaller (with each 9, the error is actually ten times smaller).
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.
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…).
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) ).
27. Specifically, if E > 10^(-X) for some positive integer X, then setting D = X will do it.
28. There is no E that is greater than zero such that E = (1 — 0.9999…).
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.
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.
31. It’s still tough to compare items when they take such different forms (like an infinite series).
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!
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?
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).
35. That must mean that one-third of it ("point 3 repeating") is also irrational.
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.
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.
38. 실제로 아이들을 가르쳐본 경험상 중학생까지는 일단 표기부터 다르니까 0.999...
39. 대략적인 원리를 살펴보자면 고등학교 방법으로는 lim(x->1) x+1 를 구해보면 2 라는 것을 알 수 있잖아?
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!
41. ε-δ 방법이 있고 극한값이 이런 개념이 있지'라고 곧바로 이해가 된다면 그 사람은 이미 이해하고 있는 사람이었겠지...아마 0.999...

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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