0.99999999... = 1 ?
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0.9999… = 1 이다.
- 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 ( ).
- And, therefore, when subtracting (with a one after the last of infinite nines).
- 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).
- One might argue that after the infinitely many zeros, there is going to be a 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.
- If we consider the surreal number 0.999… (see earlier post), we may assume that there are ω 9s after the decimal.
- Return to the Lessons Index | Do the Lessons in Order | Print-friendly page How Can 0.999... = 1?
- Whatever the reason, many students (me included) have, at one time or another, felt uncomfortable with this equality.
- The ellipsis (the "dot, dot, dot" after the last 9 ) means "goes on forever in like manner".
- 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 .
- 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.
- In other words, "0.999..." and "1" represent the same number.
- 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.
- 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.
- 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.
- 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.
- (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).
- 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.
- 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.
- 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.
- 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.
- If you stop the expansion of 9s at any finite point, the fraction you have (like .99999 = 99999/100000) is never equal to 1.
- But each time you add a 9, the margin error is smaller (with each 9, the error is actually ten times smaller).
- 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.
- 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…).
- 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) ).
- Specifically, if E > 10^(-X) for some positive integer X, then setting D = X will do it.
- There is no E that is greater than zero such that E = (1 — 0.9999…).
- 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.
- 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.
- It’s still tough to compare items when they take such different forms (like an infinite series).
- 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!
- 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?
- 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).
- That must mean that one-third of it ("point 3 repeating") is also irrational.
- 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.
- 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.
- 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).
- 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.
- The standard algebra proof (which, if you modify it a little, works to convert any repeating decimal into a fraction) runs something like this.
- Those are my first two demonstrations that our fact is true (the last one is at the end).
- The word "geometric" means that each term of the series is the identical multiple (in this case 1/10) of the previous term.
- 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).
- The documentation for the built-in round() function says that it rounds to the nearest value, rounding ties away from zero.
- 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.
- For fine control over how a float is displayed see the str.format() method’s format specifiers in Format String Syntax.
- 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).
- In fact, the probability of pulling any real subset of numbers is zero (i.e. 10000/infinity = 0).
- Proof:The Decimal 0.999... is Equivalent to 1
- Really, 0.999999… is equal to 1. Surreally, this is not so!
- How Can 0.999... = 1?
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- Does 0.99999… really equal 1?
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- Forum thread: 0.9999... = 1
- Ask Dr. Math
- Webmath.com: Doing math with fractions
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- Polymathematics: No, I'm Sorry, It Does.
- 14. Floating Point Arithmetic: Issues and Limitations — Python 2.7.2 documentation
- 0.9999 = 1, true or false and why?
많이 나오는 질문과 답변
많이 나오는 질문과 답변이 아래 링크에 정리되어 있다.
- 디씨인사이드 수학 갤러리, ε-δ작성