Does 0.999... = 1?

College Confidential Community
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<p>The limit of a function at a point is not inherently the same as the value of the function at that point.</p>

<p>Perfect example with the 1/1-x. While the limit of the function approaches 1, it never actually reaches one. Because .999… keeps on getting larger (as the 9’s are added on to the end of the decimal) it keeps on getting closer and closer to one, you are right on that point. However, the interval with which the number increases keeps getting smaller and smaller.</p>

<p>.1 to .11 increases by one hundredth.
.11 to .111 increases by one thousandth. (since .11 can be thought of as .110 or 110 thousandths)</p>

<p>Hence, as .999… increases, the interval with which .999… increases gets infinitesimally smaller, causing the rate at which it increases also to get smaller, and for that reason it will never reach 1. </p>

<p>Correct me if I’m wrong.</p>

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<p>Just to clarify, because .999… is increasing by smaller and smaller amounts each time, it is intuitively obvious that the gap between 1 and 0.999… will be getting infinitesimally smaller, but will never be equal to zero.</p>

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<p>I agree. Doing math with infinite decimals and using it to “prove” .999… = 1, is a very hard justification to accept. Maybe a math major can clarify.</p>

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<p>but the number increases at the same rate as the difference between 1 and .999… decreases.
I think if the number extends to infinity, the difference CAN reach zero…if the number were infinite, you would never be able to place the “1” at the end of it - that would make it finite. the difference between 1 and .999… would be written as .0000… </p>

<p>…well, that just brings us back to the same dispute anyway.
does .000…1 = 0? Fundamentally, that’s the same question as does .999… = 1</p>

<p>this argument can be so cyclical…</p>

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<p>It seems to to me that .9999etc extending to infinity can only continue to be nines in a universe organized scientifically If the .999999 eventually became a one, would that indicate the presence of god? how do you make that leap from the fraction to the whole without some extrahuman intervention?</p>

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<p>why would that be?
if in a scientific universe, .999… can extend to infinity, then it would logically follow that in that same scientific universe, .999… = 1. The difference between the two numbers becomes infinitely small as you add 9s to the end - that is, the difference becomes 0 because adding a 1 to end of an infinitely small number would make it finite.</p>

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<p>ok this is the last time im going to post here because im clearly out to lunch number.wise . if a real thing became infinitessimally small, it still does not disappear , according to the half life model. therefore…I just don’t see how .999999+ could become one without artificial intervention. same principal: it gets bigger but it’s never complete. i’ll stick to humanities… but I still think that the idea of a constant “single garment” time is compelling, even though freud exploited the idea for psychoanalysis. .</p>

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<p>This is not deep. It’s a matter of defintions. Infinite decimals are defined as the limit of series. The LIMIT. It doesn’t matter if the series never reaches 1, because the limit EQUALS 1.</p>

<p>The reason this debate endures is that people don’t understand infinity. It is not a number. There is no such thing as a sum of an infinite number of things. But there is this convenient idea: the limit. Thus infinite sums, which are initially meaningless, are given meaning by DEFINITION, not nature. But confusions results, because when the word “limit” is omitted, they forget that what they are dealing with is not, in fact, a sum, though it is written as if it were.</p>

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<p>Anybody who has covered infinite series, should, in fact, know the answer to this question. Like Feuler just said, this is not difficult at all.</p>

<p>Edit: I just read over the thread again, and clearly some of you do not know what you are talking about when speaking of infinity.</p>

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<p>You could also look at it like this (not really a proof, but it might be intuitive):</p>

<p>1-0.9=0.1
1-.0.99=0.01</p>

<p>etc</p>

<p>the difference between the numbers becomes infinitely small</p>

<p>because it is infinitely small, it is the smallest possible quantity</p>

<p>it must be zero</p>

<p>1-0=1</p>

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<p>Yea, but the gap never stops getting smaller, since theres an infinite amount of digits.</p>

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<p>ok this is my LAST question…and this is not a disingenuous question …do you think math is an art or a science?</p>

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<p>1/3 = .3333…
1/3 = .3333…</p>

<h2>1/3 = .3333…</h2>

<p>3/3 = .99999… </p>

<p>Oh my god. I can’t believe I just participated in one of the most pointless CC debates ever.</p>

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<p>And I don’t think math is an art or a science, it’s its own category</p>

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<p>limits exist to eliminate the debate in the first place. the limit as .999… extends to infinity is 1. some would say that simply means that .999… approaches one, but can never actually reach it; others would say .999… approaches 1 as we get closer and closer to infinity, and eventually, at some infinite point (yes, it seems like an oxymoron) does actually reach one. In either case, both would agree that the limit is still 1 - but it glosses over the debate.</p>

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<p>Absolutely right. The gap never stops getting smaller. Ever.</p>

<p>For that reason, the gap will never be non-existent. There will ALWAYS be a gap. Hence why .999… does NOT equal one.</p>

<p>I don’t see how it isn’t intuitive. Since there’s always a gap, no matter how small it gets, the gap will still be there.</p>

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<p>Actually, I retract my former statement. .999… does equal one. I have forgotten the basic postulates on which calculus is based on.</p>

<p>From the Math Forum:</p>

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<p>Very right. 0.9999… is at no point 0.9 or 0.99, it is from the beginning, and always 0.999… = 1. In fact, even if speaking in terms of an infinite process, the idea is perfectly valid.</p>

<p>0.333… + 0.333… conceptually requires an infinite number of additions of 3+3, but that doesn’t mean that the addition cannot be done mathematically or rigorously - it is very easy to notice that each time, 3+3 = 6, so 0.333… + 0.333… must equal 0.666…</p>

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<p>I’m surprised no one has never questioned 1/3 being .3 repeating. Hmm.</p>

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<p>“does .000…1 = 0? Fundamentally, that’s the same question as does .999… = 1”</p>

<p>Define 0.000…1 = 0. It is not a real number. If you were to define it as a real number, than the only value which would make sense for it is 0. You could define it as an extension to the real number system, but even in this system there is still no number between 0.999 and 1. And if I am outside of the real number system I cannot use results to pass judgement on the real number system. I could define x to be a number that is between 0.999… and 1, and this is valid. But it doesn’t make sense to then say that 0.999… < x < 1 so 0.999… /= 1, since x is outside the real number system.
If you wish to talk about 0.0000…1, first rigorously define the number 0.0000…1, and then see what you get. ie, It’s perfectly reasonable to define 0.0000…1 as the number which is greater than 0 but less than any positive real number; but it won’t help you prove that 0.999… is not equal to 1.</p>