Does 0.999... = 1?

College Confidential Community
41

<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>

42

<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>

43

<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>

44

<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>

45

<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>

46

<p>
[quote]

If the .999999 eventually became a one, would that indicate the presence of god?

[/quote]
</p>

<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>

47

<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>

48

<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>

49

<p>
[quote]
Actually, 0.9999999 repeating does equal 1. 0.999999 repeating can be represented by an infinite geometric series.</p>

<p>0.9 + 0.09 + 0.009 + ... and so on. Using the formula, (t1)/(1-r) where t1 is the first term and r is the rate, plug in the numbers. You get:</p>

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

[/quote]
</p>

<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>

50

<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>

51

<p>
[Quote]
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.

[/Quote]
</p>

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

52

<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>

53

<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>

54

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

55

<p>
[quote]
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.

[/quote]
</p>

<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>

56

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

[/quote]
</p>

<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>

57

<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>

<p>
[quote]
Hi Emily and Jenny,</p>

<p>There's no doubt that this equality is one of the weirder things in
mathematics, and it <em>is</em> intuitive to think: No matter how many 9's
you add, you'll never get all the way to 1. </p>

<p>But that's how it seems if you think about moving <em>toward</em> 1. What if
you think about moving <em>away</em> from 1? </p>

<p>That is, if you start at 1, and try to move away from 1 and toward
0.99999..., how far do you have to go to get to 0.99999... ? Any step
you try to take will be too far, so you can't really move at all -<br>
which means that to move from 1 to 0.99999..., you have to stay at 1. </p>

<p>Which means they must be the same thing!</p>

<p>Here's another way to think about it. When you write something like</p>

<p>0.35</p>

<p>that's really the same as 35/100,</p>

<p>0.35 = 35 / 100</p>

<p>right? Well, you can turn that into a repeating decimal by dividing by
99 instead of 100:
__
0.35353535... = 0.35 = 35 / 99</p>

<p>Play around with some other fractions, like 2/9, 415/999, and so on,
to convince yourself that this is true. (A calculator would be
helpful.)</p>

<p>In general, when we have N repeating digits, the corresponding
fraction is </p>

<p>(the digits) / (10^N - 1)</p>

<p>Again, some examples can help make this clear:
<br>
0.1 = 1/9
_
0.12 = 12/99
___
0.123 = 123/999</p>

<p>and so on. </p>

<p>So, here's something to consider: What fraction corresponds to
_
0.9 = ?</p>

<p>It has to be something over 9, right?
_
0.9 = ? / 9</p>

<p>The <em>only</em> thing it could possibly be is
_
0.9 = 9 / 9</p>

<p>right? But that's the same as 1. </p>

<p>Ultimately, though, this probably won't <em>really</em> make sense until you
come to grips with what it means for a decimal to repeat <em>forever</em>,
instead of just for a r-e-a-l-l-y l-o-n-g t-i-m-e. </p>

<p>When you think of 0.999... as being 'a little below 1', it's because
in your mind, you've stopped expanding it; that is, instead of </p>

<p>0.999999...</p>

<p>you're <em>really</em> thinking of</p>

<p>0.999...999</p>

<p>which is not the same thing. You're absolutely right that 0.999...999
is a little below 1, but 0.999999... doesn't fall short of 1 <em>until</em>
you stop expanding it. But you never stop expanding it, so it never
falls short of 1. </p>

<p>Suppose someone gives you $1000, but says: "Now, don't spend it all,
because I'm going to go off and find the largest integer, and after I
find it I'm going to want you to give me $1 back." How much money has
he really given you? </p>

<p>On the one hand, you might say: "He's given me $999, because he's
going to come back later and get $1." </p>

<p>But on the other hand, you might say: "He's given me $1000, because
he's <em>never</em> going to come back!" </p>

<p>It's only when you realize that in this instance, 'later' is the same
as 'never', that you can see that you get to keep the whole $1000. In
the same way, it's only when you really understand that the expansion
of 0.999999... <em>never</em> ends that you realize that it's not really 'a
little below 1' at all.

[/quote]
</p>

58

<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>

59

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

60

<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>