#92803
01/27/2003 8:18 PM
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#92804
01/28/2003 4:42 PM
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sin, cos, tan, sec, csc, cot? Anyone else with math and another language background care to comment?
Russian + math: sin, cos, tg, sec, (?), ctg... I am not sure what is csc... cosecans? then cosec
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#92805
01/28/2003 4:54 PM
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Ridicolous to say, I am the last person apt to give a definitive answer. Mostly because I don't care about names, and simply try to understand from the context.
A rose without its name, wouldn't be a rose?
So I can ask the same question about numbers in "high" or "low" way:
example:
how many different decimal representation can a real number have?
Or
0,9999999999999999999....is or not equal to one?
To understand the second question implies to have understood the core of the first one.
(Anyway, Bean, you were perfect in your list of sets of numbers. Just a comment about irrational numbers . I have always seen them written as "R\Q" = real not rationa numbers)
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#92806
01/28/2003 4:55 PM
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I forgot to add the answer
Yes,
0,99999999...... is equal to 1.
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#92807
01/28/2003 5:15 PM
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0,9999999999999999999....is or not equal to one?
I've seen it argued that, since 1/3*3=1 and 0.33333333333…+0.33333333333…+0.33333333333…=0.99999999999… then 0.99999999999…=1. On the other hand, I would maintain that 0.99999999999… is in the open interval between 0 and 1, but 1 is not, therefore 0.99999999999… does not = 1
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#92808
01/28/2003 6:00 PM
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Dear Faldage: and I still think parallel lines can never meet.
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#92809
01/28/2003 6:16 PM
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"Infinity" was once *disproved to me by this explaination:
Take your thumb and your forefinger and spread them apart as far as possible. This represents a distance. Now make that distance one half as large. Continue to cut the distance between them in half.
Theoretically, your thumb and forefinger will never actually® touch each other.
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#92810
01/28/2003 6:38 PM
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0,99999999...... is equal to 1.
Does this mean that 0.9999999999… is not in the open interval from 0 to 1?
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#92811
01/28/2003 6:46 PM
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#92812
01/28/2003 7:08 PM
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in both places
1 is in the closed interval but not the open interval; 0.99999999999999999999999 (without the ellipsis) is in the open interval, no matter how many 9s you tack onto the end. Or so I thought.
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#92813
01/28/2003 7:45 PM
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said the Queen. "When I was younger, I always did it for half an hour a day. Why,
sometimes I've believed as many as six impossible things before breakfast."
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#92814
01/28/2003 7:53 PM
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no matter how many 9s you tack onto the end
I guess that's true only as long as the number of 9s you tack onto the end is finite.
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#92815
01/28/2003 10:38 PM
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Pooh-Bah
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I disagree that 0.9999999999... is equal to the integer 1. All integers are understood to be exactly 1 or exactly 2, etc, as if they had an infinite number of zeros after a decimel point. Of course, they are practically the same in any conceivable everyday context, but they differ by that tiny amount. For example, 1/(1-1) is not defined because one cannot divide by zero. 1/(1-0.9999999.....) approaches infinity asymptotically the more nines you tack on at the end.
This is basic calculus. You'd express the above as the limit of 1/(1-x) as x approaches 1 from zero.
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#92816
01/28/2003 11:46 PM
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old hand
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I disagree that 0.9999999999... is equal to the integer 1Why not, if you will accept the infinite number of zeros, also accept the infinite number of nines?  Something to think about. But I do remember learning, and being upset by, and finally agreeing with, what emanuela said above. Edit: To clarify: you have to differentiate between how we write down a number, and what it is. What emanuela is saying is that for the number which we commonly understand to be the integer 1, there are two ways to write it: 0.9999999999999999... or 1.00000000000...
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#92817
01/29/2003 12:11 AM
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"Math is not an exact science."
-ron obvious
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#92818
01/29/2003 12:16 AM
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#92819
01/29/2003 2:04 AM
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Pooh-Bah
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In reply to:
We would still say we had one couch.
...and that is why I say that they are the same in any conceivable, everyday aspect. Of course if you are talking about real couches, 0.99999999999999 of a couch is macroscopically indistinguishable from 1.00000000 of a couch.
What I am saying is that strictly speaking, in the abstract world of mathematics, there is a difference between the following numbers:
2.99999
3
3.00001
To prove my point, here is a function which will return a different value depending on which of the three above numbers you plug in: f(x)=3/(x-3)
Now if those three numbers were truly the same, then you should get the same result from the function above, but you don't.
3/(2.99999-3) = 3/(-0.00001) = (-300,000)
3/(3 -3) = 3/(0) = not defined/infinity/or, you go to jail for dividing by zero and your mother is very disappointed
3/(3.00001-3) = 3/(0.00001) = 300,000
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#92820
01/29/2003 11:05 AM
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f(x)=3/(x-3)
Which is certainly true if we have a finite number of 9s after the decimal point. If we have an infinite number of 9s, there is an infinite number of 0s before we get to that pesky 1 in the .000000…0001 meaning that we'll never get there.
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#92821
01/29/2003 11:10 AM
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Yeah, that's the difference here, which maybe wasn't clear. The ellipsis after a number in math means "going on to infinity" so 0.99999999... means nines going on to infinity, that is, you never run out of digits. That is emphatically NOT the same as 0.99999999. Which means any kind of numerical example given with a finite number of nines isn't talking about the same thing.
Anyway, I looked it up in my analysis book last night, and you can prove pretty straightfowardly (if you've read the first three chapters of the book) that both decimal representations (that is, 0.999999.... and 1.000000...) are, indeed, EQUAL to the number 1.
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#92822
01/29/2003 11:14 AM
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Another thing I thought of, Alex, is If you don't believe that 0.9999999... is equal to 1, then please tell me how far from 1 it is. That is, what is 1 - 0.9999999... = ???? If you are right, the difference must be equal to some probably quite small but finite number. 
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#92823
01/29/2003 11:50 AM
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0.999... = 1
however
0.99999 <> 1
The repeating decimal is identical to 1, whereas the fixed decimal is not.
let x = 0.999... (where the ellipsis denotes infinite repetition of '9')
so 10x = 9.999...
Clearly, 10x - x = 9x
10x = 9.999...
- x = - 0.999...
------ ----------
9x = 9
Dividing both sides by 9 gives x=1.
Therefore 0.999... is identically equal to 1, mathematically, theoretically, and, in fact. (Well, that's what I think anyway.)
k
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#92824
01/29/2003 1:17 PM
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I have seen in various posts both
0,9999 and
0,9999...(meaning that we are imaginig infinite 9's)
Well, here is the question.
What do you mean by saying that a given number is given by an infinity of digits? In other words, which is the meaning of the decimal representation of a number, when it is not finite?
The number is the limit of the partial ways of writing. In our example, to say that 0,9999999........ = 1
means exactly that 1 is the limit of the succession
0,9
0,99
0,999
0,9999
........and so on
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#92825
01/29/2003 1:31 PM
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that puzzled me a lot, indeed. But the explanation is so easy to seem obvious: roughly speaking, I would explain this way:
Suppose that you are working in the plane - the usual plane, the plane you can imagine , the plane of the blackboard prolonged to the infinity - the so called "affine plane".
Consider there two parallel lines: don't worry, they don't meet.
But,if you consider the so called "projective plane" , obtained from the above plane by adding more "points", each one representing a "direction", the two lines will have the same "point",or, in other words, they have the same direction.
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#92826
01/29/2003 1:48 PM
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Posts: 1,819
Pooh-Bah
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In reply to:
If you don't believe that 0.9999999... is equal to 1, then please tell me how far from 1 it is. That is, what is 1 - 0.9999999... = ???? If you are right, the difference must be equal to some probably quite small but finite number
That is exactly what I am saying. 1- (0.999999...) is an infinitesimally small value not quite equal to zero. When we talk about 0.99999999... we are talking about the limit of x as x approaches 1, which is represented lim(x) x --> 1. This is understood to be infinitesimally close to 1 but not quite one.
The example I offered suffered from a lack of ellipses, but it holds true in differentiating 3.000...0001 from 2.9999999.... In fact, the more zeros or nines in the decimal point, the greater the difference between the two defined outcomes.
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#92827
01/29/2003 2:10 PM
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That is exactly what I am saying. 1- (0.999999...) is an infinitesimally small value not quite equal to zero. When we talk about 0.99999999... we are talking about the limit of x as x approaches 1, which is represented lim(x) x --> 1. This is understood to be infinitesimally close to 1 but not quite one.
I don't think so. The limit is the limit. That is, the limit is taken as exactly equal to one - well, that's the way I learned (but as Bean pointed out, we learn a lot of incorrect stuff).
lim x = 1 (identically)
x->1
For example, we don't say that integrals solved exactly are "approximately equal to blah blah."
The integral of cos(x) is sin(x) + C, not approximately sin(x) + C (and what is an integral, but a limit?).
OTOH, we don't say that differentials are identically equal to zero. It's a bit confusing. Back in HS, before actually taking calc, I read this book by a guy named Boyer called "The History of Calculus and its Conceptual Development." From what I recall (of more than 20 years ago), these concepts of differentials and limits that we take for granted today were a really big deal at one time - very controversial.
I'm a bit envious of Bean for actually having taken Real Analysis. I'm sure she's got a much better handle on the particulars because of it. (I think maybe Real Analysis is - ahem, approximately - to calculus as number theory is to arithmetic.)
k
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#92828
01/29/2003 2:36 PM
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I think my problem with this concept is that saying 0.999999999…=1 seems inelegant to me.
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#92829
01/29/2003 2:45 PM
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1- (0.999999...) is an infinitesimally small value not quite equal to zeroOK, if it's not zero, then what is it? If you're so sure that it's a nonzero value you must be able to tell me what it is! And the leaving-out or writing-in of ellipses makes all the difference here! No ellipsis means that the number ends just where you leave off. When we talk about 0.99999999... we are talking about the limit of x as x approaches 1, which is represented lim(x) x --> 1. This is understood to be infinitesimally close to 1 but not quite one. No, we are talking about the decimal representation of a number, that is, how do we write it? We all have a concept of what the integer "one" means. We are discussing whether it is more valid to write 1.00000.... or 0.999999... to express that number. And the point here is that either way is equally acceptable, both decimals (yes! including the 1.0000000...) must be extended to infinity to correctly represent the integer we all call 1. The proof that shows that writing 0.99999... means the exact same thing as writing 1.00000... shows how to write it as an infinite series (because that's what decimal really is), that is, an infinite sum of numbers. The value of the series approaches 1 as the number of the terms in the sum becomes infinite. For any small difference from 1 that you choose to inquire about, I can always add enough terms that I am closer to 1 than the difference you've specified. (This is more or less the epsilon-delta definition of a limit: you prove that for ANY small difference you are interested in, no matter show small, you can make your function/number/decimal representation get closer to its limit than that specified difference.)
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#92830
01/29/2003 2:49 PM
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0.999999999…=1 seems inelegant to me.So does 0.33333... = 1/3 also seem inelegant? The problem is these silly, finite sheets of paper on which we are supposed to write!
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#92831
01/29/2003 2:58 PM
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I think the problem some of us (non-mathematicians) have with this argument is that we view 1 as a 'pure' number, being unity. we don' need no stinking infinite string of zeros after the dot. it's just one. and to say that .9999999999999999999999999999999999999999999999999999999999999999999999999999... is equal to 1 is just yentzing with perfection.
but then, math isn't an exact science.
-ron o.
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#92832
01/29/2003 3:12 PM
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No one anywhere said it was a better or more convenient way to write it. Certainly the most convenient way to write it is just 1. But the point is that the other way is equally valid. That's all. Even a mathematician probably might not find a good USE for it. But it's important to know it's there!
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#92833
01/29/2003 3:16 PM
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does 0.33333... = 1/3 also seem inelegant?
No it doesn't. The reason that 0.33333333…=1/3 doesn't seem inelegant is that there is no other way to give the decimal representation of 1/3. 1.00000000… is vastly superior to 0.99999999… as a way of representing 1.
There is also the minor problem that there seems to be a conception that adding 9s to 0.99999, even an infinite number of 9s doesn't get to 1. The difference when you get the infinite number of 9s on the 0.9999999… is 0.0000000…0001.
You run into another communication problem using the term integer with computer types who may not have the math background. In computer parlance, 0.99999999… cannot be the representation of an integer. Neither can 1.0000000… There is no provision for representing a fractional part of an integer since there can be no fractional part.
Teach me to take a half hour to finely craft a response
OK, twenty minutes
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#92834
01/29/2003 3:54 PM
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The difference when you get the infinite number of 9s on the 0.9999999… is 0.0000000…0001.
No, it can't be. If you know where to put that last "1" then you don't have an infinite number of digits anymore. That is the difference between the infinite and finite number of digits.
1.00000000… is vastly superior to 0.99999999… as a way of representing 1.
Is it? That's an opinion. When you're talking math, you're not really dealing with opinions. Therefore, as I said to tsuwm above, it may be ugly to use 0.999999..., but that doesn't make it untrue.
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#92835
01/29/2003 4:13 PM
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Pooh-Bah
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Bean's arguments are compelling and elegant but nobody seems to have an answer as to why, if (2.9999.....), the integer 3, and (3.00000...0001) [that is, an infinite number of zeros before the final one]) are the same and truly equal, they produce different answers when plugged into certain functions.
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#92836
01/29/2003 4:18 PM
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>(3.00000...0001) [that is, an infinite number of zeros before the final one])
you lose me with this. how can you have a *final one after an *infinite number of zeros??
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#92837
01/29/2003 4:32 PM
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I think there has been some confusion here. (Which explains why I feel like I am not being understood.) 3.000....00001 is not the same as 3. I meant that 3.000...000...000...000000... is the same as 3, the integer. (no final nonzero digit, just zeros to hell and back and then some more) Just as 2.9999999... is another way to write 3.
I think the crucial thing here is the infinite digits; you simply can't allow yourself to imagine the digits having an ending. Which is super-difficult because we as humans are on the whole pretty bad at imagining infinity. I mean really, really trying to imagine infinity.
I'm not very good at imagining infinity but I can work with it fairly well. Just as, say, a doctor can't really sit down and visualize everything that happens in the human body, all at once in real-time like some kind of bizzarre movie, but they will believe in it anyway and then work with the part(s) that needs working on.
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#92838
01/29/2003 4:33 PM
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a *final one after an *infinite number of zeros?
It ain't easy, but when has Aint Easy stopped a determined mathematician?
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#92839
01/29/2003 5:04 PM
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Pooh-Bah
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In reply to:
>(3.00000...0001) [that is, an infinite number of zeros before the final one])
you lose me with this. how can you have a *final one after an *infinite number of zeros??
Well it seems as easy to visualize (4 minus 0.99999...) as it does to visualize (2.99999999...). They are each numbers that are very close to three, but on opposite sides of the integer three. I was expressing (4-0.99999...) as 3.000...0001. Maybe it would have been more precise to have expressed it as simply (4-0.99999...).
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#92840
01/29/2003 5:32 PM
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blergh.
ya, I'm stoopit, and the ekwazhuns are real purty, but I jus don't see how .999...… anything could be equal to 1
seems to de-feat the purpose of the dessimal point. but then, maybe it ain't no dessimal point neither. could be some other ar-cane cymbal.
I think it's all hocus-pocus.
formerly known as etaoin...
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#92841
01/29/2003 6:04 PM
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Pooh-Bah
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In reply to:
a *final one after an *infinite number of zeros?
Oh yeah I forgot to mention. You just add them on at that middle, right behind the decimal. More seriously, you could do it this way:
1/10^n where n is infinity = 0.0000000000000000...0001
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#92842
01/29/2003 6:06 PM
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1/10^n where n is infinity
Is that equal to zero?
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