June 8, 2011 at 7:35 am
venoym (6/8/2011)
so the definition that allows 0.999... to be equal to 1 is that there is not a number between the two. And I also believe that this is what's called a postulate. Assuming that, could it not be said that any number in between will also be equal to either of those numbers? Using that as a basis, you can argue that 1.00... = 2.00... if only considering integers (yes, I know that we're dealing with infinity which changes any equation) since there is no integer between 1.00... and 2.00... when you get to the infinity position. One could go further and say that there is no difference at the infinity position between any 2 arbitrary numbers because they are equal to all numbers that come in between them...
That's just pure nonsense.
In the words of Wolfgang Pauli:
Das ist nicht nur nicht richtig, es ist nicht einmal falsch!
"Not only is it not right, it's not even wrong!"
June 8, 2011 at 9:41 am
This whole thread has a lot of nonsense in it :hehe:
June 8, 2011 at 10:14 am
Steve Jones - SSC Editor (6/8/2011)
This whole thread has a lot of nonsense in it :hehe:
Speaking of nonsense, an instructor of mine said that people used to ask philosophers and doctors why morphine made them sleepy. Their response would be "...because of morphine's dormative nature." The questioners were happy with that and went their merry way. But "dormative" simply means "to make sleepy" essentially. So actually, they were getting no explanation whatsoever.
The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge. - Stephen Hawking
June 8, 2011 at 10:50 am
mtillman-921105 (6/8/2011)
Steve Jones - SSC Editor (6/8/2011)
This whole thread has a lot of nonsense in it :hehe:Speaking of nonsense, an instructor of mine said that people used to ask philosophers and doctors why morphine made them sleepy. Their response would be "...because of morphine's dormative nature." The questioners were happy with that and went their merry way. But "dormative" simply means "to make sleepy" essentially. So actually, they were getting no explanation whatsoever.
That's like the classic answer a doctor gives to explain the cause of death when they don't really know: heart stoppage.
Doesn't really tell you why the heart stopped: heart attack, gunshot wound, strangulation, poisoning, etc. all produce "heart stoppage".
June 9, 2011 at 6:39 am
Michael Valentine Jones (6/8/2011)
mtillman-921105 (6/8/2011)
Steve Jones - SSC Editor (6/8/2011)
This whole thread has a lot of nonsense in it :hehe:Speaking of nonsense, an instructor of mine said that people used to ask philosophers and doctors why morphine made them sleepy. Their response would be "...because of morphine's dormative nature." The questioners were happy with that and went their merry way. But "dormative" simply means "to make sleepy" essentially. So actually, they were getting no explanation whatsoever.
That's like the classic answer a doctor gives to explain the cause of death when they don't really know: heart stoppage.
Doesn't really tell you why the heart stopped: heart attack, gunshot wound, strangulation, poisoning, etc. all produce "heart stoppage".
Then again, the number 0 was pure nonsense at one point. So was the Arabic numeric system... the roman numeral system... the idea of an atomic bomb. Science declares everything nonsense until it can be explained by science, then it's "well of course!" At one point "modern" science thought the earth was flat and that there was no such thing as gravity (if you disagree with the flat earth definition, the Bible refers to the earth as round in about 1000 BC, while science as late as 1200 AD thought the earth was flat). Just because science or philosophy doesn't understand something doesn't mean it's pure nonsense. The point of introducing nonsense into a conversation or classical debate is to make people think and defend their answers. Labeling something nonsense is avoiding defending your position with the philosophical and logical points needed. An example is this: since when does a scientific consensus mean a debate is over [looking at Al Gore]? It doesn't, it just means the scientific brainiacs of the day agree on something, not that they are right. Science had a consensus that the Doctrine of Humors was the only way to practice medicine... does anyone in a modern hospital practice that now?
June 9, 2011 at 1:19 pm
venoym (6/9/2011)
Michael Valentine Jones (6/8/2011)
mtillman-921105 (6/8/2011)
Steve Jones - SSC Editor (6/8/2011)
This whole thread has a lot of nonsense in it :hehe:Speaking of nonsense, an instructor of mine said that people used to ask philosophers and doctors why morphine made them sleepy. Their response would be "...because of morphine's dormative nature." The questioners were happy with that and went their merry way. But "dormative" simply means "to make sleepy" essentially. So actually, they were getting no explanation whatsoever.
That's like the classic answer a doctor gives to explain the cause of death when they don't really know: heart stoppage.
Doesn't really tell you why the heart stopped: heart attack, gunshot wound, strangulation, poisoning, etc. all produce "heart stoppage".
Then again, the number 0 was pure nonsense at one point. So was the Arabic numeric system... the roman numeral system... the idea of an atomic bomb. Science declares everything nonsense until it can be explained by science, then it's "well of course!" At one point "modern" science thought the earth was flat and that there was no such thing as gravity (if you disagree with the flat earth definition, the Bible refers to the earth as round in about 1000 BC, while science as late as 1200 AD thought the earth was flat). Just because science or philosophy doesn't understand something doesn't mean it's pure nonsense. The point of introducing nonsense into a conversation or classical debate is to make people think and defend their answers. Labeling something nonsense is avoiding defending your position with the philosophical and logical points needed. An example is this: since when does a scientific consensus mean a debate is over [looking at Al Gore]? It doesn't, it just means the scientific brainiacs of the day agree on something, not that they are right. Science had a consensus that the Doctrine of Humors was the only way to practice medicine... does anyone in a modern hospital practice that now?
Just spinning words together is not scientific debate. Making a testable hypothesis, performing an experiment to test the hypothesis, and evaluating the results to see if they support the hypothesis is the heart of scientific debate.
In the realm of math, providing a proof to a hypothesis is the way things are done, and people have been proving that .999... = 1 for about 3 centuries, so if you have a disproof to offer, you should publish it.
If you make the assertion that "1.00... = 2.00" then I think it is up to you to offer some proof that a concept as established as 1+1=2 is not true if you don't want if called nonsense.
If you are going to say "One could go further and say that there is no difference at the infinity position between any 2 arbitrary numbers because they are equal to all numbers that come in between them." you need to explain what that even means if you don't want it called nonsense.
And while we are at it, this statement is just wrong: "...while science as late as 1200 AD thought the earth was flat". Eratosthenes of Cyrene proved the earth was round and calculated the circumference of the earth around 200 BC. Ancient people were aware that the earth was a sphere even before then, and educated people knew since ancient times the shape and size of the earth. People were not stupid; they could just climb to the top of a mountain and see the curve of the earth or look up in the sky to see the round shadow of the earth crossing the moon during an eclipse.
June 9, 2011 at 3:30 pm
Science declares everything nonsense until it can be explained by science, then it's "well of course!"
Science is a process for discovery, a methodology for explaining phenomenon, not a organizational body that makes declarations.
At one point "modern" science thought the earth was flat
There are still people who think this, look up the Flat Earth Society.
and that there was no such thing as gravity
When did anyone think this?
the Bible refers to the earth as round in about 1000 BC,
Umm, what?
while science as late as 1200 AD thought the earth was flat).
MVJ addressed this appropriately.
Just because science or philosophy doesn't understand something doesn't mean it's pure nonsense.
It also doesn't mean that nonsense doesn't exist. 🙂
... An example is this: since when does a scientific consensus mean a debate is over [looking at Al Gore]?
What are you referring to? Consensus?
______________________________________________________________________________
How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.
June 9, 2011 at 3:32 pm
Steve Jones - SSC Editor (6/8/2011)
This whole thread has a lot of nonsense in it :hehe:
Don't dodge the question, Steve Jones, does 1=0.999...?
😀
______________________________________________________________________________
How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.
June 10, 2011 at 12:56 pm
toddasd (6/8/2011)
GSquared (6/8/2011)
toddasd (6/7/2011)
Richard Warr (6/6/2011)
toddasd (6/6/2011)
Answer this: do you consider this to be true: 1/3 = 0.333...?No, however that is the only way you can represent 1/3 as a decimal.
That's a great reply, Richard. It shows where you draw the line at what you are willing to accept. How do you explain pi as a non-repeating, non-terminating decimal? An approximation?
Yes. It's an approximation untill you get to the infinitieth decimal place. Since that's impossible by definition, it is and will always be an approximation.
No. Any decimal representation of pi is an approximation. The number pi is an exact, specific number (a determined single point on a number line.)
Exactly. He asked about "a ... decimal". I answered about that. If we take pi as a separate number all by itself, it's mathematically exact. Nobody is quite sure what it's mathematically exact value is, but it is exact. It can be approximated by fractions and decimals, but they are approximate.
- Gus "GSquared", RSVP, OODA, MAP, NMVP, FAQ, SAT, SQL, DNA, RNA, UOI, IOU, AM, PM, AD, BC, BCE, USA, UN, CF, ROFL, LOL, ETC
Property of The Thread
"Nobody knows the age of the human race, but everyone agrees it's old enough to know better." - Anon
September 4, 2011 at 6:15 am
I wish I'd known about this thread when it was going on - it's my kind of nonsense.
Personally, I think the question "does .999... = 1 ?" is actually two questions., first a question about language (or about notation) and second a question about limits of sequences.
I answer the only complicated bit of the language question by saying that ".999..." means "the limit of the infinite series whose nth member is the sum of the n-term (hence finite) series whose 1st member is .9 and whose j+1-th member for every integer j greater than 0 is its j-th member deivided by 10".
If recurring decimals were introduced in school after the concepts of series and limits so that this sort of explanation could be given for them, maybe fewer kids would get confused about them; but of course most people teaching teaching kids young enough to easily grasp these concepts are mathematically illiterate, and anyway the curricula appear to be designed by math-hating lunatics bent on reducing each child's chances of understanding to a minimum, so things are almost always introduced in the wrong order. Historically, infinte series, their limits, and the series formed by their partial sums, were around a very long time before any decimal positional notation (Al Kashi wrote the first surviving material about this in the early 15th century AD, but one essential factor - the point to mark the base position - was not introduced until the early 17th century, while you can find limits of series of sums in Archimedes' work).
The question about limits is easy: it's easy to see that on the real number line with its natural topology there is exactly one point which has no open neighborhood that does not contain a member of the sequence, and every open neighborhood of that point contains all but a finite number of the elements of that sequence; and equally easy to see that this point is the one for the number 1.0. So the sequence has a limit and the limit is 1.
Since the limit of that sequence is what ".999..." means, ".999..." and "1" mean the same thing as each other, and this fact is expressed by writing ".999... = 1".
This is an exact equality, not any kind of approximation. ".999..." means the limit of the sequence, i.e. 1, not any of member of the sequence.
If one wants this to work in nonstandard analysis (where there are infinite integers as well as finite ones - new mathematics in the early 60s) as well as in classical real analysis one has to be careful in translating the definition of ".999..." - just as one has to be careful with any translation from one universe of discourse to another: for example a naive translation of the continuity definition from classical analysis to nonstandard results in the interesting result that all continuous functions are uniformly continuous (which may be regarded as somewhat undesirable since the whole point of nonstandard analysis was to provide a basis for real analysis without all the pain introduced by Weierstrass to avoid infinitesimals). See Abraham Robinson's "NonStandard Analysis" for all the gory detail.
I was much taken by Gus's approach to this - definitely starting from the "mathematics is a language" position: I disagree slightly on that, as I think mathematics is a collection of languages, not a single language. But I disagree with him about the relation between mathematics and logic - formal logic is a branch (or several branches) of mathematics, a mathematical tool-set for the examination of mathematics; and Russel's communication to Frege was the communication of a failure of the axiomatisation of set theory which Frege wanted to use (in that the axioms could not be modelled - the axioms were inconsistent, contradictory) which was not something caused by confusing maths and logic. The interaction of the general dislike of Hilberrt's new style of existence proof (existence proofs that gave not a clue how to get one's hands on the thing proven to exist) with this failure of axiomatisation led to one of the great mathematical movements of the 20th century, the constructivist program, using a new logic with infinitely many different truth values; while those who wanted to continue using two valued logic produced many new axiomatisations of set theory in attempts to get rid of the paradox while retaining most of the meat. Anyone who wants to see one of the most spectacularly complicated attempts to do the latter could read Russell and Whitehead's Principia Mathematica volumes one, two, and three but only if they have (a) a very high boredom threshold, (b) a deep pocket or access to a decent library, and (c) a lot of spare time. (I read it when I was 17 - I was too young to know better - I could more usefully have spent the time on Latin or Russian, but mathematical logic had hooked me.)
I disliked Gus's oversimplification of a topological limit in the reals (or in the rationals, it doesn't matter which) to someting like "there isn't a number in between them", not least because someone immediately applied it to the integers (where that isn't equivalent to a statement about limits); if he had used the topological formulation of limit applying it to the integers would have produced the correct result (1 is not the limit of the sequence <2,2,2......> because the natural topology of the integers is the discrete topology, so the set {N: N is an integer other than 2} is an open neighbourhood of 1 that contains no element of the sequence; in fact it's an open neighborhood of every integer except 2, and contains no member of the sequence, so no number other than 2 could be a limit of the sequence). So using a more general statement instead of the one specialised to the case where "between" was relevant might have spared us that particluar oddity (which was presumably raised specifically to kick at Gus's simplification anyway).
Tom
September 6, 2011 at 8:49 am
"I was much taken by Gus's approach to this - definitely starting from the "mathematics is a language" position" ~ Tom
Tom, since beginning with Nietzsche, who was also a philologist, the study of language as our tool of reasoning began in philosophy in general. Other philosophers ran with the idea and it can be blamed, in part, for the deconstructionist movement that you mentioned in the another thread.
With this idea, each and every word we use is a mini-theory. Words are, after all, what we use to reason with (and symbols are included here - they just stand for words). So, with this view, it encompasses not just math, but every word we utter is "a language."
But getting back to the deconstrucionists, I don't know a lot about them, but so far I'm not very impressed. I agree that their nihilism is unfounded although, speaking of skeptics in general, Hume was right about almost everything he said. (I would encourage everyone to understand Hume's destruction of causality, it's one of the most profound insights ever, if you ask me).
The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge. - Stephen Hawking
September 6, 2011 at 11:00 am
Evil Kraig F (6/2/2011)
mister.magoo (6/2/2011)
For anyone who thinks that an infinitely recurring number can simply be redefined as another number, I ask this question:Does PI (3.14159 etc ) which never repeats = 3.141.... (which is the same accuracy as the original question)?
I hope for the future of engineering that the answer is a resounding no!
There is a difference in the .999_ = 1 and the never-ending Pi. The Pi equivalent of that same theorum is
Pi = pi +/- 0.0_1, with the 0 repeating to infinity. It's the negligible space at the tail that throws in the whirlygig. .00059 is not negligible.
There is a difference only in the previously agreed to tolerance. What's "negligible" is an implementation question, NOT an absolute answer.
both representations carry the same "rounding" problem if you want to look at it. without the tolerance, both representations are equally worthless (from an engineering point of view).
----------------------------------------------------------------------------------
Your lack of planning does not constitute an emergency on my part...unless you're my manager...or a director and above...or a really loud-spoken end-user..All right - what was my emergency again?
September 6, 2011 at 11:40 am
Tom.Thomson (9/4/2011)
I wish I'd known about this thread when it was going on - it's my kind of nonsense.Personally, I think the question "does .999... = 1 ?" is actually two questions., first a question about language (or about notation) and second a question about limits of sequences.
I answer the only complicated bit of the language question by saying that ".999..." means "the limit of the infinite series whose nth member is the sum of the n-term (hence finite) series whose 1st member is .9 and whose j+1-th member for every integer j greater than 0 is its j-th member deivided by 10".
If recurring decimals were introduced in school after the concepts of series and limits so that this sort of explanation could be given for them, maybe fewer kids would get confused about them; but of course most people teaching teaching kids young enough to easily grasp these concepts are mathematically illiterate, and anyway the curricula appear to be designed by math-hating lunatics bent on reducing each child's chances of understanding to a minimum, so things are almost always introduced in the wrong order. Historically, infinte series, their limits, and the series formed by their partial sums, were around a very long time before any decimal positional notation (Al Kashi wrote the first surviving material about this in the early 15th century AD, but one essential factor - the point to mark the base position - was not introduced until the early 17th century, while you can find limits of series of sums in Archimedes' work).
The question about limits is easy: it's easy to see that on the real number line with its natural topology there is exactly one point which has no open neighborhood that does not contain a member of the sequence, and every open neighborhood of that point contains all but a finite number of the elements of that sequence; and equally easy to see that this point is the one for the number 1.0. So the sequence has a limit and the limit is 1.
Since the limit of that sequence is what ".999..." means, ".999..." and "1" mean the same thing as each other, and this fact is expressed by writing ".999... = 1".
This is an exact equality, not any kind of approximation. ".999..." means the limit of the sequence, i.e. 1, not any of member of the sequence.
If one wants this to work in nonstandard analysis (where there are infinite integers as well as finite ones - new mathematics in the early 60s) as well as in classical real analysis one has to be careful in translating the definition of ".999..." - just as one has to be careful with any translation from one universe of discourse to another: for example a naive translation of the continuity definition from classical analysis to nonstandard results in the interesting result that all continuous functions are uniformly continuous (which may be regarded as somewhat undesirable since the whole point of nonstandard analysis was to provide a basis for real analysis without all the pain introduced by Weierstrass to avoid infinitesimals). See Abraham Robinson's "NonStandard Analysis" for all the gory detail.
I was much taken by Gus's approach to this - definitely starting from the "mathematics is a language" position: I disagree slightly on that, as I think mathematics is a collection of languages, not a single language. But I disagree with him about the relation between mathematics and logic - formal logic is a branch (or several branches) of mathematics, a mathematical tool-set for the examination of mathematics; and Russel's communication to Frege was the communication of a failure of the axiomatisation of set theory which Frege wanted to use (in that the axioms could not be modelled - the axioms were inconsistent, contradictory) which was not something caused by confusing maths and logic. The interaction of the general dislike of Hilberrt's new style of existence proof (existence proofs that gave not a clue how to get one's hands on the thing proven to exist) with this failure of axiomatisation led to one of the great mathematical movements of the 20th century, the constructivist program, using a new logic with infinitely many different truth values; while those who wanted to continue using two valued logic produced many new axiomatisations of set theory in attempts to get rid of the paradox while retaining most of the meat. Anyone who wants to see one of the most spectacularly complicated attempts to do the latter could read Russell and Whitehead's Principia Mathematica volumes one, two, and three but only if they have (a) a very high boredom threshold, (b) a deep pocket or access to a decent library, and (c) a lot of spare time. (I read it when I was 17 - I was too young to know better - I could more usefully have spent the time on Latin or Russian, but mathematical logic had hooked me.)
I disliked Gus's oversimplification of a topological limit in the reals (or in the rationals, it doesn't matter which) to someting like "there isn't a number in between them", not least because someone immediately applied it to the integers (where that isn't equivalent to a statement about limits); if he had used the topological formulation of limit applying it to the integers would have produced the correct result (1 is not the limit of the sequence <2,2,2......> because the natural topology of the integers is the discrete topology, so the set {N: N is an integer other than 2} is an open neighbourhood of 1 that contains no element of the sequence; in fact it's an open neighborhood of every integer except 2, and contains no member of the sequence, so no number other than 2 could be a limit of the sequence). So using a more general statement instead of the one specialised to the case where "between" was relevant might have spared us that particluar oddity (which was presumably raised specifically to kick at Gus's simplification anyway).
Definitely agree with you that math is a multitude of languages. After all, the Calculus is no more the same "language" as simple arithmetic, than English is the same as Latin. They're relatives, so to speak, but not the same.
Later point: I don't think I'm the one who pointed out the lack of distance between the two numbers. I think that's a slightly sloppy way to express the point, since it doesn't map to "common human thought" directly. After all, if you clasp your hands together, there really isn't "space" between them, to common perception, but they are easily differentiable from each other even so. Hence, my argument on the equality of the two was not based on the numeric sets theory representation, it was based on common definition. If 1/3 = .33..., then 1 = .99..., by definition. It's a terminology point. Much easier for most people to grasp, and very easy to demonstrate. I've been able to successfully convince people with very sub-normal intelligence (as measured by IQ and scholastic aptitude; so-called "special needs students"), using a very simple demonstration using that methodology. I only mention that because it demonstrates the simplicity of the concept and how easily it can be grasped. To me, that's an advantage.
I'm not sure what precisely you're disagreeing with on the relationship between math and logic. Too much water under the bridge in this discussion for me to pinpoint what I asserted that you're disagreeing with, so I can't really respond cogently to your data on that point.
- Gus "GSquared", RSVP, OODA, MAP, NMVP, FAQ, SAT, SQL, DNA, RNA, UOI, IOU, AM, PM, AD, BC, BCE, USA, UN, CF, ROFL, LOL, ETC
Property of The Thread
"Nobody knows the age of the human race, but everyone agrees it's old enough to know better." - Anon
September 6, 2011 at 12:51 pm
Steve Jones - SSC Editor (6/7/2011)
Craig Farrell (6/6/2011)
calvo (6/6/2011)
GSquared (6/6/2011)..."Billion" means two different things on different sides of The Pond.
wow. I was about to question this when I looked it up, totally threw me for a loop.
How can
1 billion apples = 1,000,000,000 apples in the US
1 billion apples = 1,000,000,000,000 apples in the UK
How is this possible! We're not even dealing with infinity here, it's a definite finite amount! Is this just a difference of meaning in the same word?
*blinks* What.
Brits can't count properly. Or spell color.
Quite right.
We can't spell "color" because there's no such word in the English language (which is neither spoken nor written in the USA).
And we can't count properly because in 1974 our stupid government adopted the rotten American system in which a billion is a mere thousand million (a milliard) instead of a million million (along with the other stupidities that entailed, like trillion = million million instead of million million million, and so on).
But we CAN spell "colour" and until 1974 we used to count properly.
Tom
September 6, 2011 at 1:21 pm
Richard Warr (6/6/2011)
Mathematically we've seen how this can be "proved". But using pure logic we're essentially saying that a number that is definitely less than 1 is the same as 1.Or to put it another way, when you get to infinity, throw away the rulebook.
EDIT - Does the following disprove it?
Is 0.9 = 1? No
Is 0.99 = 1? No
Is 0.999 = 1? No
So, at what point does 0.999.... = 1? Never.
The first three lines don't have repeat marks, so you are comparing notational apples with notational oranges. Using • as decimal point and ¯ as infinite repetiton mark what you should be looking at is
a)Is 0•9 = 1? No
b)Is 0•9 = 1¯? Yes
c)Is 0•99 = 1? No
d)Is 0•99 = 1¯? Yes
e)Is 0•999¯ = 1? Yes, because it is means the same as lines b and d above and nothing like lines a and c above.
0•999¯ isn't 0• followed by any finite string of 9s; it is 1. that's what it means. If you want it to mean something else, go away and invent a new language, a new notation - don't ask mathematicians to change their language and notations which have gained world-wide acceptance over centuries. And please don't tell us that you know better than mathematicians what expressions in mathematical language written using mathematical notation mean.
Of course if you really think that 1 is "definitely less than 1", as you seem to be asserting if you do understand the notation, you probably need help of a non-mathematical kind.
Tom
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