June 3, 2011 at 7:49 am
calvo (6/3/2011)
...-".999 repeating will never equal 1, it's always .000...1 short"
This is the problem at hand for your colleagues. They are thinking 0.999... as a *process* instead of simply a number. They are picturing in their mind's eye a piece of paper doing the division, coming up with a remainder, and continuing. So when asked to compare, they have to stop the process and compare *that* number (which is not 0.999... but some terminating decimal that starts with 0.999). There never is a 0.000...1 anywhere. You have to consider the infinite repeating as a whole.
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How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.
June 3, 2011 at 8:25 am
mister.magoo (6/3/2011)
This is definitely going to be my last post on this thread because I am getting an ache from having my tongue in my cheek for so long 😛...are not the same, I would like to see your disproof of the simple algebraic proof from my first post, and your disproofs of the numerous other proofs on the link I provided.
Are you really that confident of your working here ?
showing two different notations, one of which is an approximation and saying they are the same does not really constitute a proof does it?
Or if they are really not the same, then what is the result of this expression?
1 - .999...
After all, if they are different, then the difference between the two must be some non-zero value.
Yes it is a non-zero value, but I do not have the symbology to write it down, however that does not negate it's own reality.
Or look at this variation or the proof:
1/3 = 0.333...
so 3*(1/3) = 3*0.333... = 0.999...
since 3*(1/3) = 3/3 = 1,
then 1 = 0.999...
Again, just writing different numbers as equal and multiplying them by 3 does not prove anything to me. 0.333... is an approximation of 1/3, not an equality.
Agreed, it can be considered to be "equal enough" in most circumstances, but it needs to be accompanied by a degree of accuracy statement.
Finally, consider the following axiom:
Q: How many mathematicians does it take to screw in a lightbulb?
A: 0.999999....
A: Show me how they got into the light bulb!
Seriously, I will not post again to this thread because I do not want to get into a war over this - everything I have said is just my opinion and I was only joking when I said that anyone who disagreed was wrong, I apologise for the hurt this has caused.
I hope you all continue to enjoy the discussion and I will read it with interest.
Humbly ....
You are just playing word games without offering any supporting proof.
Of course you don't have the "symbology" to write the non-zero value for 1 - .999... since the value is zero. It's easy to show that any non-zero difference you might propose is not possible.
0.333... is not an approximation of 1/3, 0.999... is not an approximation of 1, just like 1.000... is not an approximation of 1, and 0.000... is not an approximation of 0. They are simply alternate ways of representing the same number. The fact that .999... represents an infinite number of digits does not mean the value is not 1; it is just expressing it as the sum of an infinite series.
Seriously, there are so many well known proofs for this that I feel silly getting involved, especially since this is just a classic Internet board troll, and I'm sure that was the intention of the OP.
June 3, 2011 at 8:33 am
GSquared (6/2/2011)
toddasd (6/2/2011)
GSquared, I think it would be a wonderful thing to sit at a bar and have some beers with you. We would probably spend the whole night debating when a "beer" ceases to be a "beer". 😉Oh, and on the beer thing, my only philosophy on beer is that the human kidney is the world's most efficient machine for turning European beer into American beer.
Honestly though, I'm going to take that whole bit as a compliment. Thank you.
And, yes, I love discussing things like this.
Yep, same, I need flavor in my beer. I can't stand most mass-brewed American beers that most drink just cause they need to have the image of drinking a beer.
I have a weakness for draft Killians Irish Red (yes, bought out and now made by Coors, but that's different.) Shiner Bock from Texas is a decent brand. Abita here in Louisiana makes some pretty good flavors.
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How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.
June 3, 2011 at 9:03 am
Michael Valentine Jones (6/3/2011)
Seriously, there are so many well known proofs for this that I feel silly getting involved, especially since this is just a classic Internet board troll, and I'm sure that was the intention of the OP.
Posting in the appropriate forum, I'm not trolling. As I've said, it's a very intriguing conversation. I'm more interested in the theory, philosophy, and tightly held beliefs of those unwilling to accept the truth than the proofs themselves.
It is very interesting to hear very smart people deny logic and to themselves sound so sure and be quick to just end it with "well, you're wrong and I'm right." and not even offer the typical "let's agree to disagree." I'm more interested in why people think the way they do than what they think at all.
June 3, 2011 at 9:36 am
calvo (6/3/2011)
Michael Valentine Jones (6/3/2011)
Seriously, there are so many well known proofs for this that I feel silly getting involved, especially since this is just a classic Internet board troll, and I'm sure that was the intention of the OP.Posting in the appropriate forum, I'm not trolling. As I've said, it's a very intriguing conversation. I'm more interested in the theory, philosophy, and tightly held beliefs of those unwilling to accept the truth than the proofs themselves.
It is very interesting to hear very smart people deny logic and to themselves sound so sure and be quick to just end it with "well, you're wrong and I'm right." and not even offer the typical "let's agree to disagree." I'm more interested in why people think the way they do than what they think at all.
It's not actually an "agree to disagree" situation. It's a "define your terms" situation.
If you use the standard definitions from the language "mathematics", then .9... = 1. If you state it does not, then you need to state, (a) what language you are using other than "mathematics", and (b) what the postulates (definitions of terms) are in that language.
"Postulates" are data that are true by their very definition. "All triangles have three sides" is true because triangles are defined as having three sides. Needs no further proof.
Any number is can be defined as the sum of its component parts is a postulate, because that's what numbers are by definition. (2 is defined as 1 + 1. There is no way for 2 != 1 + 1, unless you redefine 2 or 1 or + or = symbols.)
Any number divided and multiplied by the same finite number is equal to the orginal number because of that. 3 thirds is equal to 1 by definition of what a "third" is and by definition of what "1" is. No further proof needed, those are "by definition" postulates. To violate them, you have to redefine terms.
Redefining terms isn't "wrong", but it doesn't disprove a prior postulate, it just creates a new one. At one point in time, Greek mathematicians argued over whether 1 was a valid number or not.
- 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
June 3, 2011 at 9:56 am
mister.magoo (6/3/2011)
Finally, consider the following axiom:
Q: How many mathematicians does it take to screw in a lightbulb?
A: 0.999999....
A: Show me how they got into the light bulb!
great answer!!
June 6, 2011 at 9:14 am
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.
June 6, 2011 at 10:00 am
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.
Yes! I like the try, Richard, but you didn't go far enough in your proof. Add
Is 0.9999 = 1? No
Is 0.99999 = 1? No
Is 0.999999 = 1? No
...TO INFINITY. YES.
This is what I'm saying in my previous post. The mind wants to consider this a process of adding another 9 to the end of the decimal. But you have to consider the number at infinity *as the entire number*. You've already made up your mind by saying "...a number that is definitely less than 1 is the same as 1". Definitely??? :unsure:
Also, do you consider "mathematics" != "pure logic"? Then that's an underlying problem that needs to be fixed first. 😉
Answer this: do you consider this to be true: 1/3 = 0.333...?
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How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.
June 6, 2011 at 10:02 am
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.
It's not definitely less than 1; you are just making an unfounded assertion. It's not reasonable to say it is mathematically proved and then assert that pure logic says otherwise.
I think your doubts have more to do with the following:
http://en.wikipedia.org/wiki/0.999...
"Skepticism in education
Students of mathematics often reject the equality of 0.999... and 1, for reasons ranging from their disparate appearance to deep misgivings over the limit concept and disagreements over the nature of infinitesimals. There are many common contributing factors to the confusion:
Students are often "mentally committed to the notion that a number can be represented in one and only one way by a decimal." Seeing two manifestly different decimals representing the same number appears to be a paradox, which is amplified by the appearance of the seemingly well-understood number 1.
Some students interpret "0.999..." (or similar notation) as a large but finite string of 9s, possibly with a variable, unspecified length. If they accept an infinite string of nines, they may still expect a last 9 "at infinity".
Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit."
June 6, 2011 at 10:09 am
toddasd, then how do you resolve one of Zeno's paradoxes? Basically (and mathematically), if one runner is catching up with another by half the distance per minute, then how long will it take for the runner to catch up with the other? According to strict math, the runner would never catch up. But we know that it isn't true from a practacle standpoint. The difference becomes so small that it makes no discernable difference. There are other paradoxes like that too by the way.
The greatest enemy of knowledge is not ignorance, it is the illusion of knowledge. - Stephen Hawking
June 6, 2011 at 10:27 am
mtillman-921105 (6/6/2011)
toddasd, then how do you resolve one of Zeno's paradoxes? Basically (and mathematically), if one runner is catching up with another by half the distance per minute, then how long will it take for the runner to catch up with the other? According to strict math, the runner would never catch up. But we know that it isn't true from a practacle standpoint. The difference becomes so small that it makes no discernable difference. There are other paradoxes like that too by the way.
It is true from a practical standpoint. If the runner follows the rule of "run half the distance", then he will never get there. Will he do that? No. He will run normally and catch up. But has he broken the laws of mathematics? Of course not. The conclusion of this paradox is that ALL motion is an illusion, no one can go anywhere because you always have to go half the distance first! :w00t:
I think Michael Valentine nailed it with the "Skepticism in education" post. My friend asked me the same thing at lunch on Friday, "Can you give me 0.999... of a penny? " I handed him a penny. He didn't get it.
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How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.
June 6, 2011 at 10:28 am
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.
June 6, 2011 at 10:30 am
toddasd (6/6/2011)
That's where we differ. I would have taken a penny, shaved an infinitessimally small piece off it, and given him that 😉
June 6, 2011 at 10:39 am
mtillman-921105 (6/6/2011)
toddasd, then how do you resolve one of Zeno's paradoxes? Basically (and mathematically), if one runner is catching up with another by half the distance per minute, then how long will it take for the runner to catch up with the other? According to strict math, the runner would never catch up. But we know that it isn't true from a practacle standpoint. The difference becomes so small that it makes no discernable difference. There are other paradoxes like that too by the way.
There's an infinity of difference between "...so small that it makes no discernable difference. ..." and "zero difference". Unless we assume space has an absolute minimum value, in which case, once the fraction of difference is less than half of that value, it will be forced to become zero difference, because of rounding. But that's an unnecessary assumption.
Again, the problem is assuming "mathematics" has some sort of idependent existence in some theoretical "objective universe". Math is a language. It is arbitrary, as languages must be. There is no math in whatever "objective reality" is. It's a mental construct, and it's terms are based on postulated definitions.
Assuming Zeno's paradox applies to this also assumes that 1 != 1 for any value of 1. If you can't see that, then you need to simply study up on the postulated definitions of numbers. (I've already gone over this in prior posts in this thread to the extent that I care to for this purpose. Further study is warranted if my explanation isn't complete enough for you.)
- 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
June 6, 2011 at 10:42 am
(Breaking my own promise to keep out - sorry 😛 )
This thread is RBAR... I have a set based solution:
No tally tables!
MM
select geometry::STGeomFromWKB(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iewing 15 posts - 31 through 45 (of 158 total)
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