June 7, 2011 at 3:43 pm
I don't know mathematical theory, or philosophy, or physics - or epistemology for that matter - but what a great thread. Thought provoking and entertaining. I would add my own 1.999... cents, but I fear I would simply reveal my own naivete. 😛
Chris
June 7, 2011 at 7:10 pm
Just for arguments sake, how can .999..... = 1?
It will always be .111...short of 1.
It may come close but not equal 1.
*runs and hides*
June 7, 2011 at 9:31 pm
robert.williams 82816 (6/7/2011)
Just for arguments sake, how can .999..... = 1?It will always be .111...short of 1.
It may come close but not equal 1.
*runs and hides*
Nice try at the absurd. :ermm:
Actually, adding .999... and .111... is another path to proof that .999... = 1
Adding the two series together gives:
(.9+.1) + (.09+.01) + (.009+.001) + (.0009+.0001) + ...
which reduces to:
(1) + (.1) + (.01) + (.001) + ...
which reduces to:
1.111...
Since .999... + .111... = 1.111...
subtracting .111... from both sides of the equation gives:
(.999... + .111... ) - .111... = (1.111...) - .111...
which reduces to:
.999... = 1
June 7, 2011 at 10:39 pm
Barry Mazur explains the concept of why 0.999... = 1 in his book Imagining Numbers: (particularly the square root of minus fifteen). If you had to place 0.999... and 1 on a number line, they would occupy the same position because no matter the scale, there is no value between 0.999... and 1.
June 7, 2011 at 11:07 pm
So based on what I'm seeing with my simpleton eyes, 1.111... is also equal to 1.
June 7, 2011 at 11:23 pm
robert.williams 82816 (6/7/2011)
1.111... is also equal to 1.
No, because 1.01 and 1.011 and 1.001 (you get the idea) can be placed between 1.111... and 1
June 7, 2011 at 11:27 pm
Ah....there comes my argument then, that there is a point-something like .01.. or .011.. between .999... and 1.
June 7, 2011 at 11:40 pm
remember that the ... (repeating) means repeat for infinity, so it impossible to find the value between 0.999... and 1. Unless we use another notation, does 0.000...1 + 0.999... = 1? Not sure that 0.000...1 is accepted notation, though 0.000...1 would be an interesting value. So if we accept the concept of 0.000...1 then yes there is something between 0.999... and 1 otherwise 0.999... = 1.
However if we did accept 0.000...1 could you say that 0.000...1 = 0?
June 8, 2011 at 6:11 am
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.
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June 8, 2011 at 6:12 am
robert.williams 82816 (6/7/2011)
Just for arguments sake, how can .999..... = 1?It will always be .111...short of 1.
It may come close but not equal 1.
*runs and hides*
Buy a better calculator while you're hiding! 😀
- 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
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June 8, 2011 at 6:13 am
robert.williams 82816 (6/7/2011)
So based on what I'm seeing with my simpleton eyes, 1.111... is also equal to 1.
Better glasses needed maybe?
- Gus "GSquared", RSVP, OODA, MAP, NMVP, FAQ, SAT, SQL, DNA, RNA, UOI, IOU, AM, PM, AD, BC, BCE, USA, UN, CF, ROFL, LOL, ETC
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"Nobody knows the age of the human race, but everyone agrees it's old enough to know better." - Anon
June 8, 2011 at 6:24 am
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.
Not saying that I agree with either proposition, but I understand how lay people see the argument as well as understanding the philosophical background of the argument.
June 8, 2011 at 6:29 am
Jason L (6/7/2011)
remember that the ... (repeating) means repeat for infinity, so it impossible to find the value between 0.999... and 1. Unless we use another notation, does 0.000...1 + 0.999... = 1? Not sure that 0.000...1 is accepted notation, though 0.000...1 would be an interesting value. So if we accept the concept of 0.000...1 then yes there is something between 0.999... and 1 otherwise 0.999... = 1.However if we did accept 0.000...1 could you say that 0.000...1 = 0?
Actually, there's a guy in England who argues that 0.0...1 does equal 0, calls it "nullity", teaches it to middleschoolers as "the greatest advance in mathematics in centuries", and uses it to solve all kinds of equations. The minor detail that he's (a) wrong, (b) didn't bother to look up centuries of work on the subject, (c) doesn't know what the whole point of "0" is in the first place, (c) solves equations by saying, "as soon as we don't know the value for both sides, they are ballanced and thus solved", and (d) well, there really isn't a point (d), I just got carried away there for a second.
0.0...1 is one of those imaginary numbers that's mathematically impossible, like the square root of -1. It has some interesting implications, but (unlike i) I don't think any practical applications have been found yet. The problem with it is that it violates the basic definition of numbers, which includes being able to resolve them as the sum of their component parts.
If it weren't both impossible, undefinable, imaginary, and a violation of the whole point of counting/numbers, it still wouldn't reside between .9... and 1, by the definition of what 1 is.
The whole .9... thing, as well as in the infinity of pi's decimal places, et al, is caused by weaknesses of various numerical systems.
1/2 is easy to represent in a decimal system, since 5 is half of 10, and decimal is base-10. 1/3 isn't, because there's no whole number that's 1/3 of 10. That's really all it breaks down to. In a base-3 system, 1/3 = .1, and it's very easy to add .1 + .1 + .1 and get 1 in that system. There is no approximation or repetition issue. On the other hand, just try working out one-half in base-3. (The Sumerians used a base-6 and base-12 counting system, which is where we ended up with our hours of the day from. 1/3 was no problem for them to represent. They had issues with other numbers.)
Of course, you have to keep in mind that "half" and "third" and such are arbitrary constructs as well. Some just don't communicate well in decimal format. Some are fine in numerator-denominator format, but that adds complexity to abacuses and digital computers, and engineers haven't been able to solve that problem yet.
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June 8, 2011 at 6:59 am
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.)
______________________________________________________________________________
How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.
June 8, 2011 at 7:18 am
robert.williams 82816 (6/7/2011)
Ah....there comes my argument then, that there is a point-something like .01.. or .011.. between .999... and 1.
That's not really an argument, since that would require some suggestion of how to prove it.
It is easy to show that any number that you might suggest is between .999... and 1 is not.
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