jman19
Lifer
- Nov 3, 2000
- 11,225
- 664
- 126
Oh great, here comes the bogus "but people once thought that blah blah blah was true." Yes, not all things that we think are true are actually true, but if we just choose to ignore things without trying to discover the truth through rigorous methods then we are just fooling ourselves...
Your arguments are philosophical, not mathematical.
Mathematically, .9999999... = 1.
Philosophically, I suppose you could state that since they are written differently, they comprise different entities.
But that's like saying that 3 != 3-0, because they're different.
Here's a question for you: is infinity-1 less than infinity?
Well, the common number system is constricted to the real (Archimedian) field... so there are no nonzero infinitely small numbers.
mb
Lifer
- Jun 27, 2004
- 10,233
- 2
- 71
I'm the close minded one because I refuse to believe that two different things are the same because they can end in the same result some of the time?
Right.
I give up. There is no solid proof that they are always equal and I'm not going to waste anymore time arguing about it.
If you seriously believe that all of the world's mathematicians are wrong about 0.99999.... = 1 (as people were wrong about the world being flat centuries ago), then you need to publish your proof in a mathematical journal ASAP, so other mathematicians can see the error of their ways. The results of your study would be earth-shattering, to say the least.
Also, 0.0000....1 is NOT the difference. You are assuming there is an end to the nines, but there is no end. You are using this:
1 - 0.9999999 = 0.0000001
Whereas we are using this:
1 = 0.9999............
Notice in your case you are imposing an end to the number of nines, whereas in our case we are not. The nines really do go on forever without end.
I agree. There is no proof that they are always equal. When working in different (math) fields, lots of things change. But in the real number field, they have been set as always equal.
The same logic that you discredit for the informal proof that 1 = 0.999... is the same logic you are using in that argument...
By taking 1>0.999... as a priori knowledge, you are assuming that 1 != 0.999..., and therefore using the concept you are trying to prove to prove itself.
There is no 1 after 0.999... By the very definition of 0.999..., there are nines and nines and nines, and nothing else, on into infinity. That's what you don't seem to understand. You can't stop the series at some arbitrary point and say "now put a 1 at the end of this". As soon as you stop it, it is no longer 0.999..., it becomes whatever number you decided to use to illustrate your (invalid) point.
hah, thats what happens with any sort of .999... = 1 threads.
At least I answered your OP riddle
At least I answered your OP riddle
Hey supafly...
How would you feel if I told you that 0.000...1 is actually zero?
If the number 0.000...1 equals zero then it also means that .999 repeating does equal one. :laugh:
Let's think about what 0.000...1 really is.
First think about 0.1. It is 1/10.
Next 0.01. That's 1/100.
Ok how about 0.001? 1/1000
See a pattern?
So 0.000...1 is the same as 1/(some insanely huge number) or 1/infinity.
You have to trust me on this one, but 1/infinity is equal to zero.
So 0.000...1 equals zero!!!
This means that the difference between 0.999... and 1 is zero, and therefore 0.999... must equal 1.
Hyperreal numbers are real numbers. Newton and Leibniz used infinitesimals, some mathematicians prefer their use to limits in the Calculus, and they often provide a simpler analysis to Calc problems. But even if you don't use them, the result does not make 0.999... equal to 1.
Here is a quote from a common Calculus book:
"The limit of an infinite series is sometimes called its 'sum at infinity,' but of course this is not a sum in the usual arithmetical sense when the number of terms is finite. You can't obtain the "sum" of an infinite series by adding because the number of terms to be added is infinite. When we speak of the "sum" of an infinite series, this is just a short way of naming its limit...
So what you're really saying when you say 0.999... = 1, is that the limit of the infinite series (denoted by '...')0.999 is 1. This is, strictly speaking, not the same thing as equality.
I believe that saying the Limit of ___ is 1 is the same as saying the Limit of ___ = 1. ('is' == 'equals')
Correct me if I'm wrong, but what you just quoted there indeed says that .999... is equal to 1.
I believe that book is saying that the notation .999...(repeating) is not a number but a representation of a Limit. This limit, as we've all read about to death, equals 1.
Or perhaps what you're saying is that the Limit does not equal 1, but converges at 1 (is that the proper term?). And that convergence is not the same as equality. I can't vouch for that one... might have to ask a mathematician.
Hmm... The number '.999...' itself is defined as a limit (in the reals), so would the evaluation of it (using its own limit) not be the same as its equality?
"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 the sequence never reaches its limit. Those who accept the difference between a sequence of numbers and its limit might read "0.999?" as meaning the former rather than the latter."
-Tall and Schwarzenberger p.6; Tall 2001 p.221
For the infinitesimal stuff... see this property of the real field:
http://en.wikipedia.org/wiki/Archimedean_property
But hey, you may be right... I know very little math and I haven't studied the exact definition of 'equality' yet...
Aquila76
Diamond Member
You give your keys to your best friend to drive the sick old woman to the hospital, and spend your final moments with the girl of your dreams.
I don't know what Conan the Mathematician is... I just got that source from Wikipedia and it seems reputable (http://scholar.google.com/scholar?q=Tal...Schwarzenberger&hl=en&lr=&btnG=Search).
IT DOES NOT TAKE OFF. The aircraft has no LIFT flowing over the wings, and therefore cannot take off!
It's the limit of the function that equals 1, not the function itself.
In other words 'f(x) = y' is not the same as 'lim f(x) as "the counter" approaches infinity =y'
Read this:
http://www.math.wisc.edu/~keisler/chapter_9.pdf
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