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0.999... = 1 ?
Nerdtendo
Registered User regular
Registered User regular
An interesting thread that popped up on Ebaums forums:
The question is simple, really. Are 0.999... and 1, for all intensive purposes, equal in value?
The question is simple, really. Are 0.999... and 1, for all intensive purposes, equal in value?
Nerdtendo on
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Also, it's "all intents and purposes".
Also, it's all intents and purposes.
Edit: I am not quick enough for this forum.
Just... to mix things up.
Mathematically, though? Yes, aside from in set theory where they would be two unique elements with the same properties except for name. They'd be different there.
But uh, for most intents and purposes, they're the same-- the situations where they aren't are contrived just for the sake of being contrary.
Edit: Even 'in set theory' they're the same numbers. As subsets of the real numbers, {1} = {0.999...} = {1, 0.999...}. (Because sets can't have the same thing in more than once, so since 0.999... is 1, {1, 0.999...} = {1, 1} = {1}.)
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No, they're not. Mathematically, if you achieve .9 repeating, you can round it, and claim it's pretty much 1, but it's wholly illogical to say .9 = 1
1.1 =! 1
So there.
No one's saying .9 = 1.
It's not .9, it's 0.999...
I've had people who took A-level Mathematics convinced that it wasn't the case, though. Some odd arguments arose from that...
Yes, this. The trick is that the set of real numbers does not require that each member have a unique decimal expansion.
1-0.999...=0.0....
0.0...=0
1-1=0
1-0.99...=0
1=0.999...
They are exactly the same
That's still not 1.
What's with all this dumbing down? .999 =/ 1 mathmattically.
For the price of gas? Yes, since our money only rounds to the 100th, $5.999 = $6.00. If an experiment requires measuring beyond the 1000th place, then .999 =/ 1.
That's the math I had used.
There was quite a bit of debate and general stupidity over at EBW. I figured that this community would have a clearer idea, and be able to disprove my theory if I was wrong.
Thanks!
Edit:
0.999... was simply used to represent 0.9 repeated. I couldn't think of a better way to represent it.
but yeah you are right
your numbers just don't mean anything, or if they are held as self-evident, all you needed to say was that 0.999... = 1 because you only name that lemma, you never prove it
X = 0.999...
10*X = 9.999...
10*X - X = 9*X = 9
=> X = 1
Again, no one is claiming .9 = 1, .999 = 1, or .999...999 = 1. If there are an infinite number of nines after the decimal, however, it is equal to 1.
I think it's fair to hold axiomatically that an infinite series of zeroes is equal to zero.
There is nothing to debate or discuss here. Two ways of expressing the same concept are the same. Job done.
That won't stop us though!
1/3 = 0.333...
=> 3*1/3 = 3*0.333... = 0.999...
=> 1 = 0.999...
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I used to know the actual proof for this, but calculus is ass. Can we stop using really dumb arithmetic proofs for it, though? It's grating on my nerves. ><
Not precisely the same thing. One of them is only one syllable, the other can be as many as you dream. One also has a lot fewer visible digits. One is also sexier.
This is is about .999..., aka .999 recurring, not just .999.
EDIT: Beaten by Mojo. Tis fate.
On vaguely related grounds, this is the proof I usually try and demonstrate with.
1/3 = 0.333...
3/3 (1) = 0.999...
1 - .999... =! 0
There's an infinity there if ya didn't notice.
It's like you're trying to say 1 - i = 0.
You're all being silly.
I mean, .999... is equal to one because it's static, or something?
32.999... =! 33
Pfft.
Because of infinity.
They're the same thing though. You're only pointing out differences in the descriptors of those things.
Puzzle League: 073119-160185
Actually, yes. 32.999... would equal 33.
I always found this easy to wrap my head around in high school: we are dealing with the infinite here, but the "difference" between .999 recurring and 1 is both infinitely large and infinitely small: therefore, they cancel each other out.
B.net: Kusanku
if you want to wait for someone to drag out the calculus proofs, that's fine but
in the context of arithmetic and real numbers, this notation means what you are debating
it's like asking if there is a difference between apples and apples
This graphical representation should solve all your problems. Notice that .999...'s line and 1's line don't meet.
Also, .999... might not even earn a place on this static number line, because .999... is infinity. Infinity can't be graphed. It's almost magical.
Real numbers may not look infinity in the eye, even if infinity attempts to trick the real numbers.
It's just that simple.
Not true, there is all sorts of interesting math you can do with infinity.
For instance, did you know that there are different infinities with different "sizes"? It's true!
Fixed.