Hard Light Productions Forums
Off-Topic Discussion => General Discussion => Topic started by: Polpolion on December 01, 2008, 10:04:47 pm
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Is 0.999999999 (repeating) equal to 1?
I've noticed that 1/9=.1(repeat), 2/9=0.2(repeat),...8/9=0.8(repeat), so wouldn't the next logical step be 0.9(repeat), but 9/9 is one.
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that one's always confused the **** outta me. anyone clarify for the both of us?
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This has to be one of the most common math questions on forums. :D
Yes, they are the same number, one that happens to have two decimal representations. There are a couple of different ways to see this, using decimal shifts or limits.
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so it is 1 then :nervous:
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Yep.
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that's silly :ick:
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Interesting question... the answer is certainly counter-intuitive
Think of it as the limit of the difference between 1 and 0.999...
For any given number of decimal places (n), the difference is 1/(10^n). The limit of that is 0 as n tends to infinity.
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Here's another:
Any finite number divided into infinity parts comes out to be zero.
But zero multiplied by infinity is still zero.
Interesting, eh? :D
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But zero multiplied by infinity is still zero.
Not necessarily. It's an indeterminate quantity that can be anything, depending on how "strong" the zero and infinity are.
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I think I'll stick to College Algebra. I'm really hoping Journalism never sees me having to know what the hell you just said.
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those 1s and 0s are really just a difference of signal voltages in the hardware, and the bool operation is depending on your relation to a particular threshold. over it is one, under its zero. because there are fluctuations that occurs in the electronics as paths open and close changing the power requirements. oh and im high :D
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depends on who you ask mostly, most mathematicians say yes, I say no, my position is one mostly of semantics, as I believe the other position is as well.
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I'm with Bobboau here, mathematically speaking, it's one, but you only have to look at the two numbers to see there is a difference, it might purely be a conceptual one, due to the nature of infinity, but a difference is there in my opinion.
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I'm with the 1 = 0.999... part, because, IMO, it makes sense, also it's the only way 1/3 = 0.333... works when 3/3 = 1.
1/3 = 0.333...
2/3 = 0.666...
3/3 = 0.999...
But 3/3 = 1
Therefore 0.999... = 1
There is no definite difference between 1 and 0.999..., therefore it's an infinite difference.
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But is infinite difference the same as no difference whatsoever? We all know that 1/3 = 0.3333333etc is an estimate, that the true answer is infinite in size, we can get the answer to any sum up to any resolution if we have the tools to calculate with it.
However 1/3 is a symbol, like Pi or 6.02x1023 - They are not numbers, they simply represent much longer numbers. You could say the same about 0.999999999, it can be represented as '1' but in truth 1 is only a symbol to represent a number too large to be written.
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But is infinite difference the same as no difference whatsoever? We all know that 1/3 = 0.3333333etc is an estimate, that the true answer is infinite in size, we can get the answer to any sum up to any resolution if we have the tools to calculate with it.
That is what I meant.
However 1/3 is a symbol, like Pi or 6.02x1023 - They are not numbers, they simply represent much longer numbers. You could say the same about 0.999999999, it can be represented as '1' but in truth 1 is only a symbol to represent a number too large to be written.
Yes, but 1 is also an integer. Numbers are usually used as representations of transcendentals when they come from numbers, aren't they? Like the square root of two - √2 is an exact value, but 1.4142135623730950488016887242097 is an approximation. Therefore, it is possible that 1 is an exact value and 0.999... is an approximation.
[pedantic] And it should be written as 0.999... , as 0.999999999 is also an exact value. Even though we know what you mean. [/pedantic]
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Actually, I thought about using the three dots and thought 'Oh, they'll know what I mean' :p
Now you mentioned it, there is actually one tangible difference between the two. and that is that the integer 1 is part of a set of integers, whereas the value 0.999... is not.
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X = 0.999... / multiply by 10
10X = 9.999... / -X (and since X = 0.999...)
9X = 9 /divide by 9
X = 1
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-> QED
Of course this relies on assumption that you can agree that 9.999... - 0.999... = 9
So it's still semantics, but it demonstrates nicely that the obfuscation is in your head, and not in math.
BTW: If you really want to stirr up some serious forum drama about math/physics, start a discussion about the the goddamn airplane on the goddamn treadmill (http://blag.xkcd.com/2008/09/09/the-goddamn-airplane-on-the-goddamn-treadmill/)
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Have you seen Mythbusters? That myth has been disproved. The plane will take off because the propeller/jet, not the wheels, are providing the propulsive force.
Do not let this devolve into flame war, otherwise I will be ostracized. (Don't really know what that means... I shouldn't put that, it might mean something evil...)
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LOL It'll be Slide Rules at Dawn ;)
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Gah, brain failure. . . . .
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depends on who you ask mostly, most mathematicians say yes, I say no, my position is one mostly of semantics, as I believe the other position is as well.
It's not a matter of semantics if you agree that the ... means the 9s repeat forever. As portej05 said, it's a limit of increasingly closer approximations to 1, and limits can be understood without having to consider anything infinite.
There is actually a fringe area of math called nonstandard analysis, which uses a different set of axioms that allows for the existence of infinitely small numbers. However, analysis (calculus) and basically all other mainstream math is based on the classical Zermelo-Fraenkel framework that doesn't allow such quantities.
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Not my field but if 0.9999... = 1 would that mean that 0.099999... = 0.1 , 0.19999... = 0.2 etc. ?
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I believe that's true.
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And what about 0.99999... + 0.11111... ?
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You mean 1 + .11111111111...?
I should think it equals 1 and 1/9th.
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Here's my point: 0.999 + 0.111 = 1.110 not equal to 1 + 0.111 = 1.111
It must be semantics.
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Those aren't the same numbers as in your last post.
Notice that every time you tack on an extra 9 and 1 to the first two numbers, the sum gets closer and closer to 10/9. It's the limit that matters.
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Yes, i understand this. I just think that the source of the confusion is that we symbolize a limit as a decimal.
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X = 0.999... / multiply by 10
10X = 9.999... / -X (and since X = 0.999...)
9X = 9 /divide by 9
X = 1
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-> QED
Of course this relies on assumption that you can agree that 9.999... - 0.999... = 9
So it's still semantics, but it demonstrates nicely that the obfuscation is in your head, and not in math.
Isn't that circular reasoning?
Of course 9.999... - 0.999... = 9, but if X = 0.999... and you subtract X from 10X, you wouldn't get 9X unless X = 1.
X = 0.999... / multiply by 10
10X = 9.999... / -X (and since X = 0.999...)
9.000...1X = 9 <-- and that's false unless 0.000...1 = 0
Which it would if 0.999... = 1.
I'm not arguing that 0.999... isn't the same as 1, I'm just arguing with your argument.
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It's correct, although he could have written it more clearly. The idea is that since there are an infinite number of 9s, shifting all of them towards the decimal point (which is what multiplying by 10 does) will not change the fractional part. So you get 10X=9+X, which gives X=1.
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But with 10X = 9 + X, you still have to subtract X from 10X which is still 0.000...1.
Of course, 0.000...1 is 0 since you can't really put a one after an infinite number of zeroes.
Which is why 0.999... = 1.
Because 0.999... + 0 = 1.
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But with 10X = 9 + X, you still have to subtract X from 10X which is still 0.000...1.
You're just solving it as a normal linear equation at that point, without making any further assumptions on what X is. You show that the number, whatever it is, satisfies 10X=9+X, and just from that it follows that X=1.
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10X-X = 10*X- 1*X (1 is neutral element for multiplication, hence the identity) = (10-1)*X (distributive law) = 9*X
I dont get it, what has that to do with 0,999...? Thats simple math that is universally true for ALL numbers, as it is basically part of the definition of (common) numbers itself.
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Whatever is true, I'm satisfied with the notion that nothing in/from this world can tell the difference between 1 and 0.999... :D
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X = 0.999... / multiply by 10
10X = 9.999... / -X (and since X = 0.999...)
9X = 9 /divide by 9
X = 1
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-> QED
Of course this relies on assumption that you can agree that 9.999... - 0.999... = 9
So it's still semantics, but it demonstrates nicely that the obfuscation is in your head, and not in math.
Isn't that circular reasoning?
Of course 9.999... - 0.999... = 9, but if X = 0.999... and you subtract X from 10X, you wouldn't get 9X unless X = 1.
X = 0.999... / multiply by 10
10X = 9.999... / -X (and since X = 0.999...)
9.000...1X = 9 <-- and that's false unless 0.000...1 = 0
Which it would if 0.999... = 1.
I'm not arguing that 0.999... isn't the same as 1, I'm just arguing with your argument.
Ah OK, sorry for the confusion, I was doing two things in one step. All written out it goes like this:
X = 0.999... / multiply by 10
10X = 9.999... / subtract X
9X = 9.999... -X / substitute the X on the right side with 0.999..., as it was defined on the first line
9X = 9.999... - 0.999... / execute the subtraction on the right side
9X = 9 /divide by 9
X = 1
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10X-X = 10*X- 1*X (1 is neutral element for multiplication, hence the identity) = (10-1)*X (distributive law) = 9*X
(10 - 1) * X = (9) * X
This works.
X = 0.999... / multiply by 10
10X = 9.999... / subtract X
9X = 9.999... -X / substitute the X on the right side with 0.999..., as it was defined on the first line
9X = 9.999... - 0.999... / execute the subtraction on the right side
9X = 9 /divide by 9
X = 1
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This is still circular reasoning.
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It's easier to prove with limits.
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This is still circular reasoning.
I fail to see why this should be circular reasoning, care to elaborate ? (I'm not arguing, I'd really want to know if there's logical flaw in my solution)
"10X - X = 9X" is always true, not matter what X is, this is basic algebra.
X can be 43782487, and yet if you subtract one X from 10X, you still get 9X.
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X = 0.999... / multiply by 10
10X = 9.999... / subtract X
9X = 9.999... -X / substitute the X on the right side with 0.999..., as it was defined on the first line
9X = 9.999... - 0.999... / execute the subtraction on the right side
9X = 9 /divide by 9
X = 1
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When you subtract 0.999... from 10X and get 9X, you are assuming that 0.999... equals 1. It's circular reasoning because you are using the assumption that 0.999... = 1 in order to prove that 0.999... = 1.
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"10X - X = 9X" is always true, not matter what X is, this is basic algebra.
X can be 43782487, and yet if you subtract one X from 10X, you still get 9X.
10X-X = 10*X- 1*X (1 is neutral element for multiplication, hence the identity) = (10-1)*X (distributive law) = 9*X
(10 - 1) * X = (9) * X
This works.
Ack. I've made a mistake.
(10 - 1) * X = (9) * X doesn't have anything to do with Mr. Col. Fishguts' proof up there.
It would have to be (10 - 1) * X = (9) + X.
9 * X /= 9 + X
If X = 0.999... = 1, then you'd get 9(1) = 9 + (1), and 9 /= 10.
Let me try again...
0.999... = 0.999... //multiply both sides by 10
9.999... = 9.999... //rewrite both sides
0.999...+0.999...+0.999...+0.999...+0.999...+0.999...+0.999...+0.999...+0.999...+0.999...=9+0.999...
//subtract 0.999... from both sides
9(0.999...) = 9 //divide both sides by 9
0.999... = 9/9 = 1 //identity property
0.999... = 1
Ohhh... Okay, I was wrong, Mr. Col. Fishguts. I misunderstood your proof. ^^;;
But the X = 0.999... bit was wholly unnecessary.
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/me feels like throwing up from all the math.