0.999... = 1

Hey guys,
This has been a popular debate on a few of the previous forums I've been a part of. The question is fairly simple, but it usually leads to a good discussion. So, here it is:
Does 0.999... (repeater) equal one?
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Here's my first proof:
Let x=0.999...
Therefore, 10x=9.999...
10x-x=9x (9.999...-0.999...=9)
9x=9
Therefore x=1
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What do you think? Does it equal 1?

 

the proof seems logical, I don't see why not.

We know that 1/3=0.(3)
0.(3)*3=0.(9)
1/3*3=1
and since 1/3 and 0.(3) are equal 0.(9)=1

Again, I gues it does

 

0,999999999999999999999 is almost 1.

 

Nope, it is exactly equal to one. It's not so close that it might as well be one or anything, it is exactly equal to one. It's just another way of writing it.

 

1 - 0.99' = 0.00'...1
1 - 1 = 0(.00')
0.00' < 0.00'...1
Hence 0.99' is not equal to 1.

That's my shot at it, anyway.

 

I once saw how to prove that it isnt equal, cant remember how it was, but it was long...
And its hard to work with infinity numbers

 

if 0.999999999---->infinite amount of 9s

then its no harm adding a 0.00000 ......... 000001 to make 0.99999 equal to 1 since 0.00000....001 is microscopically unnoticeable.

 

Theirs still a difference.....
I just searched the net and im chocked...
Links:
http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html
http://mathforum.org/dr.math/faq/faq.0.9999.html

 

I'm sorry but he's a jerk. When someone says "it's getting closer and closer to 1" it means that if you take 0.9 and mark it on the number line, then you take 0.99 and put the mark there, the mark will in fact keep getting closer to 1, and you can keep doing that until infinitely many digits after the decimal point.
I just hate it when people misinterpret things deliberately just to void an argument that totally nullifies their claim.

Also, there is no such number as "number nine repeating". A number is a static representation of a value, it doesn't change.

 

There's really no room for debate, I have to say.
0.999... = 1, and there are many ways to prove it, most of which can be found here:
http://en.wikipedia.org/wiki/0.999...
My favorite is the sum:
lim(n->infinity) sum(from 1 to n) [9/(10^n)] = 0.999... == 1.
Sorry about the rough notation, but if you take the geometric series sum of that, you get 9/9 = 0.999...
which of course corroborates the fractional proof that 9 * 1/9 = 9 * 0.111... = 0.999... = 1

Oh, and 0.000...0001 is nonsense, there can't be anything ending an infinite number of zeros, or it wouldn't be infinite.

 

I first voted no, but then following my communist ideology I voted yes.

 

GM_K i voted no, but i changed it too

 

Hey, a poll, I never noticed that
I voted no because it's only a convention that we say 0.00000000000000000000001 is nothin. A 10cm long string is still there even if you look at the globe from outer space.

 

It has been proved mathemathic, check the links i showed...
But I'm going to un-prove it when i have studied math!

 

Yes it is prooved, but since you can't count with infinite numbers so there isn't much use of this but pretty interesting. This is one of the reasons I like maths.

 

I voted yes...

If 0.9999... would not be equal to 1, you could o on for an infinite amount of numbers and never reach 1, that's just weird. Also consider this:

Someone jumps of a building that is 100 meters high. The person keeps getting closer and closer to the ground, and when he is only 1 cm away, he keeps falling. He goes on and on, reaching 0.1 cm and later even 0.000001, but he never ever will touch the ground, so he won't die.

That's why 0.999... is equal to1.

 

I only checked one of the links, from which I quoted and commented on the author's ignorance. I'll stick with my unbiased opinion for now.

Also, I think that Forsaken's falling example isn't good because we can comprehend gravity and collision, plus we know the result of that scenario. We can't comprehend infinity though... I'll elaborate on the connection I mentioned earlier, with a different example:
if 0.99' = 1 then 0.00'...1 = 0, right? if that's so, and not just by convention, but in all the senses, then a 0.5mm long insect would literally vanish if someone looked at Earth from so far away that the whole thing would only have, say, a 10cm diameter. Would the bug actually vanish? Would it cease to exist and thus equal to zero? Absolutely not, it'd still be visible with the right apparatus, with a concrete length value on the aforementioned "ruler". And this is true for all distances away from Earth, upto infinity.

 

Not right. As I said earlier, 0.000...001 is incorrect, because you are saying that there are an infinite number of zeros, and then once you get done with infinity, you get a 1 afterwards. The only thing after infinity is more infinity. The zeros never stop, so there is no 1 at the end. The correct number to add to 0.999... to get 1 is 0.000...., which is actually 0 (with an infinite number of zeros after it).

As for the example with the string or the bug getting infinitely small, it is also a flawed argument. It may be imperceptible, but its physical characteristics are actually independent of the observer (the bug is still 0.5mm and the Earth is still about 6,300 km in radius). I could tell you a relatively exact value for the angle the bug fills in your vision if you were on the moon, or Mars, or in another galaxy, and I can promise the decimal would have a finite, not infinitesimal, value (theta = 2*arctan(0.0005m / d), where d = distance between your eye and the center of the bug). That does go to zero as d approaches infinity, but the bug will still be there (ignoring relativity, for simplicity's sake).

Forsaken's falling example is somewhat relevant, since it involves repeating decimals. A simpler example is the sum of (1/2)^n. Add 1/2 to 1/4 to 1/8, etc. and it keeps getting closer to 1 the more terms you add. Well, if you take the limit as n goes to infinity, it actually equals 1. If you do this with 9 * (1/10)^n, it is the same thing, 9*(0.1+0.01+0.001+0.0001...) = 0.9+0.09+0.009+0.0009... = 0.9999...
This is what the sum in my previous post was, and if you use the geometric series, you get 9 * (1/9), which we can all agree is 1 (I hope). For n/n where n is any odd number not divisible by 2 or 5 the decimal will repeat and yield 0.999..., so this works not only with 9/9 and 3/3, but 7/7, 11/11, 13/13, etc.

 

Here is my 2 cents on the topic we know the difference between 1 and .999...  is infinitely small just like 1 divided by infinity which we know from lim (as x goes to infinity) 1/x = 0, thus there is no difference between the 2 and they are the same number. Kuvasz I like the zooming out from earth example but here is the flaw I see in it while I do agree as you zoom out the bug gets smaller just like the difference between 1 and .999... as you take into account more 9s but the problem between the example is you don't take into account the fact there is always another thus there is no finite difference between the 1 and .999... unlike the fact the bug does have a finite size if you continuosly zoom out. I think a more appropiate example would be you staying the same distance from the bug as the bug is constantly shrinking and yes while you can keep getting a bigger magnifying glass you can only do that to a certain point because you can't zoom in infinitely, and as the bug gets smaller it will in effect become a mathmatical point having the dimensions of 0x0x0.  

Also going to back to your first example kuvasz

that last line is a mistake, it should be .000.... = .000...0001 because the ending is irrelevant because of the infinite number of zeros in between the ending and the beggining, in fact .000.... = .000...0001 = .000...0007 = .000...0009 would all be true because you could model them with the number/10^x as x goes to infinity (assuming the number is a finite non 0 number), because any finite number divided by a number approaching infinity equals 0.

EDIT: I finished this post before Sodium made his post, but since I really don't feel like deleting it, I apologize for any repeat points.

 

So you're saying that there are two numbers to which we can add the same number and get the same result for both? That made me smile

That's exactly the reason I said "the right apparatus". I wasn't thinking an ordinary magnifying glass, I was thinking a continuously evolving technology that let's you zoom in further and further, as the technology becomes more advanced. You will always be able to see the bug, and it will always occupy a certain amount of space on that ruler. Also, there's no such thing as a 0 by 0 by 0 thing in practice, because those dimensions give "nothing". The bug will never get those attributes, because it'll never vanish.