0.999... = 1

Kuvasz, they are not different numbers, but the very same number represented in two ways, especially since trailing zeros in a decimal have no value, that is, 0.1 == 0.100. They are identical ways of writing the same number.
0 + 1 = 1 + 0 = 1 + 0.00000 = 0.999... + 0.000... = 0.999... + 0, and so on!
They are not merely equivalent, they are the same exact number. If you can't accept that a number can be represented multiple ways, then you can't accept some very fundamental principles of mathematics.

Look at all the rigorous mathematical proofs available for proving 0.999... = 1, they are all completely valid. You are trying to reason your way out, and every time your reasoning has been flawed. Try using a mathematical proof to prove a mathematical concept.

When it comes to shrinking things down, once the bug reaches the Planck length we can't really say what would happen, but since that is delving into another field again, we can say that there is no "smallest" point, and say that the bug continues to shrink. Supposing we had a particle small enough view it with (since photons are big and clunky at these scales), we could continue to observe the bug shrinking smaller and smaller for as long as we want, and as long as it doesn't actually reach 0x0x0 meters, it will always have some measurable length, width, and height. What that means is there are always going to be a finite number of zeros before your 1 in 0.0000...001, or you will have just plain old zero (no decimal values that are nonzero). So the difference would have to be (1/10)^n, where n is a subset of the positive integers, or it is zero (where n goes to infinity).

 

Again kuvasz your keep looking at the points on the way to infinity not what happens when you actually get there. Its kind of like the mathmatical example were is you keep moving half of the distance tworards an object you will never actually get there, which is only true if you are moving a finite number of times. Once you move an infinite number of times however, the distance between you and the object is 0. The only difference between that and this is you are moving 9 tenths the distance.

 

After reading these posts, i voted for yes all of those calculations are so accurate that it has to be a yes :good:

 

I can totally understand that numbers can be represented in different ways, but I don't consider using the same expression (decimal) with different numbers valid.

And yes 11-Sodium, zeros are redundant after the decimal point if there's no other number besides zero following it, which is exactly why 0.00' = 0. You're saying that we need to add 1 to 0 to get 1 and we need to add 0.99' to 0 to get 1. I see a clear contradiction there. Now, it might be my inability to accept that a number can be expressed with something other than itself in the same expression by using the substitute infinitely many times, but I never said I was an overly intelligent person ;D

I will read those links eventually, but I wanted to try and comprehend this by myself first (which oviously phailed ).

 

No it's not. You are wrong. 0,9999999999999 is not 1

I agree with you on that. :good:


So, people 0,999999999 is not 1!

 

.9 repeating doesn't equal one, because .3 repeating doesn't equal 1/3. If repeating decimals were perfect representations of fractions, why would we ever use fractions?

.9 repeating is simply infinitely close to one, but it never quite makes it

-Loki

 

Wow people are just blindly charging this topic without proof. LT .999... is 1 just like .000... is, you are the one that is wrong and if you are so sure your not provide mathmatical proof why, I wonder if there is a correlation between the amount of proof provided vs. opinion because it seems like the people who provide mathmatical evidence (aka have some knowledge of mathmatics) seem to believe .999... = 1, while the people lacking evidence are just sure it isn't.... There are a number of ways to look at this problem probably the easiet being algebraicly and ItaHexGor's original post, where he provided an algebraic proof. Or you can look at it through the Calculus goggles, then you have to acknowledge that 1 divided by infinity = 0 or something that is infinitely small = 0, and since the difference between .999... and 1 is infinitely small (aka 0) .999.... = 1.   

There is only a contradiction if you believe .999... =|= 1, however once you believe they are the same number (which has been proven a number of ways so far) the contradiction goes away.

 

You can say what you want, but infinity numbers are crap...

 

Both Loki and I have provided reasonable arguments/evidence disagreeing with the thread title.

And of course you're right with the last quotation

ps Loki's example reminds me that my teachers used to tell me that fractions are more accurate than decimals. That can't be BS now, can it?

 

Actually kuvasz I have liked your arguments but you seemed to keep commiting the same mistake, in the examples you give are effecttively talking about why .99999 or .999999999 or .99999999999999999999 =|= 1 (which is true) but you never seemed to deal with .999... which does equal 1. Understanding is hard because the way infinity can play with our minds. As for Loki's comment .333... = 1/3 just like .999... = 3/3 = 1.  Are more simple to write than a repeating decimal and can be worked with a lot easier but they have the same level acuracy. 

 

Yeah I think infinity is too much for our minds to comprehend and the fact that we like to round so much doesn't help either.

ps LK, PM!

 

Okay, I have a challenge for all of those who believe 0.999... is not equal to 1.
It is a well proven fact that between any two distinct real numbers there are an infinite number of real numbers, eg between 0 and 1 there is 0.1, 0.2, 1/2, 1/sqrt(2), 1/e, 1/pi, 1/11, 1/101, etc.
This is true no matter how close the numbers are. Between
0.0000000000001 and
0.0000000000002 (not with repeating zeros, just with 12)
there are as many real numbers as there are between 1 and 2 (just take the numbers and put the 0.000000000000 in front of those values)

Now, find me 10 real numbers between 0.999... and 1. Heck, if you can find 1 you have proven that the two are unequal.

 

But still, logically it does not sound right...
Our brain haven't been made to think infinity (i changed my no to a yes after I googled)

 

Where's your proof?

The reason it's perfect is because it goes on forever. If it didn't, then it isn't a perfect fraction.
Also, if you think that 1/3 doesn't equal 0.333... what do you think it does equal?
_________________________________________________
Here's another proof:
1-0.9=0.1
1-0.99=0.01
1-0.999=0.001
1-0.9999=0.0001
etc.
1-0.999...=0.000...
Once you approach an infinite amount of 9's, there are an infinite amount of 0's. It is impossible to have 0.0...1 because there will never be room for the 1 at the end, because there is no end.

 

Still, how can 2 notations mean the same number? Because right now 0.99' seems to be the same as 1, which is actually 1.00'. Is it true that 0.99' = 1.00'? I highly doubt that.

 

Does it really seem that strange? 2/6 = 1/3 =.333... = .111... +.222... = 1/9 + 2/9 = x/3x = 10^x/(3*10^x) those are all different ways of saying the same thing, so is it that odd to say .999... = 1 = 2/2 = 3-2 = 1/3 + 1/3 +1/3 because they are all 1? It really is just a trick of the mind we think .999... =|= 1 because when we think of .999... we think something like .99999999999999999999999999 (which isn't 1 btw) we don't always comprehend those 9s are going on forever, unending. Once you can grasp what .999... really means or what the difference between .999... and 1 really is, then it starts to make sense.

 

okay so then 1+0.999.... =2?

 

I've actually given up on the whole 0.99' = 1 thing Well, partially. Look at it this way:
If two numbers are expressed in exactly the same way, they can only be equal to each other if all the digits match and thus give the same number, correct? Examples: 0.430322 = 0.430322; 7.44' = 7.44'.
So, since 1 = 1.00', then you're saying 1.00' = 0.99'. Frankly, that has infinitely many digits not matching, but even if only 1 digit mismatched, it still wouldn't be the same number.

 

Actually...
if 0.0...1 is impossible (as in it equals 0), because there's no room for that 1 after all the zeros, it would mean that 0.0...2 is too. Same with 0.0..9, or 0.0....2134

Following this logic 0.(9)05437897 = 1

Even better yet, what about 0.(9)(4) ? Is it 1 too?

 

@ kuvasz. Numbers can always be expressed in different ways. You're yet to prove that is doesn't equal one, you're just saying 'but they look different'. It doesn't matter how different the numbers are, an example of this being; 284324/284324 = 1. Do the maths and try to prove that it doesn't equal one. There are several proofs that it does, try proving it doesn't.
@ Meee.You can't get those numbers to begin with. They don't exist.