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1 - 0.9 = 0.1 " Notice this is a number! with a 1 on the end " " If it's not? then funny if you add " 0.1 to 0.9 you get 1
1 - 0.99 = 0.01 " Notice this is a number! with a 1 on the end " " If it's not? then funny! if you add " 0.01 to 0.99 you get 1
1 - 0.999 = 0.001 " Notice this is a number! with a 1 on the end " " If it's not? then funny! if you add " 0.001 to 0.999 you get 1
The Above is Only a Small sample from the Start Onwards! But I can Assure you! is Infinitely possible!
Because 1 - ( Another Number < 1 ) Will always have an Infinite Difference!!!
P.s the 1 "at the End!" and the 9 "at the End!" is a term to decribe! at the End of an Infinite Sequence! for as far as we have Calculated! and can see on paper!
We all know it's not an actual End!! but in the Above three examples,we can see them at the End Three Times!!
:I agree with your first three statements. All of those numbers do add together to get 1. Your fourth statement however ("The Above is Only a Small sample from the Start Onwards! But I can Assure you! is Infinitely possible!") is a fallacy. Your pattern holds for every number of a finite series only. Any defined finite number approaching infinite, that is <1 will have another number defining the difference between the first number and 1. However, infinite numbers are a special case where this pattern doesn't apply. I also have a problem with your first statement ("Because 1 - ( Another Number < 1 ) Will always have an Infinite Difference!!!"). Any 1 - (another number < 1) will always have a finite difference, that much is true. However again a/Infinitely repeating decimals are special, and so must be defined specially. And b/You're "proving" that 0.999... is less than 1 by saying "1 - another number < 1 has a difference, so since 0.999... is < 1, there must be a difference, so since there's a difference, 0.999... must be < 1". This is circular logic, because you're starting out by assuming your proof is correct, and then using that assumption to prove that it is correct. Finally, in response to your PS, again you can't "describe the end of an infinite sequence" because by the very definition of "infinite" ("Boundless, endless, without end or limits, uncountable, innumerable"; from the wiktionary article on infinite) there IS no end. Again, I strongly suggest you read the wikipedia article on "infinite" because the main problem here seems to be that you don't properly understand that the concept of infinite isn't "a really big number" but it is "an endlessly big number". Hopefully reading the article will help you understand that infinitely long decimal numbers must be handled differently than finitely long decimal numbers, and not all the same rules apply when you're adding or subtracting them.
How Sad!!
1 x 0 = 1
0 x 0 = 0
I need more of a Challenge!!
A.R.B
Okay, here's a challenge. I challenge you to find a single calculator that will agree that 1 x 0 = 1. Because all mine must be broken, since they all seem to agree that any number multiplied by 0 (including 1) = 0.
To Maelwys
Quote 00.43.22
"
The main one is your definition of V2. You can't have a number that contains and infinite number of 0s, followed by a 1. Infinite means "never ending". That means that as long as you can put another number at the "end" of the string of numbers, it MUST be another zero "A.R.B "There is No zero at the end of 0.999999......"
I wasn't referring to 0.999... I was referring to your definition of V2 which you said was an infinite string of 0's, with a 1 at the end, which as I said is impossible.
You are Missing the point of what I am saying!
We Cannot actually write down all the 0.9999's
We don't have to actually write them all down, we just have to accept that the 9's will continue forever.
All we know is that it starts 0.1 < 1 There is no reason possible! why at some stage Later!
The missing 0.1 is Add to 0.9
Okay then, I agree, you're completely right, 0.9 is definitely 0.1 less than 1 and not equal to 1. Unfortunately you can't solve a number equality just by looking at the first couple digits of the number, especially not when it's infinitely long. You have to consider the entire number.
" Therefor, there is no possible number that could be added to 0.999... to give 1. And THAT, is why 0.999... equals 1, because if there's no possible number that you could add to it to achieve 1, then there's no difference between the two, which means that they're the same number."
A.R.B
We are not trying to find a Number that can be Add to 0.999... to give 1. we already know 0.999.. will always have it missing! and that's fine!
As I have already said " I am Quite Happy Knowing there is a Number Infinitely Smaller than 1
But you are trying to find a number that can be added to V1 to give 1, that's exactly what your argument was... that since V2 could be added to V1 to give 1, then V1 wasn't equal to 1. So my counter argument was that since V2 couldn't possibly exist as a number (or if it does, it's equal to 0), then that means V1 must be equal to 1. I know it's completely counterintuitive to say that a number starting with 0. is equal to a number starting with 1., but in this rare case, it's actually true. You just have to stop focusing on the first couple digits of the number and consider the entire thing (which involves wrapping your head around infinite, which is obviously also hard to do).
Answers:
Exactly Maelwys (and Welcome to the Forum!), "infinite" says "no end".
Thanks for the welcome. If it's any consolation, some good has come of this conversation... I was led to these forums by a link that Anthony posted on Wikipedia (trying to cite his post on these forums as proof that his theory was true, to get me to create a page on Wikipedia explaining his theory). ;-) That's also why a lot of his arguments seem generic and never seem to directly answer any of the questions/challenges people post for him, because he's carrying on the same arguments both here and on Wikipedia.
Anyway, to Anthony: (copied from my reply to the same argument you just made on Wiki):
Sorry, there's still a couple problems with this. The main one is your definition of V2. You can't have a number that contains and infinite number of 0s, followed by a 1. Infinite means "never ending". That means that as long as you can put another number at the "end" of the string of numbers, it MUST be another zero (to fulfill the "never ending string" definition. So you can't possibly put a one at the "end" of the string, since there is no end. It seems that basically, your entire problem with accepting the proofs presented here is with understanding the definition of infinite. Infinite isn't a "really big" number. It's a limitlessly big number. So an infinite number of zeros doesn't mean that you just keep writing zeros for a long time and then eventually stop and put a 1 on the end. It means that you keep writing zeros forever. Forever and ever. You'll never be able to put a one anywhere. Therefor, there is no possible number that could be added to 0.999... to give 1. And THAT, is why 0.999... equals 1, because if there's no possible number that you could add to it to achieve 1, then there's no difference between the two, which means that they're the same number.
V1 = 0.9999999999...." = The 0.9 Value from the Start Onwards! "
V2 = 0.0000000001...." = The < 1 Value from the Start Onwards! "
V2 can be read 3 different ways, and unfortunately none of them make sense.
V2 = 0.00000000010000000000100000000001... (repeating the entire sequence to infinitely, which just makes for a really odd number when added to V1)
V2 = 0.00000000011111111111111111111111... (repeating the final digit to infinitely, which makes a different, equally odd number when added to V2)
Or, I assume that by your definition for V2 you mean to say
V2 = 0.00000000...1 (ie: an infinitely long string of zeros with a one on the end). In this case you have to consider as already mentioned above several times, that there is NO end to an infinitely long string of zeros, so there is nowhere to put that one, so it essentially doesn't exist. Because if there's room to stick a one on the "end" of the string of zeros, then that means it's not infinitely long yet and you have to put another zero there instead... and if you're able to put the one after that zero, then instead you have to put another zero there first (and so on and so on and so on). So that theory doesn't work either, sorry.