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Some questions about 0.999..., if 0.999... is less than 1.
1/ What is the result of 1 minus 0.999...?
2/ What is the result of 1 divided by 0.999...?
3/ What is the result of 2 divided by 0.999...?
4/ What is the average of 1 and 0.999...?
5/ What is the result of 0.999... multiplied by 0.999...?
6/ What is the result of 0.999... multiplied by 2?
If you believe that 0.999... < 1, please feel free to answer each of these questions above to help the rest of us understand this relationship. Thank you.
The only thing Induction could Prove about ( Pi ) is that it's possibly made of Pastry and contains some kind of Meat or/and Vegetables!
A.R.B
Sorry, that proof fails. I prefer my pie with raspberries in it. And chocolate cream pies are pretty good on occasion too. Neither of those contains any meat or vegetables. ;-)
You start with zero. you must add or subtract each and every number between one thru nine. which numbers must you add or subtract to get 16 when adding or subtracting all the numbers??
Alternate answer:
Infinite TV / Infinite TD = 9
The problem with this is that when you multiply something by Infinity, it's value becomes meaningless. Infinity x TV = Infinity. Infinity x TD = Infinity. Infinity / Infinity = ?? Maybe = Infinity, maybe = 1, maybe = undefined, depending on who you ask. But it doesn't likely = 9.
There is space in the box, and if there is no space in the box, then there is no place in the box where there be zero, and hence it is not a box.
There is no place where there is no space. Space is not zero.
There is no such thing as an empty box.
Q.E.D.
So if I have $10, and I give Sekky $10, then how much money do I have left?
To Maelwys
Quote:" Why? Why aren't I allowed to perform grade 4 math on the equation to come up with a valid answer? Why must the only possible answer to that simple equation be the complex one? "
A.R.B
Example N = 9.2837163847 now N / N = 1 to show how you can make any Number = 1
Yes, N/N=1. Any 10 year old can tell you that. What's the question? Are you meaning to imply that I artificially created the 90/90 situation to force the answer to =1? Somehow I manipulated the formula unfairly to get the answer I wanted? Can you point to the part of that calculation where I inserted any value that wasn't already in the formula that you stated? All I performed was basic fractional algebra, nothing complex, nothing sneaky. If you were to forget for a moment your assumption that the result of that formula HAD to be 0.999... and that 0.999... HAD to be less than 1, and you were to just look at that formula with no bias, you'd probably tell me that it was "obviously 1" as well.
To Maelwys
Quote:" Which is why we were asking what 1/1000 cases of THAT induction are not provable. "
A.R.B
The 1/1000 + 1 + 1 etc.......................................................................................................
Huh?
I'm not even sure what that's supposed to represent...
Oh, and by the way. An ellipsis generally only requires 3 dots, not 33. The extra copies don't make it any more or less significant.
To Maelwys
A.R.B
From your so called Start? 0.9 has to be Infinitely/Recurred Only!! as 1.111...x 0.9
Why? Why aren't I allowed to perform grade 4 math on the equation to come up with a valid answer? Why must the only possible answer to that simple equation be the complex one?
To Maelwys!
A.R.B
Dont tell me! tell Sekky! ( The Mirror Man/Person? )
Tell him what? I'm pretty sure he agrees with me. That's why he was asking you to show him a formula for a valid proof by induction [the P(n) he was asking about] that wasn't 100% proof. You were the one saying
for 1 is 0.5n(n+1) you can only prove it 99.9% of the time!
Which is why we were asking what 1/1000 cases of THAT induction are not provable.
(2) All Infinite/Recurring Number/Values that Start with Zero then a Decimal point will always have a Value less than One Example: 0.(n) < 1
Why will they? You haven't shown your proof for this statement.
(6) Infinite/Recurring Example ( 1 / 0.9 ) x ( 0.9 ) = 0.( 999...) or as 0.(n) < 1
But remember, we have to know what the "start value" for the numbers is, where they "came from". So 0.9 came from 9/10. So we could rewrite that as
For the Sequence 1,2,3,4,5,6,7,8,9.......etc. It's Impossible for Induction to Prove if the Number Sequence will End as an Odd Number or an Even Number!.....................................................
There's two reasons for that:
1/ There is no end to an infinite sequence
2/ Inductive proof fails (n=odd, n+1=even). Induction can't prove everything, and nobody is claiming that it can. All we're saying is that in cases that you CAN use mathematical induction, it IS a valid proof that is 100% correct.
All you have to do is show the Math from 0.999...(9) to 1 then you May have a Proof? so far AGAIN! no one has!
What do you mean by "show the math from 0.999... to 1"? There have been many proofs put forward on this thread showing that they both represent the same value. So there is no math from one to the other. If they have the same value, then by definition we don't need to add anything to get "from one to the other", which seems to be what you're looking for here, and what you're never going to get anymore than you could show me the "math from e^0 to 1".
To Maelwys
Quote:" Okay, what's the 1 case in 1000 that you can't prove with this? "
A.R.B
you have a shorty memory! 1,2,3,4,5,6,7,8,9..Odd or Even?
You left off part of my quote. "I don't believe it's been wrong yet, as long as it's being properly applied."
Whatever your 1,2,3... example is, it's not a proper application of mathematical induction.
for 1 is 0.5n(n+1) you can only prove it 99.9% of the time!.....................................................
which as I have said many times is only a DEMONSTRATION!
Okay, what's the 1 case in 1000 that you can't prove with this?
Call if what you will, but the rest of the math world accepts mathematical induction as a method of proof. I don't believe it's been wrong yet, as long as it's being properly applied. And the above example is a valid one.
To Maelwys
Quote:" That's the point. He didn't have to get rid of it. He started with one statement, and proved that it was equal to another statement. If that difference had been there, he wouldn't have been able to prove that without getting rid of the difference. But because he didn't have to get rid of the difference, obviously there was no difference there at all. That's how a proof works."
A.R.B
Exactly he started with the assumption that it was not there! that is the flaw!
We all know by now! that Infinite/Recurring 0.9 starts 0.1 less than 1 otherwise it would be called 1 to start with!
Infinite 1.111... x 0.9 will always have the .1 missing..............
No, he didn't start with any assumption. He started with a number, manipulated it a bit, and came to a conclusion, without ever assuming that the number at the start was or wasn't equal to 1.
You're trying to enforce the assumption that there is a difference there.
To Maelwys
Quote:" Now you're no longer proving that number is odd though. For n, the number is odd. For n+1 the number is even. So it's no longer a proof of anything. "
A.R.B
you must now be agreeing with what i'm saying! the Number Sequence is either Odd or Even depending on how far we Calculate!..............the problem cant be Reduced by Induction!
ummm... okay.
I agree that what you're trying to do doesn't prove anything by induction. But I don't agree that this means that nothing can be proved by induction.
To Maelwys
Quote:"The point is, neither of those is an inductive proof. To make an inductive proof you need to prove that something is true for both n and n+1. Those are both just single statements (basically saying that if n=9 it's true, but not proving that it's true for n+1), that don't carry any proof at all."
A.R.B
n = 1 n+1 = 1,2 n+1+1 = 1,2,3 and so on! so the same as you are saying!.
Now you're no longer proving that number is odd though. For n, the number is odd. For n+1 the number is even. So it's no longer a proof of anything.
To mikau
Post 08:00:32 Quote:" hey guys, I posted this in a 0.999... thread in the Euler Avenue forum, but it looks like this is more the official thread for this topic. So I'm moving it here.
What do you think of this proof?
A.R.B
Still cant see how you have got Rid of the Infinite/Recurring 0.001... Difference!......................
as in 1 - Infinite/Recurring 0.999...
That's the point. He didn't have to get rid of it. He started with one statement, and proved that it was equal to another statement. If that difference had been there, he wouldn't have been able to prove that without getting rid of the difference. But because he didn't have to get rid of the difference, obviously there was no difference there at all. That's how a proof works.
a good example is " I state that in the sequence 1,2,3,4,5,6,7,8,9...for as far as we can see the numbers will always end in an Odd Number!
but if I could see a bit further! " I state that in the sequence 1,2,3,4,5,6,7,8,9,10...for as far as we can see the numbers will always end in an even Number!The point is!!
Based on the two Sequences above the results are Different! because of how far we can Calculate! the same applies to any Infinite Sequence or Calculations, the end result whether using a large sample induction or a small sample as I have done,can never show a 100% Proof because there will always be some Information missing!
INDUCTION CAN ONLY EVER BE A DEMONSTRATION!
A.R.B
The point is, neither of those is an inductive proof. To make an inductive proof you need to prove that something is true for both n and n+1. Those are both just single statements (basically saying that if n=9 it's true, but not proving that it's true for n+1), that don't carry any proof at all.
Quote:" No they aren't "
A.R.B
Yes they are!!
Both are examples to show depending on how far it is possible to look at a Sequence of Numbers! the Result can be Different!
For my examples! generalization, judgment, logical reasoning has been used by showing the sequence from the Start onwards!Main Entry: induction
Part of Speech: noun 2
Definition: inference
Synonyms: conclusion, conjecture, generalization, judgment, logical reasoning, ratiocination, rationalization, reason
Source: Roget's New Millennium Thesaurus, First Edition (v 1.3.1)
Copyright © 2007 by Lexico Publishing Group, LLC. All rights reserved.
In english, you are correct. However, we're speaking about Math here, which has it's vocabulary that is much more specific about the meanings of terms.
From Wikipedia - Mathmatical Induction:
"Mathematical induction is used to prove that every statement in an infinite sequence of statements is true. It is done by
* proving that the first statement in the infinite sequence of statements is true, and then
* proving that if any one statement in the infinite sequence of statements is true, then so is the next one"
Your "proof" doesn't fit this definition, because you're only proving that the first statement is true, but not that the next statement is also true.
Any so called Proof by induction! is still only Guess work! whether it is 150 pages or 150 million pages! size don't matter if you can't see the actual end of something 100%.....
a good example is " I state that in the sequence 1,2,3,4,5,6,7,8,9...for as far as we can see the numbers will always end in an Odd Number!
but if I could see a bit further! " I state that in the sequence 1,2,3,4,5,6,7,8,9,10...for as far as we can see the numbers will always end in an even Number!
Both examples above are by induction!........but neither is 100% correct or a Proof!.............
I'm not an expert at the definitions of proof by induction, but I'm pretty sure that a single example is not a proof. Otherwise I could come up with all sorts of "proofs"
5 is a number. 5 is odd. Therefor all numbers are odd.
-4 is a number. -4 is less than 0. Therefor all numbers are less than 0.
So saying that a single sequence of numbers ends with an odd number, means that all sequences of numbers end with an odd number isn't proof of anything.
Aah i never thought of it like that, but even if we never actually reach the 8, we still know it's there don't we? so couldn't we argue that 0.999...² could not posible be 1 as a result of this 8, therefore 0.999... could not be 1 either, because 1² = 1.
"...we still know it's there..."
It's where?
It's in the "infinity+1"th position away from the 0? Well, since such a position doesn't exist within our definition of infinity, then there's nowhere for the 8 to be, meaning that it isn't anywhere. So as stated above, 0.999...² = 0.999...
And since the only case we know of in which x²=x, x>0 is where x=1, this could also be seen as another (rather long, roundabout) proof that 0.999... = 1.
;-)
JaneFairfax wrote:I got my degree in 1994.
WOW thats older than me i'm guessing you about 20, 21 22 maybe.
I think you need to practice your math a bit more... ;-) She got her university degree in 1994, so if she's 22, that means she got it when she was 9 years old. Somehow I have to expect she's a little older than that. (sorry Jane, not to harp about a woman's age, just couldn't let this "math" go uncorrected)
Of course, I'm sure she looks 21... ;-) (okay, done sucking up now)
Aha! I like your answer to that one, wish I'd thought of it.
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