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To Luca-deltodesco
Quote" you really aern't getting this anthony.
are you saying that with induction you cannot prove that sum of the first 'n' numbers starting with 1 is 0.5n(n+1) because you can't test it with everysingle possible value of 'n'? That is what induction is for, induction allows you to do so, because you dont need to prove it for every value, because you can prove that if one value is true, then the next will be true, and then from one single result, you can induce all other results "
A.R.B
You really aern't getting! IT!!
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!
CAN WE TRUST COMPUTER PROGRAM RESULTS!?
----------------------------------------------------------
The answer to the Question above is yes! Providing we can see the source code and that we have an understanding of the language used.
But as programmers/manufacturers are not willing to do this because of possible counterfeiting their programs,and worst altering the actual programs! Then we just believe what the program tells us is true!
I have shown in my first Qbasic Infinite/Recurring 0.9 program above, just how easy it is to produce the results we want,without the program actually doing any Math calculations! If the source code was hidden most people would accept that it was calculating Infinite/Recurring 0.9
So if there is a Moral message in what I'm saying! Then it is don't believe the results 100% for programs like Mathematica6 etc. unless you have seen the source code etc.
Below is my second Qbasic Infinite/Recurring 0.9 program! This time it actually Calculates! The only Restriction is the Decimal Place Length! Other Languages produce longer Results...Understanding what is being Calculated is the most Important thing!.................................
REM ***************************************************************************
DIM START AS STRING
DIM A AS SINGLE
DIM B AS SINGLE
DIM C AS DOUBLE
DIM D AS DOUBLE
DIM RESULT AS STRING
CLS
PRINT
PRINT " INFINITE/RECURRING 0.9 QBASIC PROGRAM,By Anthony.R.Brown 30/05/07 "
PRINT
PRINT " YES! THIS PROGRAM ACTUALLY CALCULATES!........................... "
PRINT : PRINT
INPUT " PRESS < ENTER > TO RUN PROGRAM! "; START
CLS
PRINT : PRINT
REM *************************************************************************
CALCULATIONS:
A = 1
B = .9
C = A / B
D = C * B
IF D < 1 AND D <> 1 THEN RESULT = " Infinite/Recurring 0.9 < 1 and <> To 1 "
PRINT RESULT; D
PRINT : PRINT
PRINT " Length of Decimal Places Restricted By The Computer Language!.........."
END
REM ***************************************************************************
If you dont have a copy of the QBASIC program,it can be download for Free from
http://members.lycos.co.uk/brisray/qbasic/qind.htm
A.R.B
To Sekky "ITS ABOUT TIME YOU SHOWED SOMETHING!"
A.R.B
To mathsyperson
Quote:" As far as I can tell, that's not even calculating anything. You're just telling it to write 0. and then 9 lots of times. "
A.R.B
It must be doing the same trick other programs use! to prove Infinite/Recurring 0.9 = 1
ie. Mathmatica6 because how else can you get from 0.999...(9) to 1 without cheating!
So far no one on this site who thinks Infinite/Recurring 0.9 = 1 has ever shown the Math! or as you have asked about the Calculations!!...........
To Ricky
Quote:" Now, can you try to use an example in which induction can actually be applied? "
A.R.B
Induction can be applied to many things! my Argument originally is that Induction can't give a 100% Proof to anything if there is any Information missing! ie. if we can't see the end of a Calculation then we can only Guess the result!!...........................................................
I thought I would copy these here! as it's the Post most Suited!
Below is a program wrote in the Basic Computer Language! that does exactly what it is asked! i.e Calculate Infinite/Recurring 0.9 with no end! ever!................
10 N = 0.9
20 INPUT " PRESS ENTER " ; START
30 PRINT ; N
40 GOTO 10
-----------------------------------------------------------------------------------------------------------------
Below is my Infinite/Recurring 0.9 QBASIC Computer program!
To Run the program copy/paste and then save the file as INFREC09.BAS or INFREC09.TXT
then open the program in the QBASIC program's window to Run.
If you dont have a copy of the QBASIC program,it can be download for Free from
http://members.lycos.co.uk/brisray/qbasic/qind.htm
REM ************************************************************************
DIM TIMES AS SINGLE
DIM DECIMALPOINT AS STRING
DIM NINE AS STRING
DIM COUNT AS SINGLE
DIM RUNAGAIN AS STRING
CLS
PRINT
PRINT " INFINITE/RECURRING 0.9 QBASIC PROGRAM,By Anthony.R.Brown 17/05/07 "
PRINT
INPUT " ENTER HOW MANY TIMES TO RUN! THEN PRESS < ENTER > "; TIMES
PRINT : PRINT
DECIMALPOINT = "."
NINE = "9"
PRINT ; DECIMALPOINT;
NINES:
PRINT ; NINE;
COUNT = COUNT + 1
IF COUNT = TIMES THEN GOTO ENDOFRUN
GOTO NINES
ENDOFRUN:
PRINT : PRINT
PRINT " THE PROGRAM WILL RUN AN INFINITE/RECURRING AMOUNT OF TIMES! "
PRINT " IF YOU JUST PRESS < ENTER > FOR HOW MANY TIMES TO RUN! "
PRINT : PRINT
INPUT " PRESS < ENTER > TO RUN AGAIN "; RUNAGAIN
PRINT : PRINT
RUN
END
REM ************************************************************************
Quote:" Whatever this means, it isn't true. "
Infinite 1.111... x 0.9 will always have the .1 missing............
A.R.B
Prove it wrong!.....................................................................................................................
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!
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..............
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!.
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...
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
Quot:"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. "
A.R.B
Every Number in the Sequence is true from 1,2,3,4,5,6,7,8,9 ....
Quote:" You just copied that from here. "
A.R.B
Wrong!! from here http://thesaurus.reference.com/
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.
To Maelwys..etc...
Deductive reasoning is the kind of reasoning in which the conclusion is necessitated by, or reached from, previously known facts (the premises). If the premises are true, the conclusion must be true. This is distinguished from abductive and inductive reasoning, where the premises may predict a high probability of the conclusion, but do not ensure that the conclusion is true. For instance, beginning with the premises "All ice is cold" and "This is ice", you may conclude that "This is cold".
Deductive reasoning is dependent on its premises. That is, a false premise can possibly lead to a false result, and inconclusive premises will also yield an inconclusive conclusion.
A.R.B
Quote:"
On topic, I don't think it's that much of a threat, considering that the last mersenne prime found was 2^32,582,657 -1 (just under ten million digits), so modular combination of sufficiently large primes should make decryption pretty much impossible. Suppose you take smaller ones and aim for a modular combination of a million digits, that's a one meg key (if my theory is wrong, shoot me down), which will be tame by tomorrow's bandwidth standards.
I don't think we're in trouble for the moment. "
A.R.B
You have no idea what they're talking about. or any others!...............................................
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!.............
To Maelwys
Quote:" 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.
;-) "
A.R.B
Exactly! pure precision accuracy is what we are after! as with Infinite 1.111...x 0.9 = 0.999....
Quote:" Just the children? "
A.R.B
Just the children? as in Before the Adults! as in FIRST!
To MathsIsFun
Quote:" Big Deal? Yes ... because the ability to factor numbers of this size means that internet security is not so secure. "
A.R.B
It is secure if Numbers are mixed with letters etc. for Passwords etc......................................
Quote:" It was, sexagesimal wasn't in recorded use until 2000BC, and even sexagesimal is a mixed radix system of decimal and heximal. "
A.R.B
Children counted with their Fingers! long before any known named counting system!..........
Quote:" I have to partition! My sentences! With exclamation! Marks! "
A.R.B
Divide! and Conquer!.........................................................................................
THE PROBLEM IS! FOR SOMETHING TO HAVE A 100% PROOF! IT HAS TO BE PROVED!!................
We all know the Sequence 1,2,3,4,5,6,7,8,9.....etc... but We cant prove what the Last Number is!?
All Infinite Math problems have the same Problem!!....................................................................
A.R.B
Quote" Oh, becuase that makes so much sense. "
A.R.B
Fools paradise!