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That is a really stupid example and proves nothing other than you can play tricks with numbers. You can substitute 5.5 (5 repeating) in place of 9.9 and it will look like .5 is equal to 1.
If you are trying to represent a floating point number in a computer, then sure you will have anomalies where floating point precision rounding errors will give inaccurate results due to bit limitations and binary representations. However, this is not the case when dealing with these numbers in the human mind. We do not store them to memory as binary digits with a bit limitation. We store and think of 0.99999.....(repeating infinitely) as just that, a repeating decimal. If you can't grasp that concept then I suggest you just forget about it and move on.
Proof by contradiction
suppose 1) 0.999999.... = 1
2) Let set a= 0.99999....
and 3) b= 0.1111....
we have
4) a+ b= 1.000000
From 1) and 4) b must be 0.000000
which is 0
and as corollary b= 1/ 9 must be 0
so 9 X 1/9 = 0 , but the axiom on the multiplicative inverse in the real field produces 9 X 1/ 9 = 1 , hence 1= 0 ( Big contradiction) ,if so the real field R must degenerate to singleton { 1=0 } , in which case multiplication and addition are identical, hence the cardinality of the real field ( set of real number empowered by addition and multiplication laws) would be equal to 1, and 1 would be equal to infinity because the cardinality of the real field is infinity, Big Contradiction again !
we end up in a circularity issue. It looks like the old question that goes as follow:
<<What was before the Big Bang ? >>.
Proof by contradiction
suppose 1) 0.999999.... = 1
2) Let set a= 0.99999....
and 3) b= 0.1111....
we have
4) a+ b= 1.000000
From 1) and 4) b must be 0.000000
which is 0
and as corollary b= 1/ 9 must be 0
so 9 X 1/9 = 0 , but the axiom on the multiplicative inverse in the real field produces 9 X 1/ 9 = 1 , hence 1= 0 ( Big contradiction) ,if so the real field R must degenerate to singleton { 1=0 } , in which case multiplication and addition are identical, hence the cardinality of the real field ( set of real number empowered by addition and multiplication laws) would be equal to 1, and 1 would be equal to infinity because the cardinality of the real field is infinity, Big Contradiction again !
we end up in a circularity issue. It looks like the old question that goes as follow:
<<What was before the Big Bang ? >>.
Huh? All of the big math vocabulary was nice except your initial premise is flat out wrong.
0.99999.... + 0.111111.... does not equal 1. Not sure where you get b = 1/9 from. The problem people have grasping is the concept of infinity. Let a = 0.999999....... and assume 0.99999........ does not equal to 1. Then there must exist a real number b such that a + b =1.00000000. Well what number is b then? The only answer you could come up with is 1 preceeded by an infinite number of decimal zeroes, which is equal to zero, because you can't have something at the end of an infinite number of zeroes. Hence 0.999...=1.
I assure you there are no tricks or a slight of hand with algebra or rounding involved, it's simply a quirk with decimal representations within our number system.
Consider this inequality.
1−.1^n<x<1+.1^n
It seems clear that there is only a single number for which this is true for ALL values of n>0, x= 1 . This is a simple statement that any number added to x results in something greater then 1, or any number subtracted from x results in something less then one. I doubt that you will find many people who will argue with the truth of the statement, x=1.
Now,using simple arithmetic, involving only valid rational numbers, one can construct this inequality.
1−.1^n<.999....<1+.1^n
notice that is the exact inequality as above, thus we have x=.999... If the original statement is correct that the inequality can only be satisfied by 1 you must be lead to the conclusion that
1=.999...
Huh? All of the big math vocabulary was nice except your initial premise is flat out wrong.
0.99999.... + 0.111111.... does not equal 1. Not sure where you get b = 1/9 from.
Not sure where 1/9 came from? well
set c= 0.111111111.... hence 10c = 1.111111....
so 10c-c = 9c = 1 so c = 1/9 . C.E.Q
Quote:
The problem people have grasping is the concept of infinity. Let a = 0.999999....... and assume 0.99999........ does not equal to 1. Then there must exist a real number b such that a + b =1.00000000. Well what number is b then?
Is the hypothesis well-defined? I mean is a= 0.999999.... being different from zero a NECESSARY
condition for the existence of some real number b such that a+b =1?
I thing the answer is no. Because Field R is closed under addition. I mean this b would be equal to 1+ (-a) and this will always be a real number because we agree that a= 0.99999....is a real number.
Quote:
I assure you there are no tricks or a slight of hand with algebra or rounding involved, it's simply a quirk with decimal representations within our number system.
Consider this inequality.
1−.1^n<x<1+.1^n
It seems clear that there is only a single number for which this is true for ALL values of n>0, x= 1 . This is a simple statement that any number added to x results in something greater then 1, or any number subtracted from x results in something less then one. I doubt that you will find many people who will argue with the truth of the statement, x=1.
Now,using simple arithmetic, involving only valid rational numbers, one can construct this inequality.
1−.1^n<.999....<1+.1^n
notice that is the exact inequality as above, thus we have x=.999... If the original statement is correct that the inequality can only be satisfied by 1 you must be lead to the conclusion that
1=.999...
this is somewhat a quick way of making sense of the so-called Sandwich ( or Squeezing) Theorem.
It would work if you replace < ( less than) , by "Less Than or Equal To".
I thing it is rather a semantic issue , I mean instead of " 0.99999999... = 1"
why can't one say " 0.99999.... ALMOST = 1 " ?
My guess is that mathematicians gave to much "power" to the set of real numbers. Maybe some so-called real numbers such as 0.999999.... should be called "ambiguous numbers".
This is just a personal inductive reasoning based on the issue over Rational and Irrational Numbers and it is not enough to be scientifically valid.
12-22-2013, 05:54 PM
Guest
n/a posts
I remember thinking about this after I dropped out of high school.
Also;
Zero is not a number...that one gave me fits.
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