I said "we are dealing with infinitesimal numbers" and 0,999... is an infinitesimal number.
I'm going to assume this is a mistake on your part and that you meant that the result of 1 - 0.999... is an infinitesimal, beucase 0.999... certainly is not.
And I also think you didn't pay close attention to what I'm saying. I realize you're talking about infinitesimals. The point is, in real numbers, there are no non-zero infinitesimals. Any mathematical operation on real numbers that would produce an infinitesimal actually produces the value 0. Since 0 is the additive identity, and since, in real numbers 1 - 0.999... = 0, that would make 1 = 0.999... in real numbers.
Has anyone ever thought about the fact that we may never know that some number is periodic? Even pi might be periodic.
Pi has been proven to be irrational, which means it cannot be periodic. The proofs are kind of grisly though so I won't bother posting them here. You can read about it at http://en.wikipedia.org/wiki/Proof_that_π_is_irrational
For the equation
x^2 = 1
there are two answers:
x = -1 V x = 1
This doesn't mean -1 and 1 are the same
For the equation
(x*10)-9 = x
there are ALSO two answers.
x = 1 V x = 0.9999999999999999999999999999999.......
This also does not mean they are the same.
The last equation is the one you all use to "proof" that 1 and 0.999999999.. are the same.
Also an experiment with zero and 0.00..........001 (1/infinite)
If you divide a pie in an infinite amount of pieces everyone will get an infinitely small pieces (0.00...001) and the pie will be gone, but when everyone gets NO pieces (0 pieces), the pie will be still there, because you didn't give anything to anyone. So the difference between 0.00...0001 and 0 can be an entire pie.
I know I'm late here, but this needed correcting.
This only sounds reasonable if you completely ignore the fundamental theorem of algebra. The usual statement of this theorem is that any non-constant univariate polynomial of degree n has exactly n roots up to multiplicity. Using this theorem, we can show why your reasoning is complete bunk.
Let's start with your x2 = 1 example. This is equivalent to x2 - 1 = 0. This is a non-constant univariate polynomial of degree 2, so it has exactly 2 roots up to multiplicity. So in this case there are 2 roots, namely x = -1 and x = 1.
For the equation
(x*10)-9 = x
there are ALSO two answers.
And this is where you went wrong. The fundamental theorem of algebra tells us that since this is a non-constant polynomial of degree 1 it has exactly 1 root, so it can't possibly have the two answers that you claim.
If you divide the pie by infinite you will give everyone an infinitely small piece(0.000000...00000001).
If you do not divide the pie you will not give anyone anything(0).
Is that infinity countable or uncountable? Do you know anything about infinity? That second question is rhetorical, I know that you don't, which is why I asked the first question.
Has anyone ever thought about the fact that we may never know that some number is periodic? Even pi might be periodic.
A decimal expansion of a real number, x, is periodic if and only if x is rational. This is actually relatively easy to prove. What's considerably more difficult is proving that pi is irrational (but this has also been done, see Arkalius's link). Therefore the decimal expansion of pi is not periodic.
There are no non-zero infinitesimals in real numbers.
If the difference between 0.999... and 1 is an infinitesimal, then when talking about real numbers, that's the same as saying the difference is 0.
If we're going to start talking about number sets broader than the reals, then the whole discussion is more complicated.
I said "we are dealing with infinitesimal numbers" and 0,999... is an infinitesimal number.
I'm going to assume this is a mistake on your part and that you meant that the result of 1 - 0.999... is an infinitesimal, beucase 0.999... certainly is not.
And I also think you didn't pay close attention to what I'm saying. I realize you're talking about infinitesimals. The point is, in real numbers, there are no non-zero infinitesimals. Any mathematical operation on real numbers that would produce an infinitesimal actually produces the value 0. Since 0 is the additive identity, and since, in real numbers 1 - 0.999... = 0, that would make 1 = 0.999... in real numbers.
Michelangelo
Pi has been proven to be irrational, which means it cannot be periodic. The proofs are kind of grisly though so I won't bother posting them here. You can read about it at http://en.wikipedia.org/wiki/Proof_that_π_is_irrational
I know I'm late here, but this needed correcting.
This only sounds reasonable if you completely ignore the fundamental theorem of algebra. The usual statement of this theorem is that any non-constant univariate polynomial of degree n has exactly n roots up to multiplicity. Using this theorem, we can show why your reasoning is complete bunk.
Let's start with your x2 = 1 example. This is equivalent to x2 - 1 = 0. This is a non-constant univariate polynomial of degree 2, so it has exactly 2 roots up to multiplicity. So in this case there are 2 roots, namely x = -1 and x = 1.
And this is where you went wrong. The fundamental theorem of algebra tells us that since this is a non-constant polynomial of degree 1 it has exactly 1 root, so it can't possibly have the two answers that you claim.
Is that infinity countable or uncountable? Do you know anything about infinity? That second question is rhetorical, I know that you don't, which is why I asked the first question.
A decimal expansion of a real number, x, is periodic if and only if x is rational. This is actually relatively easy to prove. What's considerably more difficult is proving that pi is irrational (but this has also been done, see Arkalius's link). Therefore the decimal expansion of pi is not periodic.
Mind = Blown. Quite literally.