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Will_Newsome comments on Rationality Quotes March 2012 - Less Wrong
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Eh?
QM is a solid theory that reliably predicts every known experiment dependent on it.
And it seems you need to brush up on your arithmetic theory. There is a progression in the usual number fields, Naturals (w/ or w/o 0), Integers, Rationals, Reals, Complex, Quarternions, Octonions, Sedenions, etc.
Then it gets a bit hazy in memory, but I know Quarternions sacrifice commutativity of multiplication and Octonions aren't associative but I can't remember what neat tricks you gain there. The Sedenions have zero divisors but I can't remember what they loose.
Now the point is that complex numbers are the most interesting because they have algebraic closure; you cannot construct an equation with multiplication and addition or almost any other operation in which the solution isn't a Complex number. Not so with the Reals (sqrt -1). Thus Complex numbers are completely logical to be physics rather than Reals.
Not what I would have said. Instead, I think it would be better to say that the reals sacrifice countability in order to gain completeness.
("Uniqueness of representation" isn't a big deal at all. In fact, it doesn't even hold for the natural numbers, which is why there is such a thing as "arithmetic".)
Well, the simplest way to represent a real number is with infinite decimal expansion. Every rational number with a finite digit expansion has two infinite digit expansions. Every Natural has one corresponding string of digits in any positional number system with natural base.
I think at the time of writing I considered 'completeness' to be ill defined, since the real numbers don't have algebraic closure under the exponential operator with negative base and fractional exponent, while with ordinary arithmetic it is impossible to shoot outside of the Complex numbers.
(EDIT: I cant arithmetic field theory today) The best I can come up with is uniqueness of representation, since it implies infinite representations and thus loss of countability. (insofar as I remember my ZFC Sets correctly, a set of all infinite strings with a finite alphabet is uncountable and isomorphic at least to the interval [0,1] of the reals)
EDITED to fix elementary error.
No. Every terminating number has two infinite decimal expansions, one ending with all zeros, the other with all nines.
1/3, for instance is only representable as 0.333... , while 1/8th is representable as 0.124999... and 0.125.
Oh right, thanks for catching that.
No, they don't; that's precisely the point. There are Cauchy sequences of rational numbers which don't converge to any rational number. For an example, simply take the sequence whose nth term is the decimal expansion of pi (or your favorite irrational number) carried out to n digits.
Noted and corrected.