Quote from: Alexander Kalinowski;1105065All imaginary numbers are irrational; it's a strict subset. Any number that is not a rational (aka fractions) is by definition irrational.

No, Imaginary Numbers are of the form x + iy, where x and y are real numbers and i is the square root of -1. Real Numbers are a subset of Imaginary Numbers, i.e. those where y = 0.

Rational number are, as you say, fractions, including integers that are themselves divided by 1.

Irrational Numbers are those Real Numbers that are not Rational Numbers. There are lots of different types of Irrational Numbers.

So, in terms of subsets, Imaginary > Real > Irrational > Rational > Integers.

The beauty of maths is that between each pair of Rational Numbers is an Irrational Number and between each pair of Irrational Numbers is a Rational Number, but there are "more" Irrational Numbers than Rational Numbers, as Rational Numbers are Countable whereas Irrational Numbers aren't Countable.

Don't get me started ...

Taking it back to SciFi, a lot of people use the Sciences as the basis for storylines or plot devices, depending on what is the flavour of the day. So, it used to be Black Holes, then White Holes, then Wormholes, then Nanotech, Chaos Theory, Catastrophe Theory, Dark Matter, Dark Energy and so on. However, the Mathematics behind these theories is not really understood and very rarely used. I can't think of many examples where Maths is used in SciFi games/stories, except as a means to an end.

I read a short story, might have been by Asimov, where someone managed to manifest the Imaginary part of numbers in the real world, needless to say it didn't go as planned.

Quote from: Rhedyn;1104979Psionics in Soft Sci-fi presumes that interaction with meta-dimensional energies via Faster than Light travel will cause something like psychic powers. It establishes the energy coming from outside know reality via a method that we have no way of knowing to be true or false yet. With no basis in current Science, it's not hard sci-fi, but it doesn't abandon rationality. Your example set up the premise of "effects exceeding causes" as long as that holds, your world is a Fantasy one or Science Fantasy at best.

It depends on the setting. Psionic abilities are sometimes said to have a genetic/hereditary basis and have been around for a very long time.

From a personal point of view, discussing Psionics can be difficult as I have had personal experience with it throughout my life. nothing provable, nothing scientific, usually acknowledged after the fact, but very real to me. So, for me, Psionic Abilities exist and should be part of SciFi.

Quote from: soltakss;1105277So, in terms of subsets, Imaginary > Real > Irrational > Rational > Integers.

You're right! It's been a while since I learned this and irrational numbers usually didn't get more than being mentioned in passing (other than extending rational numbers in algebra).

Quote from: soltakss;1105274I've never understood all the various camps of SciFi and why people are so rigidly protective of them.

I imagine it starts out as a need for categorization so people can talk about the kinds of SciFi they like or don't like. But people are involved so we soon get a lot of "mystuff good, your stuff bad."

Quote from: soltakss;1105277No, Imaginary Numbers are of the form x + iy, where...
...So, in terms of subsets, Imaginary > Real > Irrational > Rational > Integers.
...Rational Numbers are Countable whereas Irrational Numbers aren't Countable.

Glad to see somebody gets it.

Quote from: Alexander Kalinowski;1105281It's been a while since I learned this and irrational numbers usually didn't get more than being mentioned in passing (other than extending rational numbers in algebra).

You aren't quite getting it.

• Complex numbers were presented to you in school as an extension of the real numbers, not of the rational numbers.
  
• Imaginary numbers are introduced by talking about what we mean by the square root of -1.
• But to talk about powers and inverse powers like square roots you have to talk about the irrational numbers like the square root of 2.
• So you don't extend the rational numbers to complex numbers, you first extend the rational numbers to include the irrational numbers, then you extend that to include complex numbers.

How delightful to see so much discussion of mathematics!

One can develop the number systems starting with the natural numbers (the logician's definition, which includes zero; each is the number of natural numbers less than it; there are misguided people who use "natural numbers" to mean the positive integers).

Integers extend the natural numbers to a group with operator addition, by introducing additive inverses of each number other than 0 which is its own inverse already.

Rationals extend the integers to a field with addition and multiplication operators, by introducing multiplicative inverses of non-zero elements and filling in the products of various numbers (or by introducing ratios, but with the added complexity in that case of identifying equivalent fractions: 2/4 being the same as 1/2 and so forth). You can do this with, say, polynomials over a variable x (which are like integers a ring - additive inverses exist, but while closed under multiplication most elements do not have multiplicative inverses).

Irrational numbers are of two types: algebraic numbers that are the roots of polynomials with integer coefficients (so, the square root of 2 and other roots) and transcendental numbers (like pi and e) - the algebraic numbers are still countable. The rationals and irrationals together give us the real numbers, which are limits of convergent sequences of rational numbers (so for pi this might be 3, 3.1, 3.14, 3.141, etc).

And then you can extend real numbers to complex numbers by introducing a root i for the polynomial x^2+1 and completing the number system under the operations of addition and multiplication. Some results in calculus with real numbers make more sense in complex analysis; for example, the radius of convergence for the power series for 1/(x^2+1) at 0 is 1, but this is obvious in complex analysis because the function is undefined at +i and -i which are at distance 1 from the origin in the complex plane.

But number systems can be extended in other ways; for example, adding hyperfinite numbers N which satisfy the formula N > i for each natural number i. The prime model of the natural numbers does not include such numbers, but any finite subset of the axioms for N are supported by some large standard natural number, so the entire theory is consistent and has models - and extending these nonstandard natural numbers lead to infinitesmals (e.g., the reciprocal of N) which support the intuitive development of calculus that Newton and Leibniz used.

Set theory continues past the finite natural numbers to include ordinal and cardinal numbers; the first infinite one is typically called omega (or aleph-0). Which have already been alluded to in distinguishing countable sets of numbers (with that cardinality) from uncountable sets of numbers (with a larger cardinality). But the class of ordinals or cardinals cannot be a set or you run into Russell's paradox (unless you instead throw out some other axioms of sets).

Turing solved one of Hilbert's problems by proposing the Turing machine as a model that would cover any possible machine for solving math problems, and then showed (by the same diagonalization argument by which Cantor showed that the real numbers are not countable) that there is a problem which none of these machines could solve (the halting problem). A machine with some added functionality which could solve the halting problem for unaugmented Turing machines would itself have a halting problem, and so there's a hierarchy of complexity among the functions that such machines could compute.

And then there's the Godel Incompleteness Theorem; and that's where we start to find the magic.

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TL/DR: rawma is apparently more interested in mathematics than anyone else here, and probably more tolerable when discussing politics.

It's like it's 1989 all over again. I can just hear the B-52s playing "Love Shack."

Quote from: Bren;1105329You aren't quite getting it.

The irony here is quite delicious. I was talking about galois theory and field extensions, quite obviously so to anyone who has ever studied algebra (and, no, I wasn't referencing linear algebra). In layman's terms, you're investigating fields that are extension fields of Q and subfields of R.
[It's been decades though, so my memory on the subject is a little fuzzy.]

Quote from: rawma;1105360TL/DR: rawma is apparently more interested in mathematics than anyone else here, and probably more tolerable when discussing politics.

Don't be too sure about that. Although, as mentioned, it's been a while and my rusty memory apparently doesn't serve me to well. Mathematical memory tends to fade faster than normal memories due to the highly abstract nature of the beast, in my experience.

But then again we're being off-topic here.

Quote from: Alexander Kalinowski;1105403The irony here is quite delicious. I was talking about galois theory and field extensions, quite obviously so to anyone who has ever studied algebra

Irony? Like mathematics, your grasp of irony is limited. One certainly doesn't need to take a college level abstract algebra course covering Galois theory to remember the rational, irrational, and complex numbers, since they are covered in high school mathematics. And if you had taken more advanced mathematics you would have seen multiple presentations in multiple courses covering the different number sets, how they are constructed, and how they relate to each other. It's such a fundamental part of mathematics that it's difficult to believe someone would so completely misunderstand and forget it. And it's not like looking up info on number sets on the internet is difficult or time consuming. In any case, here's a link so you can learn or relearn about the complex numbers.

And how this relates to this or any other RPG thread? Well it's another example of someone talking about some rule that they could easily look up, but they don't bother and then they get the rule completely backwards.

Edit. If you really want to discuss Galois theory, you can start a different thread in the proper subforum and I'll dust off my copy of Herstein. It has to be in one of those boxes in the basement.

Quote from: Bren;1105328Glad to see somebody gets it.

Glad to see that my thirty-odd year-old Pure Mathematics degree was worth getting. I so very rarely, get to use any of it these days.

Quote from: Bren;1105470And how this relates to this or any other RPG thread? Well it's another example of someone talking about some rule that they could easily look up, but they don't bother and then they get the rule completely backwards.

It does relate to this thread, in particular.

People who like "Hard " SciFi will refer to the laws of Physics as something that cannot be broken, if they are broken then it isn't Hard SciFi.

However, they often don't have a grasp of the Maths behind those laws. It's almost a case of handwaving the Maths but keeping the Laws.

Quote from: soltakss;1105483Glad to see that my thirty-odd year-old Pure Mathematics degree was worth getting. I so very rarely, get to use any of it these days.

Likewise.

Quote from: Bren;1105470If you really want to discuss Galois theory, you can start a different thread in the proper subforum and I'll dust off my copy of Herstein. It has to be in one of those boxes in the basement.

But what is the proper subforum? I'll need to know before I start looking through any boxes in my basement.

Quote from: soltakss;1105278It depends on the setting. Psionic abilities are sometimes said to have a genetic/hereditary basis and have been around for a very long time.

From a personal point of view, discussing Psionics can be difficult as I have had personal experience with it throughout my life. nothing provable, nothing scientific, usually acknowledged after the fact, but very real to me. So, for me, Psionic Abilities exist and should be part of SciFi.

I find it very odd seeing 'psionic' used in a real world context to mean 'psychic'! 'Psionic' is such an SF word, AIR it originally meant technological augmentation of psychic powers.

Quote from: S'mon;1105498I find it very odd seeing 'psionic' used in a real world context to mean 'psychic'! 'Psionic' is such an SF word, AIR it originally meant technological augmentation of psychic powers.

Huh; apparently a combination of psi with -onics from electronics. A rather weird word to use in a supplement called "Eldritch Wizardry", but Gygax liked weird words; having never encountered it before that, I never never thought of it as science fiction associated.

Quote from: rawma;1105493But what is the proper subforum? I'll need to know before I start looking through any boxes in my basement.

Fair point. If we were discussing using it for a new mechanic it could go in "Design, Development, and Gameplay." Absent that? I got nothing.