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Rational, Irrational, and Integer. The correct definition.

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Alex Eisenhut:
in da house

free_electron:
-wrong video-... look for Russel peters, invention of zero

SiliconWizard:

--- Quote from: Nominal Animal on December 21, 2022, 11:54:07 pm ---
--- Quote from: SiliconWizard on December 21, 2022, 11:22:57 pm ---So it looks like we're going over it all over again - as some have already pointed out - with some (that I remember at least) new concepts introduced about decolonization and some other woke stuff. So it's getting even better than the first time. It's like fine wine. Hats off to Peter for that.

--- End quote ---
You're right.  And I'm sorry for adding to the flames.  Apologies; I should know better.

I just get so darned excited when people talk about stuff that I've found can be effectively used as a tool for both real world work as well as for investigation and learning and science, since I want them to find that utility too, and build stuff I myself never could...  :-//

When it comes to learning, I'm an incorrigible romantic optimist.

--- End quote ---

Well, can't blame. It is interesting per se and I happen to be re-diving into advanced algebra and calculus lately, great stuff. Saddening to see all this turning into a political matter, but I guess it's the sign of the times, as they say.

CatalinaWOW:

--- Quote from: Nominal Animal on December 21, 2022, 11:16:15 pm ---This will sound like nit-picking, but isn't: I'm focusing on the underlying concept, and there is an important concept-level distinction here.

I'm trying to say that there are fundamental concepts related to "number" at play here: the number itself, its representation, and its approximation.  For many irrational numbers, we can represent them exactly in many types of computations on a computer, but can only approximate them in output useful for us humans.


--- Quote from: CatalinaWOW on December 21, 2022, 10:02:25 pm ---There is some minor merit in the OPs position.  In terms of computing there is no way to correctly represent either irrational numbers or infinities.
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Not exactly correct.

We can correctly represent many irrational numbers.  For example, \$\sqrt{2}\$ as \$2^0.5\$.  To convert to a decimal or binary form, we can only show an approximation, but internal calculations can be done exactly, using this and various other representations.

(Background: I have used this exact trick for regular cubic (SC, FCC, and BCC) lattice exact distance histograms.  The distances along each coordinate axis are always integers (or halves), so the distance is an expression of form \$d = \sqrt{\sum_{k=1}^N x_k^2}\$ (possibly multiplied by \$2^{-N}\$).  Because the distance is nonnegative, we can do all computation in the squared domain, i.e. using \$s = d^2 = \sum_{k=1}^N x_k^2\$, and only use the approximations in the labels when displaying the data.  Essentially, my histogram had variable-width bins, but each such bin was exact.  Thus, the histogram itself was exact, and only needed to be approximated when output in a form useful to us humans.)

Similarly, arithmetic infinities can easily be represented; even standard IEEE 754 floating-point types do so correctly (in the arithmetic domain).


--- Quote from: CatalinaWOW on December 21, 2022, 10:02:25 pm ---The expressions and concepts that require irrational numbers can be adequately computed by rational approximations to both real and imaginary numbers.
--- End quote ---


The expressions and concepts that involve irrational numbers can be adequately approximated, yes.  But, we can also compute many of them in a closed form (including in other arithmetic expressions) that allows us to calculate the approximation to whatever precision we like, only limited by available memory and processing time.  \$\pi\$ is a perfect example of this, especially if you want to calculate its representation in binary or hexadecimal instead of decimal, because the Bailey–Borwein–Plouffe formula exists, making it trivial to calculate the k'th hexadecimal digit of \$\pi\$ without having to know any of the preceding or succeeding digits.

This does not mean we cannot use entities that represent the irrational number exactly in computations.  Like I explained for square root of two above, there are ways of representing many irrational numbers –– perhaps not all, like some of the physical constants that might turn out to be irrational; consider the fine structure constant for example –– using alternate, non-binary/non-scalar/non-rational representations in computations; and only need to be approximated for display or final result, for human use.


--- Quote from: TimFox on December 21, 2022, 10:11:17 pm ---To ship the information back to their planet, they engrave an extra line between the original two lines on a Pt-Ir alloy rod corresponding to that fraction.

--- End quote ---

;D

(It is extra funny when one happens to know that you only need 206 bits to express the size of the observable universe in units of Planck length.  If the lines were half a meter apart, there would be fewer than 2³² atoms in the lattice between the marks, so you might be able to store the first four ASCII characters that way.  Apologies to TimFox for me trying to explain the joke..  ::))


--- Quote from: magic on December 21, 2022, 10:54:14 pm ---
--- Quote from: Nominal Animal on December 21, 2022, 09:04:24 pm ---No, there are innumerable real numbers, an uncountable amount, which means there is no number – and there can be no number – that expresses how many unique real numbers there are.

--- End quote ---

And you believe it because some men in position of authority told you so.
--- End quote ---

Nope, I proved it for myself deduced the logical necessity of it being so for myself, from simple principles.


--- Quote from: magic on December 21, 2022, 10:54:14 pm ---Really, it's quite an ironic response to the question how to define real numbers.
--- End quote ---

Not really.  I've seen variants of "Have you stopped beating your wife yet?" that were better posed than any of your questions are.  If you include an error in your question, you cannot expect correct answers to conform to your erroneous logic/understanding that lead to posing the question.


--- Quote from: magic on December 21, 2022, 10:54:14 pm ---Like, take a unit square. Remove all points whose either coordinate is rational. Is the total area of the reminder a real number?
--- End quote ---

Again, a question with an erroneous concept at its core.  By definition, the area of a point is zero.  So, no matter how countably many points you remove, the area does not change.

--- End quote ---


Sorry about not taking the time to dis-assemble this quote to just the points I disagree with you, but here goes.  I completely agree that these more exotic types can be represented symbolically.  And many operations can be performed correctly and exactly on these representations.  Maple and other symbolic operation systems do this.  But both inputs and outputs are symbols and not an exact representation of the number.  And writing an expression root(2) exactly and correctly defines one of these numbers.  But if you generated a histogram with this expression what happened in the machine was that an approximation was created and compared with the data.   Unless the data was all symbolically generated.

The fact remains that the only numbers that computers can deal with are the rationals, and only the subset of the rationals which is within the numerical representation limits of the machine and language used.  Of course those limits are fairly broad, as the specialized software for computing very extended approximations of pi, and those searching for prime numbers demonstrate.

IanB:
The reason I am skipping over this thread and not reading it in detail, is that the whole premise is a fool's errand.

Mathematics is a game, or a collection of games. For each game there are rules you must follow, and for the game to be interesting the rules must be satisfying, and lead to internally consistent logic and conclusions without contradiction. Games with rules that don't meet this criterion are not interesting and are discarded.

An example of interesting rules is for Euclidean geometry, where one of the rules is that the surface we draw on is flat and not curved. There are other games where the surface may be spherical or hyperbolic, and these games are also interesting, but they are different games.

If you want to invent new mathematics that other people can engage in, you need to come up with a set of rules that leads to an interesting game. Otherwise, your new mathematics will not find a following, and you will be doomed to be the only practitioner. Which could be fine for you, but don't expect others to be interested.

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