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Rational, Irrational, and Integer. The correct definition.
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magic:

--- 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.
Meanwhile, I recall a first semester proof that there is precisely 2ℵ0 real numbers, although things get murkier when you ask what that thing actually is.


--- Quote from: Nominal Animal on December 21, 2022, 09:04:24 pm ---
--- Quote from: magic on December 21, 2022, 07:59:00 pm ---
--- Quote from: Nominal Animal on December 21, 2022, 07:06:19 pm ---The two 'real' mean completely different things: one is 'physically real', and the other is 'the set of numbers \$\mathbb{R}\$ that we call "reals"'.
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Yes, and hardly anyone knows what the latter actually are, that's my point which started this whole exchange.
--- End quote ---
Of course they do.
--- End quote ---
And yet still no proof has been posted that real numbers are a sound concept and anything more than an in-joke between a bunch of privileged European men ;D

Everybody knows how Q is built on Z and how C is built on R and somebody even namedropped quaternions in this thread, but how precisely R extends Q is apparently a mystery.
"It's just the set of all the numbers there are, trust me dude".
"But wait, except for those others, they aren't real numbers because I said so".

So what is 0.0000.... 0007 - a simple, infinite decimal number, apparently perfectly fine according to Encyplopedia Laymanica? >:D


--- Quote from: Nominal Animal on December 21, 2022, 09:04:24 pm ---If we start by agreeing on the concept of a number, then I could walk you through it.  But everything stems from that.  No philosophy or other sophistry needed.
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17th century called and wants its calculus back :P

edit
Really, it's quite an ironic response to the question how to define real numbers.
We will never agree, I could keep producing progressively more bizarre examples.

Like, take a unit square. Remove all points whose either coordinate is rational. Is the total area of the reminder a real number?
Nominal Animal:
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.
--- End quote ---
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.
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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.
T3sl4co1l:

--- 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.

--- End quote ---

Well, numerically perhaps.  They can be represented symbolically, which is as good as anything when you get into higher level forms.

To wit: irrational numbers are constructed by taking the rationals and adding an operation (a root, for example).  This makes a set of numbers adjoined with the new operation.  Then adjoin again so you can have roots of roots, plus offsets inside and outside each root.  And so on and so forth, building whatever kind of number you like.  And finally add the irrationals that don't have any radical representation at all (just straight up roots of algebraic equations; I forget how these are supposed to be otherwise constructed or ordered?).  Rather than a numeric approach, you might represent these numbers as a tree (the sequence of operations starting from rationals on the leaf nodes), or as the equations they are the roots of.  In any case, they don't take any more storage space than any other number, given that it takes countably infinite storage to represent almost any random element from that infinite set.

"Real" numbers are the standout here, with absolutely no representation other than the (transcendental, if not algebraic) equations that represent not even a minuscule fraction of them -- indeed that's only the computable reals.  There's an immesurable infinity of them which are so thoroughly and utterly unknowable as the proverbial elder gods.  We can only infer their existence from the nature of the continuum, minus the few holes we can poke in it with these mere countable sets.

And of the computables, there's no need for any of them to have any relationship to anything else; numbers like pi and e are just about unique in their promiscuity, that they show up all over the place.  But you can also define numbers that have seemingly no possible meaning in relation to anything else (say, take an arbitrary random sequence of digits, as generated by some certain process so we aren't subject to the limitations of rationals; say, the concatenated hashes of all primes?), and, there's nothing insightful about that, it doesn't connect to anything, it's not like you'll have a billion ways to compute it like you do the digits of pi.

So, imagine that, but not even any way to even begin to identify the number, to compute the digits; that's the true unknowableness of the reals.

So, "imaginary" set is only the second worst named, among the most common sets of numbers. :P

Tim
TimFox:
A simplified discussion of real numbers that starts by ignoring their formal definition:

https://abstractmath.org/MM/MMRealNumbers.htm

The important bits are found under "Properties of the Real Numbers", including mathematical concepts such as "closure" and "closed under limits".
An important mathematical concept that is not taught at the elementary level is "limit theory", which is the basis for much of the relevant discussion here.

The formal definitions can be found in the Wikipedia article https://en.wikipedia.org/wiki/Real_number in the section "Formal definitions".

Note that the concept of "icky" is not found in standard references.

The mathematical term "real" is historical yet universal among educated people.
It is not the same as the English word "real" used in politics, history, or other non-mathematical contexts.
SiliconWizard:
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.
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