Author Topic: Rational, Irrational, and Integer. The correct definition.  (Read 6481 times)

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Offline TimFox

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #25 on: December 21, 2022, 09:02:11 pm »
Although mathematics may have been dominated around AD 1920 by white males, there were also people of color at the highest end of mathematics.
See https://www.britannica.com/biography/Srinivasa-Ramanujan

Also note that the Encyclopedia Britannica is now published in the US (headquartered in Chicago for several owners, including the University of Chicago at one time), no longer in UK.
 

Online Nominal Animal

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #26 on: December 21, 2022, 09:04:24 pm »
Can the number of all real numbers be expressed as a ratio of two integers, or is it an irrational number? ;)
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.  Your question is therefore based on an erroneous assertion and thus invalid.

Even the concept of innumerability/uncountability and the cardinality of innumerabilities and infinities is over 2000 years old; the old Jain texts have been preserved to this day.  Those semi-darkie boffins, eh?

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"'.
Yes, and hardly anyone knows what the latter actually are, that's my point which started this whole exchange.
Of course they do.  You don't, but we both know you're atypical anyway.

Any philosophical examination of what the realness of real numbers might mean is as useless as wondering what the "Nominal" in my pseudonym means.  The former case is just a label and the definition is written in the language we call math; the latter is a pun that I feel close to.  Neither warrants any philosophical or sociological analysis, because they just do not have any presence in any such domain.

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.
(But please, don't get into nitpicking on how the English language allows one to construct sentences where the word 'number' is used in different senses, for example in the sense of 'something that is countable'.  Not all languages have that fault.)

Reality is a social construct now.
This is Colonization.
I know you're yanking my chain, so I'll just say that I will never, ever accept that, and nobody can make me.
« Last Edit: December 21, 2022, 09:06:42 pm by Nominal Animal »
 

Online SiliconWizard

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #27 on: December 21, 2022, 09:10:09 pm »
Well given the type of opinions that I've gathered from magic's posts so far, I'm pretty sure he is just trolling here. Although you never know. :-DD
 

Online CatalinaWOW

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #28 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.  So these mathematical concepts have no existence in that subject matter area.  The expressions and concepts that require irrational numbers can be adequately computed by rational approximations to both real and imaginary numbers.  These expressions range from simple things like the area of a circle on up through vey complicated concepts, things like contour integrals and beyond.

In the broader world of mathematics and engineering there is real use for all of these extra concepts that cannot be exactly computed.  The identities and relationships between these computable quantities are heavily dependent on theorems and proofs using these concepts.  If you choose to ignore all of this rich body of knowledge that is fine.  Most of the human race does.  But aren't you at least a little bit curious about the limitations or your "real world" view?  They can occasionally jump up and bite you.  Look up Gibbs phenomena for a relatively simple and benign example. 
 

Offline TimFox

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #29 on: December 21, 2022, 10:11:17 pm »
The Gibbs Phenomenon falls directly out from the mathematics of a Fourier series, but is always surprising the first time that one encounters it as a student.
Most digital computation is glorified (integer) arithmetic, and the precision is limited by the available word size (etc.), but can be expanded to achieve a required precision for an engineering application.

In his other post, the OP claimed that the circumference of a circle could be a rational value, if one merely cut a string to the length required to encircle a cylinder that had been machined.

A parable:  A flying saucer lands at the UN, and the little green men negotiate peacefully.
In exchange for their information on curing cancer and other diseases, humans allow them to visit the important libraries and scan the vast collection of human knowledge contained there.
The aliens convert their scans into Intergalactic ASCII binary code, then place a radix point in front of the huge binary number to make a rational fraction.
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.
 

Offline magic

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #30 on: December 21, 2022, 10:54:14 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.
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 20 real numbers, although things get murkier when you ask what that thing actually is.

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"'.
Yes, and hardly anyone knows what the latter actually are, that's my point which started this whole exchange.
Of course they do.
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

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.
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?
« Last Edit: December 21, 2022, 11:07:31 pm by magic »
 

Online Nominal Animal

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #31 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.

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.
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).

The expressions and concepts that require irrational numbers can be adequately computed by rational approximations to both real and imaginary numbers.

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.

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.
;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..  ::))

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.
And you believe it because some men in position of authority told you so.
Nope, I proved it for myself deduced the logical necessity of it being so for myself, from simple principles.

Really, it's quite an ironic response to the question how to define real numbers.
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.

Like, take a unit square. Remove all points whose either coordinate is rational. Is the total area of the reminder a real number?
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.
« Last Edit: December 21, 2022, 11:18:12 pm by Nominal Animal »
 

Offline T3sl4co1l

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #32 on: December 21, 2022, 11:16:47 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.

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
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Offline TimFox

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #33 on: December 21, 2022, 11:17:49 pm »
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.
 

Online SiliconWizard

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #34 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.
 

Online tggzzz

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #35 on: December 21, 2022, 11:40:29 pm »
I think your question is easier to understand and answer appropriately if you annotated it as shown below.

This is the new and correct definition of a number.
1/ A number must be an integer.
2/ The term rational and irrational don't apply to a number by definition 1/.

An integer is any number that can be defined wholly. For instance, 7, and 1 / 7 can be defined wholly and are therefor integers.
An equation that cannot be defined wholly, such as root 2, where we don't know what number multiplied by itself equals 2, remains a question, and is not a number by definition 1/.
A number can't be rational or irrational. These terms have no meaning.

Thankyou.  :D

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Offline TimFox

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #36 on: December 21, 2022, 11:45:03 pm »
To which, I reply "says who?".
 

Online bdunham7

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #37 on: December 21, 2022, 11:49:04 pm »
To which, I reply "says who?".

Our future dystopian leaders who will insist that all knowledge and arts should be dumbed down to the lowest common denominator so that they don't have to feel inadequate because some things are beyond their grasp.
A 3.5 digit 4.5 digit 5 digit 5.5 digit 6.5 digit 7.5 digit DMM is good enough for most people.
 

Online Nominal Animal

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #38 on: December 21, 2022, 11:54:07 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.
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.
 

Offline free_electron

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #39 on: December 22, 2022, 12:24:38 am »
non-binary binary numbers ? next we'll have transgender gender changers...
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Offline Alex Eisenhut

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #40 on: December 22, 2022, 12:27:27 am »
in da house
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Offline free_electron

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #41 on: December 22, 2022, 12:43:07 am »
-wrong video-... look for Russel peters, invention of zero
« Last Edit: December 22, 2022, 12:46:23 am by free_electron »
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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #42 on: December 22, 2022, 12:47:32 am »
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.
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.

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.
 

Online CatalinaWOW

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #43 on: December 22, 2022, 01:03:57 am »
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.

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.

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).

The expressions and concepts that require irrational numbers can be adequately computed by rational approximations to both real and imaginary numbers.


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.

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.

;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..  ::))

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.

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

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

Really, it's quite an ironic response to the question how to define real numbers.

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.

Like, take a unit square. Remove all points whose either coordinate is rational. Is the total area of the reminder a real number?

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.


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.
 

Offline IanB

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #44 on: December 22, 2022, 02:24:06 am »
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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Offline magic

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #45 on: December 22, 2022, 08:46:09 am »
Like, take a unit square. Remove all points whose either coordinate is rational. Is the total area of the reminder a real number?
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.
"Your question is trivial and the answer is one. Therefore your question is ill-posed and I refuse to answer your other questions".

 :-DD :popcorn:

And again, the reason I ask about a lot of stupid things is not because I expect a sensible answer.
It's to show that you don't know why they are stupid and can't prove them stupid.
You could equally well say they are stupid because some heterosexual dude told you so, which is apparently how good chunk of population perceives mathematics.
I'm not even surprised ;D
« Last Edit: December 22, 2022, 08:50:24 am by magic »
 

Online ledtester

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #46 on: December 22, 2022, 09:22:04 am »
Like, take a unit square. Remove all points whose either coordinate is rational. Is the total area of the reminder a real number?

What you are describing is I x I where I is the set irrational numbers between 0 and 1. The unit square is [0,1] x [0,1].

According to standard measure theory, the set I is measurable and has measure 1. The cross product, therefore, also is measurable and has measure 1*1 = 1.

 

Offline TimFox

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #47 on: December 22, 2022, 04:52:21 pm »
As stated above, the only numbers that can be used in digital computers are rationals, which are treated as binary integers.
All digital computation is binary arithmetic, a small subset of the broad universe of discourse in mathematics.
Progress in computation theory and practice is mainly development of algorithms to improve the efficiency of the arithmetic.

When I started grad school in physics, living in a dorm, one guy on my floor was a mathematics student who disliked the University's placement of the Department of Mathematics in the Division of Physical Sciences, instead of in the humanities where he thought it belonged.
That was a defendable statement;  I asked him what he was studying, and he stated topology and differential geometry, and he was proud that neither subject was of practical use.
I had to point out that physics used both:  topology in Feynman diagrams and differential geometry in General Relativity.

The word "stupid" has been flung about in this discussion.
Ignorance and stupidity are different concepts:  we are all ignorant, since there is far too much for one person to know; stupidity is pride in ones ignorance.
 

Offline Tomorokoshi

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #48 on: December 22, 2022, 04:55:06 pm »
...
All digital computation is binary arithmetic...
...

Unless one is using an IBM 650!
 

Offline TimFox

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #49 on: December 22, 2022, 05:02:22 pm »
...
All digital computation is binary arithmetic...
...

Unless one is using an IBM 650!

The first computer I used (ca. 1968) was an IBM 1620, which, along with the 650, used BCD (binary-coded decimal) internally.
That is another type of binary arithmetic, with slightly different rules.
 


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