Author Topic: Rational, Irrational, and Integer. The correct definition.  (Read 6479 times)

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Online Nominal Animal

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #50 on: December 22, 2022, 05:54:56 pm »
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.
It was all symbolically generated, yes.  That's what "representation" as opposed to "approximation" means, at its core.

The fact remains that the only numbers that computers can deal with are the rationals
No, that's my point.  Any numerical object you can represent in arithmetic operations, you can use in a computer.

For example, consider the case of representing numbers using format \$S\cdot M^{X/F}\$, where \$S = \pm 1\$, \$M\$ and \$F\$ are unsigned integers, and \$X\$ is a signed integer.  You can represent many irrational numbers exactly with this form, although the algebraic operations you can do are a bit limited (only multiplication and division being trivial and exact).  An approximation is only needed when e.g. a human wants to see what the number is in decimal.

Almost all the computer hardware we use now does use integers, fixed-point integers (\$M/B\$, with \$M\$ a signed integer and \$B\$ an unsigned integer constant), and floating-point numbers (\$M\cdot B^X\$, with \$M\$ a signed integer, \$B\$ an unsigned integer constant most typically 2, and \$X\$ a signed integer).  These are all very efficient and effectivee representations for our typical needs –– that's why they're popular now, instead of the other possibilities! ––, but they are not the only ones, and it is a grave error to think of them as somehow a "requirement" or "definition".
 

Offline TimFox

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #51 on: December 22, 2022, 08:13:50 pm »
A late reply to Nominal Animal's post above about my parable:

"(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..  ::))"

Hamlet:  "There are more things in heaven and earth, Horatio, / Than are dreamt of in your philosophy."

There is a whole lot more data extant than the extent of the universe.  There are unreliable estimates of the data content of the holdings (digitized or not) of the Library of Congress, but they range up to the petabytes, or roughly 253 bits.
 

Online SiliconWizard

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #52 on: December 22, 2022, 11:13:04 pm »
The Hamlet quote is great.
 

Online Nominal Animal

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #53 on: December 23, 2022, 12:51:46 am »
There is a whole lot more data extant than the extent of the universe.  There are unreliable estimates of the data content of the holdings (digitized or not) of the Library of Congress, but they range up to the petabytes, or roughly 253 bits.
Yup, I only meant to illustrate how little data one can encode in a single length.

If I try to consider how much information the universe contains, I get genuinely scared/overwhelmed.  :-[

For illustration, consider a cubic meter of iron, about 7874 kg, containing about 8.5×1028 ≃ 296 iron atoms.  Iron has four stable isotopes, so in theory we could encode two bits per atom just by selecting the isotope, so that cubic meter of iron would then encode 297 bits of information.

How much is that? Well, if you unraveled that cubic meter into a monomolecular thread, just one atom wide, and you could read it at light speed, your bandwidth would be about 262 bits per second, some 500 Libraries of Congress per second.  It would take about 1088 years to read the entire iron cube.

About 30% of Earth is iron, or about 269 cubic meters of it.  That's just on Earth, mind you.  Since iron is the end product of both fusion and fission chains, there is a lot of it in the universe.  And this is just a single information storage method with zero compression, just picking between four well-known stable iron isotopes.  There are much more dense methods; I just picked iron because it is utterly stable, easy to grasp, and Stephen Baxter wrote a story about it.
 

Online SiliconWizard

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #54 on: December 23, 2022, 01:04:04 am »
I know I shouldn't add anything more here and it's not unlikely that the OP is just yanking our chain. With that said...

Universe is thought to be infinite, although we don't know that for sure, nor what this would really mean. But until we can prove it finite, it looks useless to me to try and estimate how many "elementary" particles it contains. If we have even an idea of the smallest ones, which we probably don't yet.

Secondly, to me the only real crucial point about "numbers" is whether, precisely, you admit infinity or not. (Which admittedly can itself be a philosophical matter since we have no means of grasping the reality of it.)

Once you admit infinity, pretty much any number can be derived from that. If you have integers, basic arithmetic operations and infinity, then you have rational numbers, irrational numbers (which can be expressed as infinite sums of rational numbers), complex numbers, which are just an extension of that all, and higher-dimension numbers, and so on. Just a thought. So do you admit infinity?
 

Offline EPAIII

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #55 on: December 23, 2022, 08:03:36 am »
Yea, yea, yea! And the earth is flat. And the moon landings never happened. And ...



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
Paul A.  -   SE Texas
And if you look REAL close at an analog signal,
You will find that it has discrete steps.
 

Online SiliconWizard

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #56 on: December 25, 2022, 08:27:36 pm »
Some food for thought for anyone interested. (Some people call that education, but maybe saying this is oppressive now, so I dunno.)



 

Offline Circlotron

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Re: Rational, Irrational, and Integer. The correct definition.
« Reply #57 on: December 25, 2022, 08:54:04 pm »
There is a whole lot more data extant than the extent of the universe.  There are unreliable estimates of the data content of the holdings (digitized or not) of the Library of Congress, but they range up to the petabytes, or roughly 253 bits.
Yup, I only meant to illustrate how little data one can encode in a single length.

If I try to consider how much information the universe contains, I get genuinely scared/overwhelmed.  :-[

For illustration, consider a cubic meter of iron, about 7874 kg, containing about 8.5×1028 ≃ 296 iron atoms.  Iron has four stable isotopes, so in theory we could encode two bits per atom just by selecting the isotope, so that cubic meter of iron would then encode 297 bits of information.

How much is that? Well, if you unraveled that cubic meter into a monomolecular thread, just one atom wide, and you could read it at light speed, your bandwidth would be about 262 bits per second, some 500 Libraries of Congress per second.  It would take about 1088 years to read the entire iron cube.

About 30% of Earth is iron, or about 269 cubic meters of it.  That's just on Earth, mind you.  Since iron is the end product of both fusion and fission chains, there is a lot of it in the universe.  And this is just a single information storage method with zero compression, just picking between four well-known stable iron isotopes.  There are much more dense methods; I just picked iron because it is utterly stable, easy to grasp, and Stephen Baxter wrote a story about it.

Looks like Big Iron, the euphemism for large scale computer hardware, has been redefined.  :)
 


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