General > General Technical Chat
Rational, Irrational, and Integer. The correct definition.
TimFox:
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.
Nominal Animal:
--- Quote from: magic on December 21, 2022, 07:59:00 pm ---Can the number of all real numbers be expressed as a ratio of two integers, or is it an irrational number? ;)
--- End quote ---
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?
--- 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"'.
--- End quote ---
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. 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.)
--- Quote from: magic on December 21, 2022, 07:59:00 pm ---Reality is a social construct now.
This is Colonization.
--- End quote ---
I know you're yanking my chain, so I'll just say that I will never, ever accept that, and nobody can make me.
SiliconWizard:
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
CatalinaWOW:
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.
TimFox:
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.
Navigation
[0] Message Index
[#] Next page
[*] Previous page
Go to full version