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Wrote up more stuff at pi.
Incidentally, there are some statements at irrational number that look a little peculiar to me. For example:
In the early modern era, Latin mathematicians began work with imaginary numbers, which are necessarily irrational. They subsequently proved the irrationality of pi, (…)
I suppose Legendre could qualify as a “Latin mathematician of the early modern era” if we take a sufficiently broad view (e.g., he spoke a language in the Latin “clade”), but somehow I feel this is not what the author really had in mind; there were those Renaissance-era Italians who began work with imaginaries IIRC. :-) Probably it would be good to rephrase slightly.
Also this:
There is an easy nonconstructive proof that there exist irrational numbers and such that is rational; let be and let be either or , depending on whether the latter is rational or irrational. A constructive proof is much harder
Not that hard actually: take and , where if I did my arithmetic correctly. Pretty sure that can be made constructive. (Again, I think it’s probably just a case of several thoughts being smooshed together.)
How easy is it to prove that is irrational?
I’ve never heard anyone call imaginary numbers “irrational”.
It’s really easy, because there do not exist nonzero integers such that . :-)
I’ve definitely heard imaginary numbers referred to as irrational, and I think there are compelling reasons for it, although I’d have to think a little to produce them myself. But it’s definitely part and parcel of the Gelfond-Schneider result.
(Well, of course nonzero imaginaries are irrational by definition. (-: )
Looking again at the article pi, I confess that I’m stuck on the characterization of the sine function (that was written by Jesse McKeown, not me). My first idea is somehow to extract the behavior of a complex exponential from the description, but I’m not getting it yet. Where does this come from?
I googled “characterization of sine” and ran across a kind of delightful little fact:
is the only infinitely differentiable real-valued function on the real line such that and for all real and .
Wow!
Nonzero imaginaries are irrational if the definition of “irrational number” is “any number that isn’t rational”, but that’s not really a precise definition until you fix a meaning of “number”. Are infinite ordinal numbers irrational? ? How about nonstandard integers? I’ve always thought of the definition of “irrational number” as being “any real number that isn’t rational”.
Right, the relevant context here is the field of complex numbers, i.e., people do speak of irrationals within that context. Certainly that’s so for Gelfond-Schneider etc.
Okay, I’ll take your word for it that people do use the word in that way. (-: I’ve never heard it.
Re: #1, I agree, those should be rephrased.
Todd (#4), you may be interested in:
The condition given there is different to the one that you give in #4, and I’m slightly suspicious of the one that you give in light of the remarks in the above paper, but a back-of-the-envelope doesn’t give me a counter example.
I suppose intuitively one can see the difficulty in making a curve, , turn sufficiently quickly to touch while controlling the higher derivatives. See how arctan(x)’s higher derivatives go wildly out of control at .
Thanks, Andrew. I saw the assertion quoted in a 1972 paper, and I also found it pretty surprising.
I can’t imagine that anyone will care, but the thing I said I was stuck on in #4 (the nLab’s characterization of sine) has been worked through here. Only an obsessive nerd would actually work through such silliness. :-)
But I do plan on following up on Andrew’s #8. The trouble is that it seems to be stuck behind a pay-wall. :-(
You should have a copy now, Todd :)
You don't have to go all the way to the Gelfond–Schneider Theorem to see that imaginary numbers1 should be regarded as irrational; just think about the symmetries of quadratic equations. The square roots of are not fundamentally different, algebraically, from the square roots of .
Of course, the counter-argument is this t-shirt.
Imaginary numbers are also all non-zero. Zero is purely imaginary (because its real part is zero), but it's not imaginary (because it's real). Using ‘imaginary’ to mean purely imaginary (rather than non-real) is as bad as using ‘irrational’ to mean real and non-rational (rather than non-rational). ↩
I wrote that ‘Latin mathematicians of the early modern era’ bit. I figured that, after a paragraph in which such disparate figures as Abu Kamil (9th century, Egypt) and Omar Khayyam (12th century, Persia) are lumped together as ‘mediaeval Arabic mathematicians’ (a phrase that I also wrote but did not originate), it would be appropriate to treat Tartaglia and Legendre (who were closer to each other in both space and time) in a similar fashion. (Here, both ‘Arabic’ and ‘Latin’ should be regarded primarily as linguistic terms. Of course, Legendre's Latin was a debased vulgar form, but that's nothing compared to Khayyam, who wrote in Persian!)
Toby, I’m not sure what do you mean by “Legendre’s Latin”. As far as I can tell, all of his publications were written in French – is French the debased vulgar form of Latin you refer to? (For that matter, it seems that Tartaglia wrote in Italian, at least in the few works that have survived.)
I didn’t pick up on the lumping together of Abu Kamil and Omar Khayyam; those figures are far less familiar to me. So it was “Latin mathematicians of the early modern era” that stuck out as confusing, not “mediaeval Arabic mathematicians”, but after reading your argument, both phrases now sound like a stretch. I don’t have a recommendation for alternative phrasing yet, but it might be less confusing for readers if there wasn’t that lumping together.
@Toby #15: But both are fundamentally different from , and yet we call irrational.
@Mike: Sure, there are many kinds of irrational numbers. But in particular, there are the quadratic irrational numbers, of which the square roots of and are the most famous examples (followed, I guess, by the golden ratio ). Both of these ought to be considered irrational; that's the claim that I'm defending.
I gather that people who instinctively feel that ought to count as rational are interpreting ‘rational number’ in a complex context as meaning a Gaussian number. But is algebraically much more like or than like , so this is an unwise interpretation. (Of course, is yet different still, being in a sense even more irrational.)
@Todd: There is no defence for the current phrasing, it is just lazy. I naturally wrote the phrase ‘mediaeval Arabic mathematicians’, which is how people often talk about these things, and then when I got to the next paragraph I thought, ‹Well, if those guys are mediaeval Arabic mathematicians, then I guess these guys are early modern Latin mathematicians›. And so then I just wrote that.
Yes, I snarkily referred to French as debased vulgar Latin. Actually, I'm surprised that Tartaglia wrote in Italian; even Euler (a good century later) still wrote in Latin. But I guess that Tartaglia didn't publish his results; Cardano did, and Cardano wrote in Latin.
I was also surprised, while researching my comment #15, to learn that Khayyam wrote in Persian. I naturally assumed that he wrote in Arabic for roughly the same reasons that I naturally assumed that Tartaglia wrote in Latin, but no. (For a little while, I was hoping that Khayyam only wrote poetry in Persian, using Arabic for his scientific work, but I was surprised again.)
Wikipedia more intelligently writes about mathematics in medieval Islam. We could switch from linguistic to religious cultural identifiers: Islamic and Christian instead of Arabic and Latin. (Of course, there is the risk among the early moderns of meeting a nonbeliever1 who is only culturally Christian, but it’s really culture that matters.) Or geographic: near-Eastern and European (although ‘near East’ has some problems of its own).
Actually, we have that risk among the mediaeval Muslims, too. At least Khayyam was devout, but I can't find information on Abu Kamil; and as for Khwarizmi, his religious opinions seem to be a matter of some controversy (he may have been a crypto-Zoroastrian). But again, all of them were culturally Muslim; indeed, they were all ostensibly Muslim, whatever their true personal beliefs may have been. ↩
Toby: as always, your scholarship and follow-through is appreciated.
I tried to implement your suggestion and rewrote parts of irrational number, adding even more gray links in the process. Please have a look and please feel to make further adjustments as you see fit.
For what it’s worth, I don’t think is a rational number either; for me both the rational and the irrational numbers are subsets of the real numbers.
Great, Todd! I have made a few.
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