There are many standard sets of numbers, such as real numbers and integers. Some of the common sets are often hard to keep track of. In other cases, interesting sets of numbers are entirely ignored by standard math education. Sixteen such sets of numbers, their definitions, and (if applicable) their standard notations are presented here.
I think I was 14 for 16 in terms of grasping those concepts on my first read through….
Which two were tough? They probably need rewritten.
Cool stuff! Some thoughts that occurred to me while reading this:
– Those are really handy definitions of “prime” and “rational” numbers.
– I finally understand what’s meant by “transcendental” numbers. I vaguely knew that pi and e are transcendental, and that radicals aren’t, but this makes sense now.
– Regarding normal numbers: If pi is truly normal, then all these movies and books where somebody finds a message buried in the digits of pi are even sillier, since sooner or later you’ll find _everything_ in there.
– For how many decimal places does a number have to appear rational for it to be called schizophrenic? I mean, pi isn’t, but would 3+pi/10 qualify? (Does this even have a formal definition? Also, is it used for anything, or is it just neat?)
To my knowledge, schizophrenic numbers were defined just for fun, and have not been rigorously studied. I haven’t seen a formal lower limit on the repetitions, but generally speaking, rational or irrational character is determined only by looking at the decimal portion. I would think that 3+pi/10 is not included, but 3.3+pi/100 would be, as then there are a finite number of repetitions after the decimal place. Similarly, 0.027+pi/10000 would not fit the definition, but 0.027027+pi/10000000 would be, as then we have repetition to some finite number of digits.
True, but it also becomes exponentially unlikely that you will find a given arbitrary sequence of digits the longer that arbitrary sequence is, with slightly better odds of finding a “meaningful” sequence of the same length since there are going to be more of those. Statistically, to find a meaningful sequence in a normal number of the kind of length you usually see used by fiction in this way you’d likely have to have the universe’s most powerful computer theoretically possible running flat out for longer than the lifetime of the universe itself.
I suppose a hypothetical omnipotent creator trying to encode messages into universal constants that are also normal numbers would, by definition, known that mathematicians capable of finding the message would also know that every possible sequence of digits if going to be present somewhere. To avoid your message being dismissed as an extremely unlikely fluke you would therefore want to back it up with another message, some how linked to the first. For instance, to borrow from Carl Sagan’s “Contact”, if you were going to draw a circle in zeros and ones in pi, then you might make it fairly early in the sequence and maybe back it up with, say, a square at the same location in another related number such e.
OK, Sagan didn’t put a square in “e”, but I’m almost certain that the sublimity of Euler’s Formula (“e^(i *pi) + 1 = 0”) has something to do with their selection, and would probably be the numbers I’d have picked too.
Regarding the number 1 not having any designation specifically to itself, I’d have to disagree.
1 …. is the loneliest number.
Perhaps it could have the set L all to itself. :-)
Well, 2 can be as bad as one, but I guess the loneliest number is the number 1…
“Although no titles have I won… surely, I must be lonesome number one!” quoth Spouse.
She also says she loves Don Gibson, the happiest depressing music you’ll ever hear!
As I’m unfamiliar with Don Gibson, I went for Three Dog Night. J_W_W, what were you going for?
I recognized the song you guys were going for. (I have a nice cover of it by Aimee Mann.) It’s just that Spouse’s random song generator was running. :)
“Somewhat counterintuitively, there are algebraic
numbers which can only be expressed in terms of the polynomial(s) for whom
they act as roots.”
So these would have to be irrational numbers then? Otherwise we would be able to express them in other ways?
Yes, they would be irrational. They can only be expressed in terms of the polynomials they are solutions to.
I think the section on quaternions could be improved. It was so simplified that it didn’t really convey much information to me. I found the Wikipedia article to be more illuminating. Of special interest was the non-commutativity, specifically that
ij = kbut
ji = -k. I think it also would have been nice to know that the quaternions, because of how they’re defined, make up four orthogonal axes in a 4-dimensional space, and that the extra dimensions can’t be mapped to something simple like the square root of -1 but are instead defined by the results of their mathematical operations.
I’m not sure what Hamilton meant by the quotient of two vectors — perhaps you could explain?
And I’ve heard of “Hamiltonians” being used in quantum computing theory. Is that the same as quaternions?
I’ll redraft the quaternion section later, and repost.
Quaternions are often used in special relativity, though I prefer other methodologies instead, as I’ll describe when that summer school hits. The Hamiltonians that appear in kinematics and quantum theory are other constructs that represent the total mechanical (kinetic and potential) energy of a system or particle.
Regarding algebraic numbers: There actually are special functions (generalizations of elliptic functions iirc) that can be used to solve equations of 5th degree and higher.
Regarding normal numbers: There are numbers that are normal in one base but not in others (e.g., 0.12345678901234567890…). There is a proof in Hardy and Wright that almost all real numbers are normal in every base, but I do not think that anyone has exhibited even a single example of such a number.
Quaternions have many applications in computer graphics representing motions in 3-dimensional space.
There are a type of numbers called “Octionic” which have, as their name implies, eight components and which contain quaternions as a subset. These numbers lose (iirc) associativity, as quaternions lost commutativity.Interestingly enough, there are no extensions to 16.
The book “Numbers (Graduate Texts in Mathematics / Readings in Mathematics) (v. 123) [Paperback]” by Heinz-Dieter Ebbinghaus and many others, published by Springer, has discussions of all these types of numbers (except for the silly one). I liked it.
All in all, a reasonable attempt.