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Posts tagged with imaginary numbers

ratios and products of polynomials, exp, log, simple sums and differences of those. Code sketch here.

with Wegert, series and other terrible stuff becomes fun!










playing along with Elias Wegert in R:

X <- matrix(1:100,100,100)                  #grid
X <- X * complex(imaginary=.05) + t(X)/20    #twist & shout
X <- X - complex(real=2.5,imaginary=2.5)     #recentre
plot(X, col=hcl(h=55*Arg(sin(X)), c=Mod(sin(X))*40 ) ,        pch=46, cex=6)

Found it was useful to define these few functions:

arg <- function(z) (Arg(z)+pi)/2/pi*360     #for HCL colour input
ring <- function(C) C[.8 < Mod(C) &   Mod(C) < 1.2]        #focus on the unit circle
lev <- function(x) ceiling(log(x)) - log(x)
m <- function(z) lev(Mod(z))
plat <- function(domain, FUN) plot( domain, col= hcl( h=arg(FUN(domain)), l=70+m(domain)), pch=46, cex=1.5, main=substitute(FUN) )           #say it directly

NB, hcl's hue[0,360] so phase or arg needs to be matched to that.










Suppose you are an intellectual impostor with nothing to say, but with strong ambitions to succeed in academic life, collect a coterie of reverent disciples and have students around the world anoint your pages with respectful yellow highlighter. What kind of literary style would you cultivate?

Not a lucid one, surely, for clarity would expose your lack of content. The chances are that you would produce something like the following:

We can clearly see that there is no bi-univocal correspondence between linear signifying links or archi-writing, depending on the author, and this multireferential, multi-dimensional machinic catalysis. The symmetry of scale, the transversality, the pathic non-discursive character of their expansion: all these dimensions remove us from the logic of the excluded middle and reinforce us in our dismissal of the ontological binarism we criticised previously.

This is a quotation from the psychoanalyst Félix Guattari, one of many fashionable French ‘intellectuals’ outed….

scientist and polemicist Richard Dawkins, Postmodernism Disrobed. A review of Intellectual Impostures published in Nature 9 July 1998, vol. 394, pp. 141-143.

 

Above we read an assertion without evidence. Dawkins posits that an intellectual impostor with nothing to say would write in a certain way. But where’s the proof? I guess whoever’s reading this book review is assumed to already know what Dawkins (Sokal/Bricmont) are talking about and agree with his implications: namely, that postmodernists have nothing to say, and that they cultivate an obtuse literary style to obscure the fact (and that this somehow also attracts followers).

Who says “chances are”? Dawkins’ attack amounts to a flame.

 

Here is a not-unusual passage written in that other famously obtuse jargon, mathematics:

The prototypical example of a C*-algebra is the algebra B(H) of bounded (equivalently continuous) linear operators defined on a complex Hilbert space H; here x* denotes the adjoint operator of the operator x: H → H. In fact every C* algebra, A, is *-isomorphic to a norm-closed adjoint closed subalgebra of B(H)….

That’s from Wikipedia’s article on C* algebras. I think the language is similarly impenetrable to Guattari’s. But mathematics = science = good and humanities = not science = bad, at least in the minds of some.

Here is an excerpt (via @wtnelson) written for teachers of 4–12-year-olds, 40 years ago, by Zoltán Pál Dienes:

psychologically speaking, relating an object to another object is a very different matter from relating a set of objects to another set of objects. In the first case, perceptual judgment can be made on whether the relation holds or not in most cases, whereas in the case of sets, a certain amount of conceptual activity is necessary before such a judgment can take place. For example, we might need to count how many of a certain number of things there are in the set and how many of a certain number of these or of other things there are in another set before we can decide whether the first and the second sets are or are not related by a certain particular relation to each other.

Clear as mud! Clearly Z. P. Dienes was an intellectual impostor with ambitions to collect a coterie of reverent disciples.

 

I don’t know enough about postmodernism to opine on it. I just get annoyed when putatively sceptical people casually wave it off without proving their point.

(And if you’re going to point me to the Sokal Affair or Postmodernism Generator CGI, I’ll point you to At Whom Are We Laughing?.)

 

In Lacan: A Beginner’s Guide, Lionel Bailly describes his subject as “a thinker whose productions are sometimes irritatingly obscure”. He goes on:

Most Lacanian theory [comes from his]  spoken teachings…developed in discourse with…pupils…. [Various modes of presentation which are appropriate in speech] make frustrating reading. …leading the reader toward an idea, but never becoming absolutely explicit…difficult to discover what he actually said…thought on his feet—the ideas…in his seminars were never intended to be cast in stone…freely ascribes to common words new meanings within his theoretical model…Lacan, despite the fuzziness of his communication style, strove desperately hard for intellectual rigour….at the end of the day, it is … clinical relevance that validates Lacan’s model. [Lacan being a psychoanalyst and his ideas coming out of that work.]

So there’s an alternative hypothesis from an authority. Bailly admits the communication style was poor and gives reasons why it was. But rather than judging the work on rhetorical grounds, we should judge it on clinical merit—the ultimate empirical test!

Compare this to Dawkins. Besides the suppositions I already mentioned, he chooses words like: “intellectuals” within scare quotes; ‘anoint’, ‘revere’, ‘coterie’—to undermine the intellectual seriousness of his targets. Who are the empiricists here and who relies on rhetoric?

(Source: members.multimania.nl)




Consider ℂ, the field of complex numbers, as a 1-dimensional vector space. The balanced sets are ℂ itself, the empty set and the open and closed discs centered at 0 (visualizing complex numbers as points in the plane). Contrariwise, in the two dimensional Euclidean space there are many more balanced sets: any line segment with midpoint at (0,0) will do.

As a result, ℂ and ℝ² are entirely different as far as their vector space structure is concerned.

(Source: Wikipedia)




Real numbers are imaginary, and imaginary numbers are real.


[I]maginary numbers describe a physical state of something, so as much as a number can exist, these do. But … real numbers, [being ideal], are imaginary.

David Manheim

(I changed some parts that I don’t agree with but the phrasing and initiative are his.)

The “rational” numbers are ratios and the “counting” numbers are, um, what you get when you count. But “real” and “imaginary” numbers have nothing to do with reality or imagination (each is both real and ideal in the same sense).

 

How about we start referring to them this way?

  • ℝ = the complete numbers. ℝ is the Cauchy-completion of the integers, meaning that ℝ has completely fills in enough options so that any sequential pattern will be able to dance wherever it wants and never need to step its shoe on another element outside the system in order to fulfill its pattern.
  • Any field adjoined to the √−1 becomes "twisting numbers". This derives from the “twisting” feeling one gets when multiplying numbers from ℂ. For example 3exp{i 10°} • 5exp{i 20°} = 15exp{i 30°}, they spiral as they multiply outwards. Keep multiplying numbers off the zero line and they keep twisting. Tristan Needham coined the word “amplitwist” for use in ℂ.
  • ℂ = the complete, twisting numbers. Since ℂ=ℝ adjoin √−1.
  • "Complete spiral numbers" sounds nice as well.

Just to give a few examples of other acceptable numbers systems:

  • ℚ adjoin √2
  • the algebraics
  • ℚ adjoin √[a+√[b+c]]
  • sets
  • DAGs
  • square matrices … with many kinds of stuff inside
    Magma to group2.svg
  • special matrix families
  • certain polynomials (sequences) … taking many kinds of things (not just “regular numbers”) as the inputs
  • clock numbers (modulo numbers)
  • Archimedean fields and non-Archimedean fields
  • functions themselves … and the number of things that functions can represent boggles the mind. Especially when the range can be different than the domain. (Declarative sentences can have a codomain of truth value. Time series have a domain of an interval. Rotations of an object map the object to itself in a space. And more….)
  • And many, many more! Imagination is the limiting reagent here.




I. am. Telling you!! Mathematics is like tripping on acid + mushrooms + peyote + huffing ether + being wide awake the whole time and, if you’re a Real Mathematician, also being ridiculously out of the park intelligent. Can you imagine seeing this in your head and not being able to explain it to anyone. Gaston Julia was born in 1893. No computers, I don’t think they had even invented pen and paper at that point. It was just all in his head. Also the guy had no nose. Yuck.




When I was a maths teacher some curious students (Fez and Andrew) asked, “Does i, √−1, exist? Does infinity ∞ exist?” I told this story.

You explain to me what 4 is by pointing to four rocks on the ground, or dropping them in succession — Peano map, Peano map, Peano map, Peano map. Sure. But that’s an example of the number 4, not the number 4 itself.

So is it even possible to say what a number is? No, let’s ask something easier. What a counting number is. No rationals, reals, complexes, or other logically coherent corpuses of numbers.

Willard van Orman Quine had an interesting answer. He said that the number seventeen “is” the equivalence class of all sets of with 17 elements.

Accept that or not, it’s at least a good try. Whether or not numbers actually exist, we can use math to figure things out. The concepts of √−1 and serve a practical purpose just like the concept of (you know, the obvious moral cap on income tax). For instance

  • if power on the power line is traveling in the direction +1 then the wire is efficient; if it travels in the direction √−1 then the wire heats up but does no useful work. (Er, I guess alternating current alternates between −1 and −1.)
  • allows for limits and therefore derivatives and calculus. Just one example apiece.

Do 6-dimensional spheres exist? Do matrices exist? Do power series exist? Do vector fields exist? Do eigenfunctions exist? Do 400-dimensional spaces exist? Do dynamical systems exist? Yes and no, in the same way.