[HN Gopher] Where the complex points are (2016)
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Where the complex points are (2016)
Author : ColinWright
Score : 19 points
Date : 2020-05-21 10:09 UTC (12 hours ago)
(HTM) web link (blogs.adelaide.edu.au)
(TXT) w3m dump (blogs.adelaide.edu.au)
| elteto wrote:
| Not a mathematician so correct away:
|
| I think the initial confusion stemmed from not pinning down the
| domain of x and y.
|
| When he says "I don't see the complex points" in the xy graph
| that is because they aren't there: the xy graph is in R^2 and x^2
| + 1 = y has no solution in the reals for y = 0.
|
| But if you imagine, as he did, an extra dimension attached at
| each point then that is now a complex space, is it not?
| potiuper wrote:
| The domains of x and y both appear defined as the real numbers.
| But, the reals do not have a root for every non-constant
| polynomial defined in terms of x or y as the real numbers are
| an not algebraically closed field. The proposed concept
| attaches two extra (complex) dimensions to each two dimensional
| real point. This is close to visualizing R^4 by attaching an
| R^2 to each point of R^2, so in some sense the tangent bundle
| of R^2.
| cmehdy wrote:
| It's always appreciable to try to comprehend stuff through new
| ways and tools, but I fail to see the aim of the post. Isn't it a
| matter of properly defining the domain of application and
| associating a way to visualize it?
|
| Representing the result of a function from R to R requires two
| orthogonal axes because the element pre-transformation is
| 1-dimensional and the result is also 1-dimensional.
|
| For C to R, this would require 2+1=3 orthogonal axes, so it can
| be visualized with a 3D representation. Likewise for R to C.
|
| From C to C that would be 4-dimensional and becomes already
| trickier without some effort to conceptualize it, and certainly
| becomes less intuitive without resorting to alternate ways to
| conceptualize dimensions.
|
| It quite probable that beyond that, one would really encounter
| decreasing returns on trying to visualize the situation because
| the cost of abstraction would increase in order to rely on
| multisensorial approaches to compensate for our inability to
| visually perceive much beyond 3D.
|
| It's entirely possible that the whole post flew way over my head
| and I absolutely did not get it though, in which case I am truly
| just a confused commenter.
| perl4ever wrote:
| One of the things that is painful because it's out of reach for
| me intellectually and yet tantalizingly straightforward
| sounding is extending the concept of an n-dimensional space to
| infinite dimensions.
|
| The idea of a mathematical function of a function doesn't sound
| like a big deal; it sounds vaguely similar to mundane
| abstractions in programming. This sort of thing is beyond my
| ability to cope with and yet it sounds like counting 1, 2, 3,
| not like abracadabra...
|
| https://en.wikipedia.org/wiki/Functional_analysis
| https://en.wikipedia.org/wiki/Hilbert_space
| JadeNB wrote:
| One of the very important things to know is that, while an
| infinite-dimensional, separable Hilbert space is basically
| "n-dimensional space, but more so", there are more general
| kinds of infinite-dimensional spaces (I'm thinking of Banach
| spaces, but you can certainly get still more general than
| that) that are much more general than that. The prototypical
| examples of these are the L^p spaces, where p stands for [?],
| or a real number p >= 1--but p _isn 't_ the dimension, as in
| R^n (they're all infinite dimensional, at least for a
| reasonable underlying metric space). Rather, it's a parameter
| that controls, in some sense, how close the geometry is to
| being governed by the Pythagorean theorem, so that only p = 2
| gives (a Hilbert space, and hence) the 'usual' geometry.
|
| I think your point with functional programming is spot on.
| Just as one learns, to pick that bete noire, monads not by
| reading yet another clever re-packaging that somehow only
| manages to make them sounds _more_ difficult, but rather by
| finding a problem for which they 're relevant and getting a
| feel for them by using them, so too does one learn about
| infinite-dimensional space not by treating it as some sort of
| philosophical profundity, but by finding a problem for which
| it's the right setting, and realising that it's just a
| mathematical tool like any other.
| btilly wrote:
| I think about it quite differently.
|
| A function in the complex plane is a 4-dimensional thing. A graph
| of that function on the real plane is a 2-D slice of that 4-D
| thing.
|
| That said, the one visualization of complex numbers that I wish
| more understood was a complex number in polar coordinates. In
| polar coordinates, addition is complicated. But multiplication is
| simple. Every complex number is a magnitude and an angle. You
| multiply the magnitudes and add the angles.
|
| What this means is that -1 is (1, 180 degrees). Literally a turn
| halfway around the circle. And now what are its square roots?
| Well i is (1, 90 degrees) and -i is (1, -90 degrees). Now stand
| up and actually do those turns.
|
| The result is that i is a turning motion that takes you off the
| real line. But that visualization helps build intuition about why
| in the complex plane there should be a close connection between
| exponential functions and sin/cos. (Specifically e^(ix) = cos(x)
| + i sin(x) - in other words it is a turn by x radians.)
| galaxyLogic wrote:
| Does it matter what is the orientation of the "iPlane"?
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