> few of the computable numbers that are not algebraic are interesting, the main exceptions being the numbers that are algebraic expressions containing "2*Pi" and/or "ln 2".
I don’t think this is true at all. For example: the solution to a generic PDE that has no closed form solution at some point of import is likely transcendental, not algebraic, but definitely computable. (Think, say, Navier-Stokes being used for weather predictions in some specific place.)
True, but with such numbers you will normally not do anything else except computing an approximate value of them.
They are not comparable with numbers like 2*Pi or various irrational nth roots that can appear in a lot of relationships and formulae in symbolic computations.
That is what I meant by "interesting", i.e. the necessity of using symbols of such numbers, obviously for use in symbolic computations, since in numeric computations you would never use the actual numbers, but only some approximations of them.
What I have said is equivalent to saying that there are only a few transcendental numbers for which you need symbols.
The number of symbols that are really needed is much less than the number of symbols that happened to be used during the history. For instance a single symbol related to Pi is needed, and it would have been much better if it was a symbol for 2*Pi, not for Pi. When using decimal numbers, one may want to use the value of the decimal logarithm of "e", or of its inverse, but there exists no need whatsoever to use decimal numbers anywhere, this is just a historical accident. Etc., there are various other examples of superfluous constants, which are not needed in any practical application, unlike "2*Pi" and "ln 2", which are ubiquitous (because they appear in the derivation formulae for the trigonometric and exponential functions).
> True, but with such numbers you will normally not do anything else except computing an approximate value of them.
That's what I think people do with other numbers like "pi" at the end of the day, no? :)
> That is what I meant by "interesting", i.e. the necessity of using symbols of such numbers, obviously for use in symbolic computations, since in numeric computations you would never use the actual numbers, but only some approximations of them.
It's very much an encoding problem, I think. Though we probably, on aggregate, use "unnamed computable numbers" implicitly on the order of as much as we use "named computable numbers" the former just has way more of a "tail" of uses where the "encoding of the symbol" is, e.g., "here's the PDE you use to compute this number"!
(It gets a little weird since we're kind of not distinguishing between the approximation that can be used to construct said numbers to arbitrary precision vs the specific program instance that constructs one specific approximation, but the idea is mostly there.)
I don't want to put OP on blast here, but this is unfortunately just complete slop writing.
The points being made are fine, I think, but look, if it's faster for you to generate than it is for us to read, I think this qualifies as denial-of-service-lite.
Yes definitely a great extension would be to add a camera in the image plane (alternatively, defocusing the image slightly and using a photodiode would also be fun!)
yes ! but it also assumes you have: a good optical breadboard + bench + dampeners, a beautiful set of lenses, all sorts of nice lasers and kinematic mounts and linear stages etc etc
so yes, we _also_ (back in my phd lab) built equipment in that sense, but there was a pretty good foundation of Fairly Fancy stuff already sitting around !
All of those parts can also be acquired through alibaba for a stiff discount off the thorlabs pieces though. Whilst some labs have fancy stuff going around, a significant amount don't and there isn't very good equipment sharing between labs at most institutions.
I think getting an electron source and creating a robust-ish adjustable set up is v doable, but is definitely more of a Real Project(TM) than this silly little interferometer :)
> Ok, time to confess: I did cheat a little in calling it the “cheapest” Michelson interferometer, since technically even this beam splitter is like 16 USD, but it is very possible to use a microscope slide instead at the cost of some contrast, which will net out to < 20 cents, even at pretty expensive per-unit prices.
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