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Nice comparison image of the two telescopes together. The improvement is definitely noticeable.

This is a reasonable stance, but you're assuming early scientists weren't methodical, and you're ignoring the next step. Since every possibility can't be right, a scientist must start removing the ones they know won't work from the list of all possibilities. And that's what mainstream science is, the list of remaining explanations that match experiment and observation of the natural world after millions of scientists have worked their entire lives making sure these explanations are trustworthy.

I presume you're referring to the Argand plane? The Argand plane diagrams are graphing complex numbers out from the origin as vectors. We can graph vectors on our Cartesian x-y graph, too. However, the geometric interpretation of the real number line is already a construct of numbers as vectors to my mind: a number selected on the line has a magnitude, its value, and a direction, positive or negative away from 0. For the complex plane, look at this difference: To my mind, this is related to the idea that with complex numbers there is no concept of larger or smaller that can be compared to how we think of real numbers as being closer or further from 0. Imaginary is just a name, given by Descartes himself I believe. Gauss did not like it -- and thought they were on equal footing, to paraphrase his writing. +1 uncool on number sets +1 joigus on matrices

Sorry, I meant addition. A false friend tricked me. A little bit more on this fascinating --at least to me-- topic: Suppose that, for some reason, you are repulsed by numbers which are the square root of a negative real number. You can always obtain a numeric structure that's totally equivalent to complex numbers by means of the following trick: Complex numbers "secretly" are 2x2 real matrices. Now, 2x2 real matrices can always be uniquely expanded into a symmetric part and an antisymmetric part. Here's how you do it. Introduce the special matrices that are going to be respective stand-ins for 1 and i: \[ E=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right) \] \[ I=\left(\begin{array}{cc} 0 & 1\\ -1 & 0 \end{array}\right) \] and define a complex number \( z \) with real part \( x \) and imaginary part \( y \) --and its conjugate-- as "secretly," \[ z=\left(\begin{array}{cc} x & -y\\ y & x \end{array}\right) \] \[ z^{*}=\left(\begin{array}{cc} x & y\\ -y & x \end{array}\right) \] Then, the absolute value (squared) of \( z \) is, \[ z^{*}z=\left(\begin{array}{cc} x & -y\\ y & x \end{array}\right)\left(\begin{array}{cc} x & y\\ -y & x \end{array}\right)=\left(\begin{array}{cc} x^{2}+y^{2} & 0\\ 0 & x^{2}+y^{2} \end{array}\right)=\left(x^{2}+y^{2}\right)E \] The product of \( z \) and \( z' \) --another complex number-- is \[ zz'=\left(xx'-yy'\right)E+\left(xy'+x'y\right)I \] etc. Now, whether complex numbers are "secretly" 2x2 real matrices, or conversely 2x2 real matrices "secretly" are complex numbers is, of course, totally immaterial from a purely mathematical POV. Errata It should be: \[ z=\left(\begin{array}{cc} x & y\\ -y & x \end{array}\right) \] \[ z^{*}=\left(\begin{array}{cc} x & -y\\ y & x \end{array}\right) \] So I guess my answer is: Yes, we do need complex numbers. We can dress them as 2x2 real matrices if we want, but we need them is some disguise or another.

Am I the only one who regrets the passing of metal biscuit tins and their replacement by plastic ones?

The complex numbers (do you know the difference between imaginary numbers and complex numbers ? What does complex mean ? ---- imaginary numbers are in the form Yi. ---- complex numbers have two parts X+Yi, X is a real number. I only can understand complex number in that way - First we invented imaginary number which is not exist. then we try to find some uses of it.

@Philandes Just build lots of clean energy and let the coal plants close; there is nothing so good about coal power plants that we should go out of our way to preserve their viability. Quite the opposite - like so much is wrong about them that we should be going out of our way to get rid of them. We have more options now. The scale of greenhouse construction required would be staggering and coal plant operators are mostly struggling to remain economically viable and taxpayers are not going to support it... This taxpayer sure won't. I think your idea won't work - The exhaust gases won't make fertile soils - more likely they would contaminate soils or require treatment first, to be benign, without any specific benefits. Greenhouse plants can't take enough CO2 out of air to be effective as Carbon Capture; plants can't take out that much. The CO2 that greenhouse plants take in becomes CO2 again through decomposition; it doesn't remove CO2 from the carbon cycle. There is just too much CO2 - we now make more CO2 than all other waste combined, several times over. There isn't anything we make more of besides things like sand and gravel that we don't actually make. Stopping doing the things that make CO2 waste - making energy by other means - is always going to be a better option. @TheVat Agree 100%. The gas and oil industry is offering to do CCS for us IF taxpayer funding is provided to them to do it - so kind of them - but not at levels that would make a difference (apart from sounding 'green') and NEVER out of their revenues, not even revenues at the hyper profit levels currently enjoyed. They like Carbon Capture when they can pump the CO2 down oil wells to force more oil out of nearby wells - the single biggest use. They like it when they can take CO2 out of low quality natural gas they otherwise could not use, to make it more saleable (eg Australia's Gorgon Gas project, the single largest CCS project). They like it best when they get taxpayer funding to do it - and must think it hilarious (in private) when they get emissions reduction funding for activities that increase overall production of fossil fuels, that they know can't ever work at large scale to eliminate emissions. But I am not laughing. For CCS to be able to allow unrestricted use of fossil fuels without emissions it has to become the single largest industry in the world - but without any intrinsic way to be profitable. The largest industry ever, because we make more CO2 than anything else barring things like gravel that we don't actually make; for each ton of fuel burned there is 2 to 3 tons of CO2 - and it should be more than that, but for incomplete, inefficient burning. Any capture methods that combine CO2 with other materials, including plants, has to be that much larger again.

This is true in classical physics, but as it turns out it is not true in quantum mechanics. You can construct a class of experiments where real-valued QM (replace complex numbers by pairs of real ones) makes predictions that are different from complex-valued QM, thereby opening up a way to test this experimentally. Turns out, complex Hilbert spaces are an essential feature of any QM formalism that describes the world accurately (within that domain): https://arxiv.org/abs/2101.10873

Once you've got a good Faraday Cage, drop your cellphone in there, and try to call it from another phone. A pretty reliable test. (My spouse has some valuable data stored at home, so we made several FCs in case of EMP attack) Avoid things that have power cords going into them, even if they appear to be a tight metal box. Moon's trash can is a good one. Be sure it's lined with something so that your computer hardware isn't touching the walls (i.e. air gap). Antique dealers sometimes have old double boilers, or any large metal cooking vessel with a tight lid. Small items, like thumb drives, can go in old metal food tins if tightly lidded. May you never have need for this.

LoL. Apparently the moon now has an atmosphere.

Strictly speaking, no, you do not "need" the complex numbers for anything. You can write everything you do in term of pairs of complex numbers, which just so happen to have the properties of the complex numbers. You could go even further; real numbers are equivalence classes of sequences of rational numbers (under an equivalence relation of, approximately, "limits to the same number"). And further still; rational numbers are equivalence classes of pairs of integers (under an equivalence relation of "same ratio"). And yet further; integers are equivalence classes of pairs of rational numbers (under an equivalence relation of "same difference"). But it's far more convenient to not constantly be going through this entire hierarchy to make even simple statements. It's much, much easier to simply define a set of numbers once, and then deal with all of the ramifications by repeatedly using that definition. And the complex numbers turn out to be useful enough to do so - much more than you might expect, in fact.

There may be two parts; emitter and receiver, placed close together. An IR emitter keeps transmitting infrared light and when any object comes near, it is detected by the receiver by monitoring the reflected light from the object. If you are far from the sensor the emitted IR light reflected from you will not be registered by the receiver. By blocking the sensor there is no path for the emitted IR light to reach the receiver; the IR light will not be reflected to the receiver. Below is a picture, note the emitter and receiver: IR LED and photo diode. (Source: links to https://www.instructables.com/Easy-Infrared-Proximity-Sensor/ The page describes the principle in more detail) You can google for "ir proximity sensor".