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Thank you very much for that. Really cool one 
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I can further the development: see (-1)^(1/4) 
Imaginary numbers are actually from Italy. Solving the quintic, they came up, and everyone said they weren't real, but Cardano showed that you could actually plug in the answers and get zero out, so... They did give answers, but not a real solution that you could use in this world.
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Kenneth Haugland wrote: Imaginary numbers are actually from Italy. Solving the quintic, they came up, and everyone said they weren't real, but Cardano showed that you could actually plug in the answers and get zero out, so... They did give answers, but not a real solution that you could use in this world.
Actually, it was to solve the cubic. The only way to solve a real cubic that has 3 real solutions is to have an intermediate quadratic solution of complex conjugates.
The quintic cannot be solved, and that was proven by Abel, who refined the proof that Ruffini had done (and which was a mess). Later on, Galois used Abel's work to come up with the idea of permutation groups, from which spring the discipline of Group Theory. Arnold came up a with proof that does not rely on Group Theory, but instead relies on the complex plane and functions that wind around the plane.
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My comment got autotuned/autocorrected on me, but yes. Between the proofs and the Renaissance Italians, there was also the paper from Lagrange that showed the similarities in how to get the solutions to these two equations, cubic and quartic, which is basically the start of group theory.
I wanted to study complex numbers in detail but never got around to it. I mean, I know the basics, but there are a lot of neat and cool theorems you can use from them to solve real-world problems.
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Sorry for that question, but at the moment I'm too lazy to think in deep for it...
The sqrt of a positive number x has always two solution +/- (abs(sqrt(x))).
How it is about the sqrt of a negative number? Are there also two solutions?
Sorry again, to be that lazy and simply ask for it 
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x^2 - 1 = 0
x = +/- sqrt(1) => x^2 = 1
x^2 + 1 = 0
x = +/- sqrt(-1) => x^2 = -1
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The reason for the +/- is not due to imaginary numbers, it's a byproduct of how square roots are defined in general. Squaring a number removes sign information, so while we can reverse the process to determine the magnitude of the original value, we can't regain the sign information* so we put +/- because we don't know.
*: You can reason about and figure this out sometimes, but it relies on the specific problem and how it's constrained.
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Jon McKee wrote: I know in school my teachers never really explained them beyond
Unfortunately there are very many people that do not understand proofs.
And even for those that do (or claim so) they don't understand what assumptions and definitions really mean for those proofs.
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But then they find applications with the imaginary numbers in the real world. Sometimes centuries later!
Alternating Current electrical circuits are simplest to model and solve with imaginary numbers.
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You missed the joke.....
ed
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I've got the Beelink GR9 computer that evidently does not have Bluetooth capability. And I have a smartphone in which it seems the only way I can transfer photos that I take on it is to share it via Bluetooth.
Can I just a Bluetooth dongle that could be inserted into a USB drive, and would get the ability to receive files sent via Bluetooth - like these?
Amazon.com : bluetooth dongle[^]
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Form experience - a Bluetooth dongle works perfectly...
"If builders built buildings the way programmers wrote programs, then the first woodpecker that came along would destroy civilization." ― Gerald Weinberg
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ditto
"A little time, a little trouble, your better day"
Badfinger
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I use that exact same model, and it works perfectly
Happiness will never come to those who fail to appreciate what they already have. -Anon
And those who were seen dancing were thought to be insane by those who could not hear the music. -Frederick Nietzsche
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I have the Beelink N5095, which nowhere near fancy as the GR9, and it's got BT built-in. Are you sure it's not just turned off, or the driver needs to be installed?
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I don't even have a WIFI modem that works. I tried to contact Beelink technical support, but that was a joke. I bought it off of Amazon as Beelink was pretty much the only inexpensive, fast micro-desktop - and by the time I had come to the conclusion that the WIFI just doesn't work, I had already installed Windows 10 (i.e., instead of the crap Windows 11) and installed my apps & files, so I wasn't in the mood to zap the drive and return it. With a WIFI USB stick, everything works, and there are a lot of USB ports on the rig, and I have a venerable Dynex USB hub too - and I have exactly 1 free USB port for that Bluetooth. 
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I used a Bluetooth 4.0 dongle with headphones, keyboard, mouse and phone for a couple of years with no issues. Any of them should just plug in and work.
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not supposed to be the only way to tranfer photos, you miss out some tools. I can give you some names, Wondershare, FoneDog Phone Transfer. and they're faster than using bluetooth.
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Maybe the AI can tell us if we did the right thing 
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Can't see that from work. What is it?
The difficult we do right away...
...the impossible takes slightly longer.
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It's a Dilbert. You can try your phone, or enjoy delayed gratification, at home.
"If we don't change direction, we'll end up where we're going"
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