As a random aside, if anyone wants a fun read, check out An Imaginary Tale: The Story of √-1
Quantum physics falls apart without imaginary numbers
121–130 of 166 posts
Re: Quantum physics falls apart without imaginary numbers
#122My favorite property of "imaginary numbers":
i^i = 0.20787957635... (Spoiler alert: it's real!)
No, they're not imaginary, and yes they have real-world significance. They represent oscillations in control systems [1]. They're useful for accurately approximating derivatives [2]. I really dislike the term "imaginary" because of how useful they actually are.
[1]: https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf
[2]: https://mdolab.engin.umich.edu/wiki/guide-complex-step-deriv...
Re: Quantum physics falls apart without imaginary numbers
#123There is much more to the history of complex numbers, and that is also worth a read [0]. In particular, Gauss was very against the term "imaginary numbers" because it implies some mystery around them. I vaguely remember reading that he preferred the term "lateral" numbers, but that may be a mistake. Euler's formula connects them very plainly with rotations in a complex number plane. The intuition I developed with the…
I guess people felt the same when negative numbers were introduced.
Re: Quantum physics falls apart without imaginary numbers
#124Earlier quoted context omitted.
Same holds for negative numbers. There are no negative quantities in physics, negative numbers as quantities only appear if you order your equations wrong. (And one can argue against the other appearances of negative numbers and minus signs.)
But, how would one handle positive and negative charge?
You just do the bookkeeping, taking into account that if charges are opposite then they attract, etc.
Re: Quantum physics falls apart without imaginary numbers
#125I've always had problems with how complex numbers are taught. The most common explanation is a geometric one, that of the "complex plane", that seems awfully analagous to any old 2D plane. But teachers never seem to explain why you'd have a complex plane in the first place, or when you'd use it instead of a regular plane, and you slowly realize that indeed, nobody's ever using it as a dimensional "plane" at all that'…
You keep saying spiral but in a 2d spiral the value of |y| would be increasing as well. So if I'm understanding you correctly I think you mean helical? As in:
https://qph.cf2.quoracdn.net/main-qimg-1b9122546ee68a13e259e...
Re: Quantum physics falls apart without imaginary numbers
#126Earlier quoted context omitted.
> There really isn't anything weird or suspect about renormalization This is the first time I've heard anyone say that. To me, renormalization is extremely weird, if anything because it's so unrigorous and ad-hoc that I find it hard to believe it even works. Sure, it does the job it's supposed to, and I understand how it does that (for the most part anyway), but that doesn't make it any less weird.
That's always been my take too. Like, ok, we get testable answers and they match experiments but also this is _so_ hacky and I can't shake the feeling that one day someone will come along and show that there's some reason why these bad assumptions work out fine. You know, like how "to find the Schwarzschild radius for a black hole of known mass, calculate the radius at which the escape velocity is equal to the speed…
Read "Quantum Field Theory in a Nutshell."
Re: Quantum physics falls apart without imaginary numbers
#127I've always had problems with how complex numbers are taught. The most common explanation is a geometric one, that of the "complex plane", that seems awfully analagous to any old 2D plane. But teachers never seem to explain why you'd have a complex plane in the first place, or when you'd use it instead of a regular plane, and you slowly realize that indeed, nobody's ever using it as a dimensional "plane" at all that'…
If you should be interested, 3Blue1Brown has a video primer on complex numbers that includes describing multiplications by i as rotations on the plane, showing why a plane is useful for its representation. I always enjoy videos from that channel. https://www.youtube.com/watch?v=5PcpBw5Hbwo
Re: Quantum physics falls apart without imaginary numbers
#128Quantum physics is a description of reality, not reality itself.
Re: Quantum physics falls apart without imaginary numbers
#129Earlier quoted context omitted.
"And as imprecise as that sounds, many of the formulas that take complex numbers as inputs multiply them with other complex numbers in such a way that the imaginary side cancels out. " The only thing imprecise about this is "many". Really any formula for an observable of any kind (including probabilities) has to come out to a real number.
Not really! Often you get a complex solution and both the real and imaginary components are valid.
A complex solution can be valid but you never measure a complex number. I'd argue that in situations like using complex numbers to simulate time varying electrical activity the ontological status of the imaginary part of the solution is uncontroversial: the complex numbers in that situation have no ontological status at all and what is present is charges. In that case we're simply using the complex numbers as a convenient notation for a variable and its conjugate. In quantum mechanics one is more easily led to wonder about whether the ontological status of the complex numbers in that theory really can be settled so easily.
Re: Quantum physics falls apart without imaginary numbers
#130I've always had problems with how complex numbers are taught. The most common explanation is a geometric one, that of the "complex plane", that seems awfully analagous to any old 2D plane. But teachers never seem to explain why you'd have a complex plane in the first place, or when you'd use it instead of a regular plane, and you slowly realize that indeed, nobody's ever using it as a dimensional "plane" at all that'…
You keep saying spiral but in a 2d spiral the value of |y| would be increasing as well. So if I'm understanding you correctly I think you mean helical ? As in: https://qph.cf2.quoracdn.net/main-qimg-1b9122546ee68a13e259e...
The Heyser spiral or Heyser corkscrew seems to be a common name for this type of plot. https://www.google.com/search?q=heyser+spiral
And it connects circles, e/euler's formula, and sin/cos is a visually grokkable way.