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Quantum physics falls apart without imaginary numbers

scientificamerican.com

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Re: Quantum physics falls apart without imaginary numbers

#81

Post by dist-epoch »

andrew_eu wrote: post #15

There 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…

There is a really good visual and intuitive explanation of complex numbers, and the sub-title is "a tale of numbers that like to turn":

https://acko.net/blog/how-to-fold-a-julia-fractal/

Re: Quantum physics falls apart without imaginary numbers

#82

Post by Robotbeat »

andrew_eu wrote: post #15

There 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 always just thought of them as a second dimension to the number line. 2D numbers, if you will. That enables rotation as well, of course, as such a thing doesn’t make sense in 1D. And for certain situations this helps resolve ambiguities that would be difficult and messy without this extra dimension.

Like Quaternions, which add another dimension to our 3 to help solve ambiguities with Euler angles and gimbal lock. Going another dimension up makes the solutions much more elegant.

EDIT: the difference between complex numbers and 2D vectors (as I understand it) is just that you can’t really multiply 2 2D vectors together to get another 2D vector (you need to multiply a scalar times the vector to get a vector), but you CAN multiply two complex numbers together to get another complex number.

To do multiplication with 2 vectors you need to introduce new things like the “inner product” (dot product) to produce another vector (as opposed to “outer product” which produces yet ANOTHER kind of thing, a matrix) whereas with complex numbers you can use regular multiplication without introducing new types of multiplication.

Re: Quantum physics falls apart without imaginary numbers

#83

Post by anthk »

ars wrote: post #23

This video by Sabine Hossenfelder is probably more informative on this topic: https://www.youtube.com/watch?v=ALc8CBYOfkw (it discusses this paper).

Sabine's theories look like a New Age bullshiter parroting "quantumagical" nonsense to sell books, sorry.

Re: Quantum physics falls apart without imaginary numbers

#86

Post by xeonmc »

andrew_eu wrote: post #15

There 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…

For me it’s more intuitive to think of numbers as being either unsigned (magnitude-only) or signed (has magnitude and polarity). It never really made sense to me to accept the concept of negative numbers without also accepting imaginaries.

Re: Quantum physics falls apart without imaginary numbers

#87

Post by tomrod »

Robotbeat wrote: post #82
andrew_eu wrote: post #15

There 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 always just thought of them as a second dimension to the number line. 2D numbers, if you will. That enables rotation as well, of course, as such a thing doesn’t make sense in 1D. And for certain situations this helps resolve ambiguities that would be difficult and messy without this extra dimension. Like Quaternions, which add another dimension to our 3 to help solve ambiguities with Euler angles and gimbal lock. G…

Eh, RE: your edit, "multiplication" itself has several different definitions in a vector space. For example, a dot product, outer product, Hadamard product, etc.

Re: Quantum physics falls apart without imaginary numbers

#88

Post by javajosh »

Physics and computer science share the feature that it seems easier to start by explaining linear motion (linear programs) but all the really interesting stuff is circular (loops). Complex numbers are the simplest representation for describing and combining rotations in a consistent way. (Other representations like "r theta" are not as simple.)

Note also that a world with only monotonic linear motion could not possibly have life or thought or any complex behavior. Rotation is required to model any sort of accretion over time. (note that the typical finite case of "particles in a box" bouncing off the walls is, on average, circular motion too.)

See Clifford Algebra for generalization of the complex numbers

Re: Quantum physics falls apart without imaginary numbers

#89

Post by Robotbeat »

tomrod wrote: post #87
Robotbeat wrote: post #82

Earlier quoted context omitted.

I always just thought of them as a second dimension to the number line. 2D numbers, if you will. That enables rotation as well, of course, as such a thing doesn’t make sense in 1D. And for certain situations this helps resolve ambiguities that would be difficult and messy without this extra dimension. Like Quaternions, which add another dimension to our 3 to help solve ambiguities with Euler angles and gimbal lock. G…

Eh, RE: your edit, "multiplication" itself has several different definitions in a vector space. For example, a dot product, outer product, Hadamard product, etc.

Exactly. You have to introduce these new definitions, which isn’t required for complex numbers.
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