31–40 of 166 posts
Earlier quoted context omitted.
Did someone claim otherwise?
"Quantum physics falls apart without imaginary numbers." > Marco made a curious face, so Toni posed the question: “Can standard quantum theory work without imaginary numbers?”
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…
The best way I've ever had it explained to me is with electron tunneling. You ask, how did the electron "jump" that potential hill, when it actually didn't have the momentum to do so? The answer: it didn't, it quite literally "went through" the potential hill. So you ask, well, what kind of momentum (mv^2) would allow for this "tunneling" momentum? You invariably arrive at a _negative_ moment... and thus only an imag…
Earlier quoted context omitted.
The name "imaginary" was due to Descartes and it absolutely was intended as a pejorative, even though they're necessary to algebraically close the reals. Some ancient Greeks, IIRC, were similarly hostile to negative numbers. Of course the "real" numbers have never been controversial despite the whole concept being a lot weirder (and uncomputable), probably because their informal aspects just so happen to line up with…
> Of course the "real" numbers have never been controversial Apparently the existence of irrational numbers was a shock to Pythagoreans. There may be also people unhappy with transcendental numbers.
Even from a 21st century perspective, I think that the first "two dimensional number" is always going to freak people out and I can see where it's coming from. Imaginary numbers intrinsically involves leaving numbers that can be used to describe the number of apples you have in your hand, and by the time people get there, they've been pretty darned used to numbers looking like that. Real numbers nominally overshoot that too (you can't really have apples in two hands whose size only differs by 10^(-(10^1000))) but people tend to not have their faces rubbed in this until they get a math degree.
Matrices nominally are such numbers too, but they are often presented as shortcuts rather than numbers in and of themselves.
Of course in the 21st century now we have a zoo of these representations and the community as a whole is comfortable with it.... but for any given person I still think that first number that isn't something that can be a number of meters or apples is a shock.
I don't see how this is remotely true. You can't even solve the ODE for an undamped mass-spring system without imaginary numbers. More generally, most of our notion of eigenvalues falls apart if we work over the field of reals rather than complex numbers, and once you lose that, you lose most of linear algebra and with it vast swaths of engineering.
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…
The best way I've ever had it explained to me is with electron tunneling. You ask, how did the electron "jump" that potential hill, when it actually didn't have the momentum to do so? The answer: it didn't, it quite literally "went through" the potential hill. So you ask, well, what kind of momentum (mv^2) would allow for this "tunneling" momentum? You invariably arrive at a _negative_ moment... and thus only an imag…
Quantum mechanical velocity is complicated as well.
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…
The best way I've ever had it explained to me is with electron tunneling. You ask, how did the electron "jump" that potential hill, when it actually didn't have the momentum to do so? The answer: it didn't, it quite literally "went through" the potential hill. So you ask, well, what kind of momentum (mv^2) would allow for this "tunneling" momentum? You invariably arrive at a _negative_ moment... and thus only an imag…
Earlier quoted context omitted.
"Quantum physics falls apart without imaginary numbers." > Marco made a curious face, so Toni posed the question: “Can standard quantum theory work without imaginary numbers?”
By that logic, it doesn't even need real numbers. Just do everything with cauchy sequences of rationals.
Earlier quoted context omitted.
> Of course the "real" numbers have never been controversial Apparently the existence of irrational numbers was a shock to Pythagoreans. There may be also people unhappy with transcendental numbers.
In 21st century hindsight, being annoyed by irrational numbers seems a bit odd to me. I mean this very much as an opinion. I actually partially understand where they were coming from; it makes a bit more sense than the 21st century perspective would indicate, but still, obviously, not something we'd agree with today. Even from a 21st century perspective, I think that the first "two dimensional number" is always going…
That’s what Pythagoreans thought about irrational numbers, I guess. You don’t need them to denote a fraction of an apple.