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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" beca

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 them while studying physics was that, unlike "real" numbers which interact by stacking, complex numbers interact by stacking and rotating. This is bizarre to think about with single numbers in a 1D world, but we don't live in a 1D world. In higher dimensions they rotate and sheer rather than just scale. And indeed, QM (at least as it was thought to me) would fall apart without complex numbers. Whether a physical theory can be consistent without them is an interesting question, but not because a physical theory with them creates some kind of metaphysical paradox. [0] https://en.m.wikipedia.org/wiki/Complex_number#History https://en.m.wikipedia.org/wiki/Complex_number#History

reeboo 3y ago
Here is fun, and rather short, book on the history of complex numbers that I liked -- https://www.amazon.com/Imaginary-Tale-Princeton-Science-Library/dp/0691169241 https://www.amazon.com/Imaginary-Tale-Princeton-Science-Libr...
fullstackchris 3y ago
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 imaginary velocity can fit! My intuition leads me to believe that almost some sort of other dimensional effects are at play, and our feeble math just can't accurately describe it. Perhaps it's just the nature of quantum itself, and no special dimensional consideration is needed. It's been a long time since I studied any math or physics...
- tanseydavid 3y ago
  Thank you for sharing this concept. I find it very illuminating.
- mr_mitm 3y ago
  This makes no sense whatsoever. Classical momentum is mv, not mv^2. Also, momentum is a vector and generally can't be negative. Quantum mechanical velocity is complicated as well.
- whatshisface 3y ago
  Momentum is mv, mv^2 is two times the kinetic energy. (Energy is real in quantum systems as well as classical ones.)
- digging 3y ago
  I thought quantum tunneling was due to uncertainty, allowing the position of a particle to resolve on the other side of a barrier sometimes because it is undefined before the tunneling. Are we talking about different things? Is one of us way off the mark? Or are these two angles of the same phenomenon?
- lanza 3y ago
  Classical mechanics are fundamentally wrong. They are low energy approximations to reality. "How did the bal go through the hill when it actually didn't have the momentum to do so" is a nonsense question because the equations of motion you are attempting to use to describe the phenomena are wrong. You can't and shouldn't try to understand QM from a CM standpoint. If you remember Taylor series expansions, this is like trying to understand `sin(x)` by looking at it's first order Taylor series expansion `x`. Your question about momentum and a potential hill is the same question as "how did the value of sin(x) start decreasing if `x` is linear?" You're using too few terms of the series expansion. The mechanisms of Newton's laws are first order terms of a proper QM solution.
  - TechnicolorByte 3y ago
    Incredible analogy! The closer is a keeper: > The mechanisms of Newton’s laws are first order terms of a proper QM solution.
machina_ex_deus 3y ago
If you have probability distribution transition function, it makes sense to want to diagonalize it. And you can't do that without algebraic closure of complex numbers. Even a simple transition matrix of a 3 cycle has complex eigenvalues (the 3rd roots of unity). The stationary probability distribution of some transition matrix is the eigenvectors, so the stationary "probability distribution" of even very simple matrices (like cycle of 3 values) are complex. It's not as magical as they make it out to be.
detrites 3y ago
Maybe my understanding is too limited, but when I had them explained as "rotational" numbers they seemed to reduce to a simple logic shortcut to get signs changing correctly around our arbitrary axes-based coordinate system. No less useful, but kind of mundane. Are there other things they fundamentally do, or is everything else rooted in that property? (Or have I just misunderstood them?)
- bottom999mottob 3y ago
  I don't quite understand your question. Imaginary numbers are useful for modeling waves and particles, both foundational things in our universe. In quantum mechanics, we use complex numbers to describe the behavior of particles. A complex number has two parts: a real part and an imaginary part. The real part represents something we can physically measure, like the position or momentum of a particle. The imaginary part represents something that's a bit harder to grasp - it's related to the probability that the particle will be in a certain state or position. For example, let's say we're trying to describe the position of an electron in an atom. We can't know for sure where the electron is at any given moment, but we can calculate the probability of finding it in a certain region. The probability is represented by a complex number, with the real part telling us the position and the imaginary part telling us the probability. Things like Feynman Diagrams model probability distributions of particles and all possible paths they can take from A to B. They allow us to do interesting calculus
  - detrites 3y ago
    Thanks for the explanation. In engineering circles (to use an apt word) complex numbers only assist in performing rotation, eg, calculating things such as angle changes or phase changes etc. It sounds to me that it's the same in quantum mechanics. That is while at higher levels they allow modelling of things such as probability and state, the way they are fundamentally enabling this is also just the same thing - a shorthand trick that permits fast calculations of rotational characteristics. I'm not sure that they're able to do anything else. A wave is a particle as I understand it, or rather, they are like views into the same thing. But waves have identical characteristics in terms of what is tracked to enable calculations with them. Eg there's amplitude phase and frequency. Maybe they all have probabilities or are otherwise dynamic, but that's what there at the base level. The complex plane then just takes the role of enabling easier calculation and tracking of state changes. A wave is a wave whether it is a sine wave on a scope, a wave of pressure through a solid or gas, a light wave, or the path of a particle when viewed as a wave. Or, I'm misunderstanding and there are actually two, or more types of complex numbers.
TRiG_Ireland 3y ago
Or, as someone I know (Gnomon on h2g2) remarked, they're just as real as real numbers, which is to say, not real at all.
dist-epoch 3y ago
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/ https://acko.net/blog/how-to-fold-a-julia-fractal/
Robotbeat 3y ago
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.
- tomrod 3y ago
  Eh, RE: your edit, "multiplication" itself has several different definitions in a vector space. For example, a dot product, outer product, Hadamard product, etc.
  - Robotbeat 3y ago
    Exactly. You have to introduce these new definitions, which isn’t required for complex numbers.
- computerfriend 3y ago
  You've gone up a degree for each example. The inner product gives you a number and the "cross product" for vectors gives you another vector and is not the same thing as the outer product. I think it's also not really correct to compare them like this, because complex numbers give you complex structure, whereas vectors don't. Yes, you get another dimension, but also a rich algebra, Cauchy-Riemann equations, etc.
  - 3y ago
xeonmc 3y ago
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.
IIAOPSW 3y ago
Quantum physics would not fall apart. In quantum computing you can always replace the imaginary component at the cost of just one extra qubit in your circuit. The real part maps to the |0> state on your ancila, the imaginary to the |1> component. In terms of quantum mechanics as a theory, this implies you can always do away with the imaginary part at the cost of introducing just one fictitious two-level degree of freedom, like idk call it "spin" or something I guess. Oh shit. Put another way, complex numbers are a qubit the universe gives you for free.
- bollu 3y ago
  I do not believe this. What will one do about intermetidate computations which can have complex coefficients? In general, you'd need some way to change the gates/ unitary matrices themselves to be purely real. So you'd need to find an isomorpism from U(n) into a subgroup of SO(poly(n)) for this claim to work. Why does such an isomorphism exist?
  - IIAOPSW 3y ago
    Its simple. Start with your circuit and add one ancila qubit. Then in your set of basis gates (universal for quantum computation), replace every phase shifting operation with a controlled X rotation targeted on that ancila. For example, lets just use the cliffords plus arbitrary phase rotation {X,Y,Z,H,R(theta)}. X,Z and H are all real. Y is real up to an irrelevant global phase (if you really want to implement it anyway, just do X,Z on the target and then flip the ancila with an additional X). All that's left to handle is R(theta). Map R(theta) into a (real) controlled rotation in X (to wit: C-X(theta)). Thus we have a set of gates, universal for quantum computation, using only real numbers. If you don't believe in arbitrary rotations X(theta) without intermediate imaginary operations, just pretend I used the T gate instead to extend the Cliffords. Either way you can construct a set which is universal for QC with only real numbers. Why should it work? Simple. In your math, if you replace every i with |i> your equations are still the same. You've just substituted one squiggle on the page for another that works the same way. The Riemann sphere is just like the Bloch sphere. Complex numbers are a qubit the universe gives you for free.
dxuh 3y ago
I often wonder about how much mystery and skepticism would surround them if they were simply called "complete numbers" instead. Of course that's not a great name either, but it's vaguely motivated them being algebraically closed and most importantly it's a neutral, or even slightly positive name.
MengerSponge 3y ago
I prefer "subtle" over "imaginary"
albrewer 3y ago
These[0] two[1] videos are probably the best overview of imaginary numbers I've ever come across. The whole channel is a treasure trove of the history behind math, notation, and conventions used in physics. [0]: https://www.youtube.com/watch?v=CdwxpSInhvU https://www.youtube.com/watch?v=CdwxpSInhvU [1]:https://www.youtube.com/watch?v=M12CJIuX8D4 https://www.youtube.com/watch?v=M12CJIuX8D4
amelius 3y ago
I guess people felt the same when negative numbers were introduced.
lacker 3y ago
Whether a physical theory can be consistent without them is an interesting question I agree, it is interesting, and the answer is yes, a physical theory can be completely consistent while never using complex numbers. If you don't want to use imaginary numbers, you can avoid them for almost anything by using matrices instead. Use 2x2 matrices instead of complex numbers, and instead of 1, use the identity matrix: 1 0 0 1 Instead of i, use a matrix that corresponds to a 90 degree rotation: 0 -1 1 0 This means that i * i = -1, and so pretty much everything else will just work the same way as it does with complex numbers. Adding, multiplying, calculus, all the same. No need for complex numbers if you don't like them. I do like complex numbers though. They are just a bit more concise and convenient than using matrices.
winter_blue 3y ago
Can complex numbers be seen as just a tuple of length 2, containing two real numbers?

reeboo 3y ago

Here is fun, and rather short, book on the history of complex numbers that I liked -- https://www.amazon.com/Imaginary-Tale-Princeton-Science-Library/dp/0691169241 https://www.amazon.com/Imaginary-Tale-Princeton-Science-Libr...

fullstackchris 3y ago

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 imaginary velocity can fit! My intuition leads me to believe that almost some sort of other dimensional effects are at play, and our feeble math just can't accurately describe it. Perhaps it's just the nature of quantum itself, and no special dimensional consideration is needed. It's been a long time since I studied any math or physics...

tanseydavid 3y ago
Thank you for sharing this concept. I find it very illuminating.
mr_mitm 3y ago
This makes no sense whatsoever. Classical momentum is mv, not mv^2. Also, momentum is a vector and generally can't be negative. Quantum mechanical velocity is complicated as well.
whatshisface 3y ago
Momentum is mv, mv^2 is two times the kinetic energy. (Energy is real in quantum systems as well as classical ones.)
digging 3y ago
I thought quantum tunneling was due to uncertainty, allowing the position of a particle to resolve on the other side of a barrier sometimes because it is undefined before the tunneling. Are we talking about different things? Is one of us way off the mark? Or are these two angles of the same phenomenon?
lanza 3y ago
Classical mechanics are fundamentally wrong. They are low energy approximations to reality. "How did the bal go through the hill when it actually didn't have the momentum to do so" is a nonsense question because the equations of motion you are attempting to use to describe the phenomena are wrong. You can't and shouldn't try to understand QM from a CM standpoint. If you remember Taylor series expansions, this is like trying to understand `sin(x)` by looking at it's first order Taylor series expansion `x`. Your question about momentum and a potential hill is the same question as "how did the value of sin(x) start decreasing if `x` is linear?" You're using too few terms of the series expansion. The mechanisms of Newton's laws are first order terms of a proper QM solution.
- TechnicolorByte 3y ago
  Incredible analogy! The closer is a keeper: > The mechanisms of Newton’s laws are first order terms of a proper QM solution.

machina_ex_deus 3y ago

If you have probability distribution transition function, it makes sense to want to diagonalize it. And you can't do that without algebraic closure of complex numbers. Even a simple transition matrix of a 3 cycle has complex eigenvalues (the 3rd roots of unity). The stationary probability distribution of some transition matrix is the eigenvectors, so the stationary "probability distribution" of even very simple matrices (like cycle of 3 values) are complex. It's not as magical as they make it out to be.

detrites 3y ago

Maybe my understanding is too limited, but when I had them explained as "rotational" numbers they seemed to reduce to a simple logic shortcut to get signs changing correctly around our arbitrary axes-based coordinate system. No less useful, but kind of mundane. Are there other things they fundamentally do, or is everything else rooted in that property? (Or have I just misunderstood them?)

bottom999mottob 3y ago
I don't quite understand your question. Imaginary numbers are useful for modeling waves and particles, both foundational things in our universe. In quantum mechanics, we use complex numbers to describe the behavior of particles. A complex number has two parts: a real part and an imaginary part. The real part represents something we can physically measure, like the position or momentum of a particle. The imaginary part represents something that's a bit harder to grasp - it's related to the probability that the particle will be in a certain state or position. For example, let's say we're trying to describe the position of an electron in an atom. We can't know for sure where the electron is at any given moment, but we can calculate the probability of finding it in a certain region. The probability is represented by a complex number, with the real part telling us the position and the imaginary part telling us the probability. Things like Feynman Diagrams model probability distributions of particles and all possible paths they can take from A to B. They allow us to do interesting calculus
- detrites 3y ago
  Thanks for the explanation. In engineering circles (to use an apt word) complex numbers only assist in performing rotation, eg, calculating things such as angle changes or phase changes etc. It sounds to me that it's the same in quantum mechanics. That is while at higher levels they allow modelling of things such as probability and state, the way they are fundamentally enabling this is also just the same thing - a shorthand trick that permits fast calculations of rotational characteristics. I'm not sure that they're able to do anything else. A wave is a particle as I understand it, or rather, they are like views into the same thing. But waves have identical characteristics in terms of what is tracked to enable calculations with them. Eg there's amplitude phase and frequency. Maybe they all have probabilities or are otherwise dynamic, but that's what there at the base level. The complex plane then just takes the role of enabling easier calculation and tracking of state changes. A wave is a wave whether it is a sine wave on a scope, a wave of pressure through a solid or gas, a light wave, or the path of a particle when viewed as a wave. Or, I'm misunderstanding and there are actually two, or more types of complex numbers.

TRiG_Ireland 3y ago

Or, as someone I know (Gnomon on h2g2) remarked, they're just as real as real numbers, which is to say, not real at all.

dist-epoch 3y ago

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

Robotbeat 3y ago

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.

tomrod 3y ago
Eh, RE: your edit, "multiplication" itself has several different definitions in a vector space. For example, a dot product, outer product, Hadamard product, etc.
- Robotbeat 3y ago
  Exactly. You have to introduce these new definitions, which isn’t required for complex numbers.
computerfriend 3y ago
You've gone up a degree for each example. The inner product gives you a number and the "cross product" for vectors gives you another vector and is not the same thing as the outer product. I think it's also not really correct to compare them like this, because complex numbers give you complex structure, whereas vectors don't. Yes, you get another dimension, but also a rich algebra, Cauchy-Riemann equations, etc.
- 3y ago

xeonmc 3y ago

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.

IIAOPSW 3y ago

Quantum physics would not fall apart. In quantum computing you can always replace the imaginary component at the cost of just one extra qubit in your circuit. The real part maps to the |0> state on your ancila, the imaginary to the |1> component. In terms of quantum mechanics as a theory, this implies you can always do away with the imaginary part at the cost of introducing just one fictitious two-level degree of freedom, like idk call it "spin" or something I guess. Oh shit. Put another way, complex numbers are a qubit the universe gives you for free.

bollu 3y ago
I do not believe this. What will one do about intermetidate computations which can have complex coefficients? In general, you'd need some way to change the gates/ unitary matrices themselves to be purely real. So you'd need to find an isomorpism from U(n) into a subgroup of SO(poly(n)) for this claim to work. Why does such an isomorphism exist?
- IIAOPSW 3y ago
  Its simple. Start with your circuit and add one ancila qubit. Then in your set of basis gates (universal for quantum computation), replace every phase shifting operation with a controlled X rotation targeted on that ancila. For example, lets just use the cliffords plus arbitrary phase rotation {X,Y,Z,H,R(theta)}. X,Z and H are all real. Y is real up to an irrelevant global phase (if you really want to implement it anyway, just do X,Z on the target and then flip the ancila with an additional X). All that's left to handle is R(theta). Map R(theta) into a (real) controlled rotation in X (to wit: C-X(theta)). Thus we have a set of gates, universal for quantum computation, using only real numbers. If you don't believe in arbitrary rotations X(theta) without intermediate imaginary operations, just pretend I used the T gate instead to extend the Cliffords. Either way you can construct a set which is universal for QC with only real numbers. Why should it work? Simple. In your math, if you replace every i with |i> your equations are still the same. You've just substituted one squiggle on the page for another that works the same way. The Riemann sphere is just like the Bloch sphere. Complex numbers are a qubit the universe gives you for free.

dxuh 3y ago

I often wonder about how much mystery and skepticism would surround them if they were simply called "complete numbers" instead. Of course that's not a great name either, but it's vaguely motivated them being algebraically closed and most importantly it's a neutral, or even slightly positive name.

MengerSponge 3y ago

I prefer "subtle" over "imaginary"

albrewer 3y ago

These[0] two[1] videos are probably the best overview of imaginary numbers I've ever come across. The whole channel is a treasure trove of the history behind math, notation, and conventions used in physics. [0]: https://www.youtube.com/watch?v=CdwxpSInhvU https://www.youtube.com/watch?v=CdwxpSInhvU [1]:https://www.youtube.com/watch?v=M12CJIuX8D4 https://www.youtube.com/watch?v=M12CJIuX8D4

amelius 3y ago

I guess people felt the same when negative numbers were introduced.

lacker 3y ago

Whether a physical theory can be consistent without them is an interesting question I agree, it is interesting, and the answer is yes, a physical theory can be completely consistent while never using complex numbers. If you don't want to use imaginary numbers, you can avoid them for almost anything by using matrices instead. Use 2x2 matrices instead of complex numbers, and instead of 1, use the identity matrix: 1 0 0 1 Instead of i, use a matrix that corresponds to a 90 degree rotation: 0 -1 1 0 This means that i * i = -1, and so pretty much everything else will just work the same way as it does with complex numbers. Adding, multiplying, calculus, all the same. No need for complex numbers if you don't like them. I do like complex numbers though. They are just a bit more concise and convenient than using matrices.

winter_blue 3y ago

Can complex numbers be seen as just a tuple of length 2, containing two real numbers?