10 ms·

This is only mentioned in passing in the article, but the penny dropped for me when I read somewhere else that you should first think about negative numbers. Th

This is only mentioned in passing in the article, but the penny dropped for me when I read somewhere else that you should first think about negative numbers. They seem pretty real, but they're not. There's no negative quantity in nature. You can't have -5 rocks. Negative numbers are just the solution to equations of the form x + n = 0. Now after you let that sink in, imaginary numbers don't seem so weird after all. Just another kind of number defined as the solution to some equation.

mettamage 5y ago
I disagree. > You can't have -5 rocks You can have 5 rocks destroyed in the future. That to me, is what -5 physically means: a guarantee that the object associated with the number will be eliminated out of existence in the future and decrement the negative number by 1. Some people call it debt, but to me, emotionally that word feels too financial. So I prefer “a guarantee to be eliminated out of existence in the future “, or something like that. Conversely, 5 cows means: 5 cows currently in existence. 5 cows - 5 cows means: I see 5 cows and now they don’t exist any more and there is nothing. I wish my math skills were better, I am optimistic that I’d find a similar thing for imaginary numbers and maybe even complex numbers. With that said, I do get where you’re coming from and I find it a compelling perspective as well. It’s simply that I feel the perspective I described as well.
- easywood 5y ago
  >> You can have 5 rocks destroyed in the future You have 5 rocks now, and zero rocks in the future. At no point are there negative rock-shaped holes in the fabric of space. The parent's comment still stands, he is talking about reality, not our mathematical interpretation of it.
- marcus_cemes 5y ago
  > You can have 5 rocks destroyed in the future. Exactly, you use it to store some information that has no real quantity but may be converted to a real quantity in the future through some other process. I'm a polytechnic university student, we use imaginary numbers extensively in all sorts of places, especially whenever there is any oscillatory behaviour, such as an electrical signal or a light wave. A complex number is just a two-dimensional vector with real/imaginary components, whcih provides an amplitude and a phase (angle). An oscillating sinusoidal signal/wave may appear to be zero and completely static if you freeze time at the right moment, but as time progresses, it will continue oscillating, like a swing in a park. In a way, the magnitude represents the built up "momentum" of the system, whilst the real quantity is the immediate physical value at any given point in time (given by the phase). The amplitude is always the same at any given moment, even when the swing is vertical, it has momentum which will help it reach its maximum height. Personally, I still think they are just "invented", but I think the vast majority of engineers much prefer them to the alternative, manipulating trigonometric functions (every engineer's nightmare). They're a neat way to represent the exchange of potential and mechanical/electrical energy with a single value and some simplified mathematics (this is an engineer's, not a mathematician's, point of view). Like negative numbers, we could have chosen to have two positive quantities, balance and debt, instead we find use in merging these definitions, whether negative values make sense or not. We have become used to to negative numbers representing the "inverse" action, which makes sense when representing a phyiscal quantity such as velocity.
  - tsimionescu 5y ago
    > A complex number is just a two-dimensional vector with real/imaginary components, whcih provides an amplitude and a phase (angle) Note that, while the complex numbers are isomorphic to two-dimensional vectors, they are not used the same way. In particular, there is no equivalent to complex multiplication or division that is normally used with two dimensional vectors (though you could define them of course, they are not normally used). The difference is also reinforced by the fact that there is no equivalent of the complex numbers for 3-dimensional vectors, or really any other n-dimensional vectors. That is, you can't define a multiplication and addition operation for n-tuples of real numbers for n>2, with the usual semantics of multiplication and addition (associativity, distributivity, inverse, neutral element).
    - xyzzyz 5y ago
      You can with n=4, you get quaternions. These are not commutative, but that’s really not too bad.
      - hgomersall 5y ago
        You can for any positive n with geometric algebra (which encompasses the complex and quaternion algebras, the latter being much easier to understand in GA).
- taco_emoji 5y ago
  You're not actually disagreeing. OP is saying the negative numbers are useful conceptually, which is what you're demonstrating here. Also thinking about a number line is useful when talking about both negative numbers AND complex ones: negative numbers are to the left of 0, but complex numbers are up and down from the number line.
  - simonh 5y ago
    Never underestimate the ingenuity and imagination some HN will deploy towards the sacred goal of disagreeing with someone.
redler 5y ago
You could have five exactly rock-shaped holes in the ground, and by adding seven rocks, only two remain above the surface.
TheOtherHobbes 5y ago
You can owe someone five rocks (or dollars, to make it only slightly less abstract.) You can go five miles east instead of west (more of a vector, but still makes sense if your movements are limited to a number line instead of a number plane). Imaginary numbers are more of the same with a twist. i is just a 𝛑/2 rotation on a plane instead of a negative number, which can be thought of as rotation by 𝛑 on a line. The big mistake with imaginary numbers is calling them imaginary. There's nothing imaginary about them. They're a very specific kind of operation which can be expanded with very little thought or effort to complex numbers, which have incredibly useful properties in engineering. Calling them "imaginary" is cripplingly confusing for almost everyone, and many never get over it.
- exclipy 5y ago
  > You can go five miles east instead of west You can also go five miles south instead of west
- dr_dshiv 5y ago
  All numbers are imaginary. Perhaps except for one. Is that worked out philosophically? It’s a damn old problem.
ska 5y ago
Lots of replies to you are missing the point I think. You are right, negative numbers are a useful concept which can be mapped to a number of real world operations, but that in some fundamental way not the same thing as positive integers. The same can be said of zero, of course. This is probably why it took humans ages to come up with such concepts, the jump is bigger than it seems after you think of this as "normal". On a related note, real numbers are far weirder than most people think....
vishnugupta 5y ago
> You can't have -5 rocks. Negative numbers are just the solution to equations of the form x + n = 0. The best mental model of negative numbers that works for me is to treat it as direction. In your example, if I have "-5" rocks it means I owe +5 rocks to someone else, let's say John. If, after a few days Rachel were to give me +5 rocks and then you square it off with John you are left with no rocks. Directionally Rachel → (5 rocks) You → (5 rocks) John; So +/- stand for things flowing into and away from you respectively.
- CamperBob2 5y ago
  Exactly. Signed numbers for direction, complex numbers for 2D rotation, quaternions for 3D rotation (if you want to stretch the analogy to the breaking point.)
  - Qem 5y ago
    Octonions?
    - CamperBob2 5y ago
      Over my pay grade.
joppy 5y ago
Negative numbers are more obvious though: if you have half a number line (positive numbers) and you want to talk about relative positions of things, negative numbers make a lot of sense.
mihaic 5y ago
My big problem with imaginary numbers has always been their classification as "numbers" and not some other higher level structure with its own name. We don't call a matrix a number, even if it can be used in equations for instance. "Abelian group elements" ironically don't have a catchy ring to them. Numbers in laymans terms imply something that can be quantified and compared, which in the end I think leads to a lot of confusion when introducing the term.
- deleted 5y ago
  [deleted]
- Jtsummers 5y ago
  Should we also stop calling negative numbers numbers?
  - garmain 5y ago
    Maybe. Consider for example Hossack (2020) Knowledge and Philosophy of Number. He indeed lays out a framework in which negative "numbers" aren't considered numbers, precisely because negative numbers combine a magnitude with a direction.
    - Jtsummers 5y ago
      Let's throw out every textbook and start over, then.
      - garmain 5y ago
        Maybe we should. Irrespective of any concerns on the philosophical underpinnings of mathematics, there is an active discourse on the shortcomings of pedagogical resources. See for example Hung-Hsi Wu and his negative view of "Textbook School Mathematics (TSM)". His critique is aimed at the US. I am unfamiliar with any cohesive argumentation across for example European curricula.
- ravi-delia 5y ago
albrewer 5y ago
Negative numbers show up in nature all the time - they represent how much smaller a particular set of things is than what was defined as the 0 set. Sure, you can't have -5 rocks, but if you're counting how many more rocks you have than when you started counting gains and losses, -5 rocks is a reasonable quantity.

mettamage 5y ago

I disagree. > You can't have -5 rocks You can have 5 rocks destroyed in the future. That to me, is what -5 physically means: a guarantee that the object associated with the number will be eliminated out of existence in the future and decrement the negative number by 1. Some people call it debt, but to me, emotionally that word feels too financial. So I prefer “a guarantee to be eliminated out of existence in the future “, or something like that. Conversely, 5 cows means: 5 cows currently in existence. 5 cows - 5 cows means: I see 5 cows and now they don’t exist any more and there is nothing. I wish my math skills were better, I am optimistic that I’d find a similar thing for imaginary numbers and maybe even complex numbers. With that said, I do get where you’re coming from and I find it a compelling perspective as well. It’s simply that I feel the perspective I described as well.

easywood 5y ago
>> You can have 5 rocks destroyed in the future You have 5 rocks now, and zero rocks in the future. At no point are there negative rock-shaped holes in the fabric of space. The parent's comment still stands, he is talking about reality, not our mathematical interpretation of it.
marcus_cemes 5y ago
> You can have 5 rocks destroyed in the future. Exactly, you use it to store some information that has no real quantity but may be converted to a real quantity in the future through some other process. I'm a polytechnic university student, we use imaginary numbers extensively in all sorts of places, especially whenever there is any oscillatory behaviour, such as an electrical signal or a light wave. A complex number is just a two-dimensional vector with real/imaginary components, whcih provides an amplitude and a phase (angle). An oscillating sinusoidal signal/wave may appear to be zero and completely static if you freeze time at the right moment, but as time progresses, it will continue oscillating, like a swing in a park. In a way, the magnitude represents the built up "momentum" of the system, whilst the real quantity is the immediate physical value at any given point in time (given by the phase). The amplitude is always the same at any given moment, even when the swing is vertical, it has momentum which will help it reach its maximum height. Personally, I still think they are just "invented", but I think the vast majority of engineers much prefer them to the alternative, manipulating trigonometric functions (every engineer's nightmare). They're a neat way to represent the exchange of potential and mechanical/electrical energy with a single value and some simplified mathematics (this is an engineer's, not a mathematician's, point of view). Like negative numbers, we could have chosen to have two positive quantities, balance and debt, instead we find use in merging these definitions, whether negative values make sense or not. We have become used to to negative numbers representing the "inverse" action, which makes sense when representing a phyiscal quantity such as velocity.
- tsimionescu 5y ago
  > A complex number is just a two-dimensional vector with real/imaginary components, whcih provides an amplitude and a phase (angle) Note that, while the complex numbers are isomorphic to two-dimensional vectors, they are not used the same way. In particular, there is no equivalent to complex multiplication or division that is normally used with two dimensional vectors (though you could define them of course, they are not normally used). The difference is also reinforced by the fact that there is no equivalent of the complex numbers for 3-dimensional vectors, or really any other n-dimensional vectors. That is, you can't define a multiplication and addition operation for n-tuples of real numbers for n>2, with the usual semantics of multiplication and addition (associativity, distributivity, inverse, neutral element).
  - xyzzyz 5y ago
    You can with n=4, you get quaternions. These are not commutative, but that’s really not too bad.
    - hgomersall 5y ago
      You can for any positive n with geometric algebra (which encompasses the complex and quaternion algebras, the latter being much easier to understand in GA).
taco_emoji 5y ago
You're not actually disagreeing. OP is saying the negative numbers are useful conceptually, which is what you're demonstrating here. Also thinking about a number line is useful when talking about both negative numbers AND complex ones: negative numbers are to the left of 0, but complex numbers are up and down from the number line.
- simonh 5y ago
  Never underestimate the ingenuity and imagination some HN will deploy towards the sacred goal of disagreeing with someone.

redler 5y ago

You could have five exactly rock-shaped holes in the ground, and by adding seven rocks, only two remain above the surface.

TheOtherHobbes 5y ago

You can owe someone five rocks (or dollars, to make it only slightly less abstract.) You can go five miles east instead of west (more of a vector, but still makes sense if your movements are limited to a number line instead of a number plane). Imaginary numbers are more of the same with a twist. i is just a 𝛑/2 rotation on a plane instead of a negative number, which can be thought of as rotation by 𝛑 on a line. The big mistake with imaginary numbers is calling them imaginary. There's nothing imaginary about them. They're a very specific kind of operation which can be expanded with very little thought or effort to complex numbers, which have incredibly useful properties in engineering. Calling them "imaginary" is cripplingly confusing for almost everyone, and many never get over it.

exclipy 5y ago
> You can go five miles east instead of west You can also go five miles south instead of west
dr_dshiv 5y ago
All numbers are imaginary. Perhaps except for one. Is that worked out philosophically? It’s a damn old problem.

ska 5y ago

Lots of replies to you are missing the point I think. You are right, negative numbers are a useful concept which can be mapped to a number of real world operations, but that in some fundamental way not the same thing as positive integers. The same can be said of zero, of course. This is probably why it took humans ages to come up with such concepts, the jump is bigger than it seems after you think of this as "normal". On a related note, real numbers are far weirder than most people think....

vishnugupta 5y ago

> You can't have -5 rocks. Negative numbers are just the solution to equations of the form x + n = 0. The best mental model of negative numbers that works for me is to treat it as direction. In your example, if I have "-5" rocks it means I owe +5 rocks to someone else, let's say John. If, after a few days Rachel were to give me +5 rocks and then you square it off with John you are left with no rocks. Directionally Rachel → (5 rocks) You → (5 rocks) John; So +/- stand for things flowing into and away from you respectively.

CamperBob2 5y ago
Exactly. Signed numbers for direction, complex numbers for 2D rotation, quaternions for 3D rotation (if you want to stretch the analogy to the breaking point.)
- Qem 5y ago
  Octonions?
  - CamperBob2 5y ago
    Over my pay grade.

joppy 5y ago

Negative numbers are more obvious though: if you have half a number line (positive numbers) and you want to talk about relative positions of things, negative numbers make a lot of sense.

mihaic 5y ago

My big problem with imaginary numbers has always been their classification as "numbers" and not some other higher level structure with its own name. We don't call a matrix a number, even if it can be used in equations for instance. "Abelian group elements" ironically don't have a catchy ring to them. Numbers in laymans terms imply something that can be quantified and compared, which in the end I think leads to a lot of confusion when introducing the term.

deleted 5y ago
[deleted]
Jtsummers 5y ago
Should we also stop calling negative numbers numbers?
- garmain 5y ago
  Maybe. Consider for example Hossack (2020) Knowledge and Philosophy of Number. He indeed lays out a framework in which negative "numbers" aren't considered numbers, precisely because negative numbers combine a magnitude with a direction.
  - Jtsummers 5y ago
    Let's throw out every textbook and start over, then.
    - garmain 5y ago
      Maybe we should. Irrespective of any concerns on the philosophical underpinnings of mathematics, there is an active discourse on the shortcomings of pedagogical resources. See for example Hung-Hsi Wu and his negative view of "Textbook School Mathematics (TSM)". His critique is aimed at the US. I am unfamiliar with any cohesive argumentation across for example European curricula.
ravi-delia 5y ago

albrewer 5y ago

Negative numbers show up in nature all the time - they represent how much smaller a particular set of things is than what was defined as the 0 set. Sure, you can't have -5 rocks, but if you're counting how many more rocks you have than when you started counting gains and losses, -5 rocks is a reasonable quantity.