4 ms·

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 assoc

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.

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.