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I have always felt like "imaginary" was a poorly-chosen name. After all, I can plot, in two dimensions, a function that has "imaginary" roots, and yet I can see

I have always felt like "imaginary" was a poorly-chosen name. After all, I can plot, in two dimensions, a function that has "imaginary" roots, and yet I can see those roots in the graph. There is no discontinuity.

mjhay 3y ago
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 everyday intuition.
- mtlmtlmtlmtl 3y ago
  Irrational numbers have been known about since ancient Greece, but they were in fact controversial back when discovered/invented because they challenged conventional wisdom in Greek mathematics at the time.
- kgwgk 3y ago
  > 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.
  - jerf 3y ago
    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 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.
    - kgwgk 3y ago
      > that first number that isn't something that can be a number of meters or apples is a shock. That’s what Pythagoreans thought about irrational numbers, I guess. You don’t need them to denote a fraction of an apple.
    - C-x_C-f 3y ago
      Plus Pythagoreans had a much more religious attitude towards number than any present-day mathematician. In fact I think it's almost misleading to remember them chiefly for their mathematical contributions (many of which are disputed anyway) while they were first and foremost mystic philosophers.
- tokai 3y ago
  >"real" numbers have never been controversial Some controversy does exist. NJ Wildberger is most famous for not believing in the real numbers.
- eternityforest 3y ago
  Why don't imaginary numbers ever show up "in real life" outside of STEM? It's interesting everyone seems to think they are fundamental to everything, but we don't see them. In fact we only see plus/minus/times/divide before getting into "You'll probably use a computer for that, and you probably don't need to unless you're an engineer" stuff. Does anything ever happen outside of science that we could use imaginary numbers to understand? Are they just fundamentally outside ordinary experience, or do we not see them because we have workarounds that hide them? It would be interesting to read a SciFi novel where regular people commonly encounter imaginary numbers and other similarly far from everyday life stuff like calculus.
  - p_j_w 3y ago
    Man, "outside of science" is doing a lot of work here. We also don't see quantum mechanics "outside of science," why would they also not be fundamental? The fact of the matter is that there are A LOT of things we can't explain without complex numbers or quantum mechanics, so people believe that they're fundamental. That's not weird.
    - eternityforest 3y ago
      Yeah, it definitely is. I'm not denying that they're fundamental, I have no reason to doubt that, I just think it's odd that they are so deeply embedded into everything, simple enough to kind of understand the basic idea of, and yet totally separate from everyday experience. Like, a construction worker may know hundreds of techniques. A painter uses hundreds of colors. A while many people likely only use four operations on real numbers their whole lives. Math is everywhere, essential for engineering, but almost totally hidden from most people. Probably just a consequence of how we kind of jumped straight from "experience and guesswork" to "Hire a pro if you want to build a barn and let them use computers" though.
  - RugnirViking 3y ago
    real things heating up and down (i.e not an idealized particle but a thing with volume), the motion of a pendulum, electricity flowing through basically anything (same as heating, when you consider actual volume), these are a few of the things I have worked with where I literally have no option but to use imaginary numbers to model them. Modelling here means predicting what a change would do, for example if I want to make a robot that can balance a stick upright on its hand like you might do with a broom, I use maths related to pendulums to predict the movement of the stick, which require imaginary numbers to show how moving in a certain direction will cause the pendulum to fall (in what direction & how fast). I must emphasise here that I have done these things with real robots, and they do indeed heat/electricity flows/balance as the maths predicts. Does this imply your brain is doing the same maths, imaginary numbers and all, when you balance a broom on your hand? Is there an alternate mathematical notation that doesn't include such strange unintuitive things? probably, thats for the mathematicians to figure out. But as an engineer, they definately do work.
pcrh 3y ago
I have the same attitude towards the use of the term "significant" in statistics, where it has a technical meaning that doesn't correspond to the common language use of the term to mean "important" or "large". Similar to "imaginary" numbers, it's curious how influential/distracting the terms are, though.

mjhay 3y ago

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 everyday intuition.

mtlmtlmtlmtl 3y ago
Irrational numbers have been known about since ancient Greece, but they were in fact controversial back when discovered/invented because they challenged conventional wisdom in Greek mathematics at the time.
kgwgk 3y ago
> 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.
- jerf 3y ago
  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 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.
  - kgwgk 3y ago
    > that first number that isn't something that can be a number of meters or apples is a shock. That’s what Pythagoreans thought about irrational numbers, I guess. You don’t need them to denote a fraction of an apple.
  - C-x_C-f 3y ago
    Plus Pythagoreans had a much more religious attitude towards number than any present-day mathematician. In fact I think it's almost misleading to remember them chiefly for their mathematical contributions (many of which are disputed anyway) while they were first and foremost mystic philosophers.
tokai 3y ago
>"real" numbers have never been controversial Some controversy does exist. NJ Wildberger is most famous for not believing in the real numbers.
eternityforest 3y ago
Why don't imaginary numbers ever show up "in real life" outside of STEM? It's interesting everyone seems to think they are fundamental to everything, but we don't see them. In fact we only see plus/minus/times/divide before getting into "You'll probably use a computer for that, and you probably don't need to unless you're an engineer" stuff. Does anything ever happen outside of science that we could use imaginary numbers to understand? Are they just fundamentally outside ordinary experience, or do we not see them because we have workarounds that hide them? It would be interesting to read a SciFi novel where regular people commonly encounter imaginary numbers and other similarly far from everyday life stuff like calculus.
- p_j_w 3y ago
  Man, "outside of science" is doing a lot of work here. We also don't see quantum mechanics "outside of science," why would they also not be fundamental? The fact of the matter is that there are A LOT of things we can't explain without complex numbers or quantum mechanics, so people believe that they're fundamental. That's not weird.
  - eternityforest 3y ago
    Yeah, it definitely is. I'm not denying that they're fundamental, I have no reason to doubt that, I just think it's odd that they are so deeply embedded into everything, simple enough to kind of understand the basic idea of, and yet totally separate from everyday experience. Like, a construction worker may know hundreds of techniques. A painter uses hundreds of colors. A while many people likely only use four operations on real numbers their whole lives. Math is everywhere, essential for engineering, but almost totally hidden from most people. Probably just a consequence of how we kind of jumped straight from "experience and guesswork" to "Hire a pro if you want to build a barn and let them use computers" though.
- RugnirViking 3y ago
  real things heating up and down (i.e not an idealized particle but a thing with volume), the motion of a pendulum, electricity flowing through basically anything (same as heating, when you consider actual volume), these are a few of the things I have worked with where I literally have no option but to use imaginary numbers to model them. Modelling here means predicting what a change would do, for example if I want to make a robot that can balance a stick upright on its hand like you might do with a broom, I use maths related to pendulums to predict the movement of the stick, which require imaginary numbers to show how moving in a certain direction will cause the pendulum to fall (in what direction & how fast). I must emphasise here that I have done these things with real robots, and they do indeed heat/electricity flows/balance as the maths predicts. Does this imply your brain is doing the same maths, imaginary numbers and all, when you balance a broom on your hand? Is there an alternate mathematical notation that doesn't include such strange unintuitive things? probably, thats for the mathematicians to figure out. But as an engineer, they definately do work.

pcrh 3y ago

I have the same attitude towards the use of the term "significant" in statistics, where it has a technical meaning that doesn't correspond to the common language use of the term to mean "important" or "large". Similar to "imaginary" numbers, it's curious how influential/distracting the terms are, though.