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Try forgetting a lot of what you know about numbers then, and maybe then you might understand better. Real numbers are quite easy for people to "understand"--y

Try forgetting a lot of what you know about numbers then, and maybe then you might understand better. Real numbers are quite easy for people to "understand"--you can point out the ratio between a circle's circumference and its diameter will always be constant, and name it, while not being exactly expressible as a ratio between two integers, and people will understand that as a concept. Yes, real numbers have all sorts of weird properties, are challenging to build in a set-theoretic principle, etc., but most people aren't going to reach a level of mathematics where those issues come up. By contrast, complex numbers are invariably introduced as a mathematical gimmick. The imaginary number 'i' is sqrt(-1)... what does that even mean? What would an i-meter long rod look like? What can you point to as a physical construct that encapsulates complex numbers? Especially as is often taught, complex numbers just feel like someone arbitrarily decides that it means something, and you're then manipulating something without any justification for what it actually is or why you should care. I'll note that negative numbers are similar to complex numbers: a rod of -1 meters doesn't make sense as a physical construct. Indeed, historically, negative numbers aren't in much use until about the same time complex numbers are invented. However, the value of negative numbers is readily apparent: you can use it to keep track of directionality; a credit of -$100 is clearly the same as a debit of $100, for example. With complex numbers, examples of why you might want to do the work are rarely forthcoming. But to give the example no one gave me in school: complex numbers encode magnitude and phase--it will tell if you adding two things that are equally strong will cancel each other out completely, intensify each other, or something in between. (Of course, this is not readily apparent from the usual representation of complex numbers as a + bi, which may be why it's not really covered).

onethought 5y ago
I think you invalidated your main criticism with i not being physically representable just like negative numbers which people have no problem comprehending, despite also suffering that same problem. Our "comfort" with mathematics isn't because some concepts are better and some aren't. It's probably more how often we encounter it in everyday life. How often people need to apply complex numbers is WAY less than postive/negative/real numbers. ALL of the concepts are made up though. There isn't a set of math that isn't entirely made up, with no relation the physical world except for the relationships we also make up.
- jcranmer 5y ago
  > I think you invalidated your main criticism with i not being physically representable just like negative numbers which people have no problem comprehending, despite also suffering that same problem. The point is that, when negative numbers are introduced, they're invariably done so in a way that helps people get over that no-physical-meaning hump. Complex numbers generally aren't, and often, if you try to push for examples, the people teaching themselves don't give anything that helps. Compare to the article on "Negative number" on Wikipedia to "Complex number". For the former, you are immediately greeted in the first paragraph three separate example use cases for negative number. Turn to the latter, and... it's basically all just mathy justification: complex numbers make this bit of math work. Oh, there's a sentence on how they are "fundamental in many aspects of the scientific description of the natural world"--but there's no examples! Even scrolling down to the applications section, the descriptions tend more towards "complex numbers are used in this field" rather than "this is what a complex number represents." Put differently, if someone asks a question "what is a negative number? why would I use it?", you'd get a reasonable response. But swap "negative" for "complex", and the answer usually becomes "just shut up and do your math!"
  - onethought 5y ago
    That’s just a familiarity thing. Negative numbers are typically taught in primary school, complex numbers are perhaps only encountered in university. If you want to see some interesting applications of them, check out some explanations of the Riemann Hypothesis on YouTube.
  - Jtsummers 5y ago
    While you probably need to follow some links to grok all of it, in the introduction on Wikipedia they talk about the geometric interpretation which is probably one of the most easily graspable answers to the question "what does a complex number represent?": > This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm. I went to a middling high school, and even our algebra teacher covered the geometric interpretation of complex numbers.
CogitoCogito 5y ago
I don’t understand how “i” could be called a gimmick, if “sqrt(2)” is not. Or pi. Or as you point out “-1” for that matter.
stainforth 5y ago
>magnitude and phase Can you go a bit further? Magnitude or strength is not just the value (+5 or -5) itself?
- deleted 5y ago
  [deleted]
brandmeyer 5y ago
> The imaginary number 'i' is sqrt(-1)... what does that even mean? Relating to phase is also hard for some people, so I try to keep it even simpler. In the same way that negative numbers provide directionality on a line, the complex numbers give directionality in rotation about the plane. The unit `i` is a counter-clockwise quarter-turn about the plane. -1 is a half-turn. -i is a a quarter-turn clockwise (or 3/4 turn counter-clockwise). `exp(i * tau) = 1` or `exp(i * pi) = -1` are now simple identities, referring to the fact that a full turn doesn't change orientation, and a half-turn inverts it.

onethought 5y ago

I think you invalidated your main criticism with i not being physically representable just like negative numbers which people have no problem comprehending, despite also suffering that same problem. Our "comfort" with mathematics isn't because some concepts are better and some aren't. It's probably more how often we encounter it in everyday life. How often people need to apply complex numbers is WAY less than postive/negative/real numbers. ALL of the concepts are made up though. There isn't a set of math that isn't entirely made up, with no relation the physical world except for the relationships we also make up.

jcranmer 5y ago
> I think you invalidated your main criticism with i not being physically representable just like negative numbers which people have no problem comprehending, despite also suffering that same problem. The point is that, when negative numbers are introduced, they're invariably done so in a way that helps people get over that no-physical-meaning hump. Complex numbers generally aren't, and often, if you try to push for examples, the people teaching themselves don't give anything that helps. Compare to the article on "Negative number" on Wikipedia to "Complex number". For the former, you are immediately greeted in the first paragraph three separate example use cases for negative number. Turn to the latter, and... it's basically all just mathy justification: complex numbers make this bit of math work. Oh, there's a sentence on how they are "fundamental in many aspects of the scientific description of the natural world"--but there's no examples! Even scrolling down to the applications section, the descriptions tend more towards "complex numbers are used in this field" rather than "this is what a complex number represents." Put differently, if someone asks a question "what is a negative number? why would I use it?", you'd get a reasonable response. But swap "negative" for "complex", and the answer usually becomes "just shut up and do your math!"
- onethought 5y ago
  That’s just a familiarity thing. Negative numbers are typically taught in primary school, complex numbers are perhaps only encountered in university. If you want to see some interesting applications of them, check out some explanations of the Riemann Hypothesis on YouTube.
- Jtsummers 5y ago
  While you probably need to follow some links to grok all of it, in the introduction on Wikipedia they talk about the geometric interpretation which is probably one of the most easily graspable answers to the question "what does a complex number represent?": > This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm. I went to a middling high school, and even our algebra teacher covered the geometric interpretation of complex numbers.

CogitoCogito 5y ago

I don’t understand how “i” could be called a gimmick, if “sqrt(2)” is not. Or pi. Or as you point out “-1” for that matter.

stainforth 5y ago

>magnitude and phase Can you go a bit further? Magnitude or strength is not just the value (+5 or -5) itself?

deleted 5y ago
[deleted]

brandmeyer 5y ago

> The imaginary number 'i' is sqrt(-1)... what does that even mean? Relating to phase is also hard for some people, so I try to keep it even simpler. In the same way that negative numbers provide directionality on a line, the complex numbers give directionality in rotation about the plane. The unit `i` is a counter-clockwise quarter-turn about the plane. -1 is a half-turn. -i is a a quarter-turn clockwise (or 3/4 turn counter-clockwise). `exp(i * tau) = 1` or `exp(i * pi) = -1` are now simple identities, referring to the fact that a full turn doesn't change orientation, and a half-turn inverts it.