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I think you invalidated your main criticism with i not being physically representable just like negative numbers which people have no problem comprehending, des

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.

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.