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> So what are the complex numbers, really? If by reality we mean the real world (not mathematics or some abstract model) then the article does not actually ans

> So what are the complex numbers, really? If by reality we mean the real world (not mathematics or some abstract model) then the article does not actually answer the question what the complex numbers are really -- they describe what complex numbers mean mathematically (geometrically, algebraically, number theory). Here are some interpretations of imaginary and real parts in terms of the real (physical) world: o Time vs. space. This means that they have to be measured using different constituents with different properties. For example, space is reversible and periodic (we can return to the same point where we were before many times while it is not possible to return to previous point in time). o Periodic vs. monotonic behavior. For example, rotation frequencies or oscillators (electric or physical) vs. linear motion.

jonsen 9y ago
Well the article does mention: If beauty doesn't motivate you sufficiently, then take solace in the fact that the properties of some differential equations which arise in physics and engineering are determined by polynomials you can build out of them, and the systems modelled by these differential equations are stable if the real part of all the roots are negative. Google for control theory to see lots more. and This turns out to be a useful way of thinking about alternating current, and you can analyse AC circuits using complex arithmetic in much the same way as you use real algebra to analyse DC circuits. and ... did you read it all?
- jwilk 9y ago
  From the HN guidelines: Please don't insinuate that someone hasn't read an article. "Did you even read the article? It mentions that" can be shortened to "The article mentions that."
- techno_modus 9y ago
  This turns out to be a useful way of thinking about alternating current, and you can analyse AC circuits using complex arithmetic in much the same way as you use real algebra to analyse DC circuits. "Userful way of thinking" and "analyse" something is not a real object strictly speaking - it is a process or phenomenon (no doubt complex numbers are useful here). But can you say what kind of real object or property a complex number represents or measures? For example, an integer can represent the number of apples and their weight is represented by a real number. What property of a real object can be expressed, for example, as 50i ?
jondubois 9y ago
I think the best description of what a complex number really is was 'a bookkeeping device'. The other descriptions were merely circular references to other mathematical inventions. The bookkeeping explanation is still not satisfying for me though.
- akvadrako 9y ago
  Why not? Everything you can do with complex numbers, you can also do without them - they just make things easier.
  - snaky 9y ago
    "By making our numbers 'more complicated', the methods we use to work with those numbers may become more simple."
    - ColinWright 9y ago
      By making our programming languages more complex, we make our programs easier. By pushing the common complexity into a shared location, problems become easier to solve. Complex numbers really do factor out a lot of the complexity from many problems, at comparatively little cost. When people say they don't "get" complex numbers, usually they mean they've seen them mentioned, but have never actually done anything with them. It's like staying that you've read a book on lisp and don't get it. Using these things is an essential part of coming to understand them.
      - techno_modus 9y ago
        > When people say they don't "get" complex numbers, usually they mean they've seen them mentioned, but have never actually done anything with them. What people complain about is that they do not have a (real or hypothetical) device which can display its measurements as a complex number while they do have devices which measure properties in terms of integer, rational and real numbers.
mathgenius 9y ago
> Periodic vs. monotonic behavior. Another way to say this: imaginary versus real exponentials.
rom16384 9y ago
In Physics the complex i is used for different purposes, which makes it seem more complex that it really is. My favorite explanation of complex numbers is the paper "Imaginary Numbers are not Real – the Geometric Algebra of Spacetime" [1] which shows what the complex i actually is in terms of Clifford Algebras. [1] http://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are-not-real-the-geometric-algebra-of-spacetime/ http://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are...
- asavinov 9y ago
  I like this idea of representing inner (dot, scalar) product and outer product as one construct with two constituents which behaves like a complex number. This analogy with geometry makes complex numbers more realistic.

jonsen 9y ago

Well the article does mention: If beauty doesn't motivate you sufficiently, then take solace in the fact that the properties of some differential equations which arise in physics and engineering are determined by polynomials you can build out of them, and the systems modelled by these differential equations are stable if the real part of all the roots are negative. Google for control theory to see lots more. and This turns out to be a useful way of thinking about alternating current, and you can analyse AC circuits using complex arithmetic in much the same way as you use real algebra to analyse DC circuits. and ... did you read it all?

jwilk 9y ago
From the HN guidelines: Please don't insinuate that someone hasn't read an article. "Did you even read the article? It mentions that" can be shortened to "The article mentions that."
techno_modus 9y ago
This turns out to be a useful way of thinking about alternating current, and you can analyse AC circuits using complex arithmetic in much the same way as you use real algebra to analyse DC circuits. "Userful way of thinking" and "analyse" something is not a real object strictly speaking - it is a process or phenomenon (no doubt complex numbers are useful here). But can you say what kind of real object or property a complex number represents or measures? For example, an integer can represent the number of apples and their weight is represented by a real number. What property of a real object can be expressed, for example, as 50i ?

jondubois 9y ago

I think the best description of what a complex number really is was 'a bookkeeping device'. The other descriptions were merely circular references to other mathematical inventions. The bookkeeping explanation is still not satisfying for me though.

akvadrako 9y ago
Why not? Everything you can do with complex numbers, you can also do without them - they just make things easier.
- snaky 9y ago
  "By making our numbers 'more complicated', the methods we use to work with those numbers may become more simple."
  - ColinWright 9y ago
    By making our programming languages more complex, we make our programs easier. By pushing the common complexity into a shared location, problems become easier to solve. Complex numbers really do factor out a lot of the complexity from many problems, at comparatively little cost. When people say they don't "get" complex numbers, usually they mean they've seen them mentioned, but have never actually done anything with them. It's like staying that you've read a book on lisp and don't get it. Using these things is an essential part of coming to understand them.
    - techno_modus 9y ago
      > When people say they don't "get" complex numbers, usually they mean they've seen them mentioned, but have never actually done anything with them. What people complain about is that they do not have a (real or hypothetical) device which can display its measurements as a complex number while they do have devices which measure properties in terms of integer, rational and real numbers.

mathgenius 9y ago

> Periodic vs. monotonic behavior. Another way to say this: imaginary versus real exponentials.

rom16384 9y ago

In Physics the complex i is used for different purposes, which makes it seem more complex that it really is. My favorite explanation of complex numbers is the paper "Imaginary Numbers are not Real – the Geometric Algebra of Spacetime" [1] which shows what the complex i actually is in terms of Clifford Algebras. [1] http://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are-not-real-the-geometric-algebra-of-spacetime/ http://geometry.mrao.cam.ac.uk/1993/01/imaginary-numbers-are...

asavinov 9y ago
I like this idea of representing inner (dot, scalar) product and outer product as one construct with two constituents which behaves like a complex number. This analogy with geometry makes complex numbers more realistic.