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Comment by techno_modus | Hacker News Reader

> 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.

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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?

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."

"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 ?

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.

Why not? Everything you can do with complex numbers, you can also do without them - they just make things easier.

"By making our numbers 'more complicated', the methods we use to work with those numbers may become more simple."

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.

> 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.

Real numbers almost certainly are just as fictional as complex numbers. Physics doesn't seem to admit infinitely precise measurements, so literally no theoretical device can operate with real numbers.

Even if real numbers are "real", only a countable number of them are computable, even in theory. The rest are probably completely inaccessible, barring hypercomputation. (Interestingly, access to true real numbers gives you hypercomputation as a side-effect)

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...

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.

> Periodic vs. monotonic behavior.

Another way to say this: imaginary versus real exponentials.

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