So what exactly is a complex number?

[Grzegorz Słodkowicz]

>  I was trying to motivate the idea by means of analogy. This is a
>  legitimate thing to do.  It helps lead people to a conceptual
>  understanding long before they understand the minutae.
You're mixing terms again. Analogy is saying 'something is like 
something else.'  What you are saying is 'Something is used to represent 
something.' You also make a scalar - vector distinction which is neither 
here nor there and implies that complex numbers are vectors, as opposed 
to real numbers.

Are you seriously saying that the imaginary unit is a small detail when 
explaining complex numbers? If you don't even mention the form 'a + bi' 
how can anyone make heads or tails of it? You omit many logical steps in 
what you think to be a simple explanation.

In fact, a proper vector in physics has 4 features: point of 
application, magnitude, direction and sense. In case of a vector in two 
dimensions (a special case, which you also fail to stress not to mention 
that you were talking about space) the magnitude and sense can be 
described by one number and the direction as another.  Since a complex 
number is composed of two numbers it can be used to describe those 
features of a vector.
Numbers are commonly used to represent breast sizes. This doesn't mean 
numbers are breasts, alas. If someone asks you what numbers are and you 
say only that they are used to denote breasts sizes, you might create 
the wrong impression.

>  I just didn't think that was the best place to start explicating the
>  *concept*. I find concrete examples that then can lead to theoretical
>  underpinnings a better way to go than the reverse.
1. Actual answer to the question can hardly be regarded as 'theoretical 
underpinnings'
2. You don't mention that answer at all.
3. Your 'concrete example' is equally sloppy and full of omissions.

>  It is so hard to grasp that learning happens in layers, and that each
>  layer need not be complete or even precise?
Is it so hard to grasp that you completely omit certain important layers?

>  Evidently you've either never taught (or were very bad at it).  You
>  have to motivate concept and interest before you can get to the
>  precise detail.
If someone asks a question their interest is evident, I think. I also 
reckon they expect a simple answer which in this case is either a link 
or something along the lines:

'As you know, square roots of negative numbers aren't real numbers. We 
are taught at school that they "don't exist". However, if we denote 
square root of -1 as i we can represent any such root as a product of a 
real number and i.
Eg.
sqrt(-9) = sqrt(9 * -1) = 3 * i
sqrt(-0.25) = sqrt(0.25 * -1) = 0.5i etc.

Since sqrt(-1) has little representation in the surrounding world 
(saying "I have 2i apples" makes no sense) it's called the imaginary 
unit. Now, complex numbers are numbers of the form
a + bi
where a and b are real numbers. They have been invented because they 
have many applications in mathematics, physics, engineering etc. They 
have many useful properties, perhaps most important of which being that 
i2 = -1. <Explanation of complex plane as an example of application>

Note that real numbers are a subset of complex numbers. They are complex 
numbers with imaginary part equal to 0.
Also, since i also means current in physics and electrical engineering, 
the imaginary unit is denoted as j in those contexts. This is also the 
case in Python (presumably because i is a common name for an integer 
variable). Speaking of which, they are a built-in type in Py. Try 
performing operations on them in the interactive mode:
> >> c = 4 + 1j d = 1 + 2j c + d
(5+5j)
> >> c = 1j # 1 is necessary because 'j' is a legit name for a
> >> variable c * c
(-1+0j)'

And no, I've never taught professionally. Have you?

>  For instance, you start with Newtonian physics, not quantum physics.
>  The entry level physics classes ignore things like the the non-linear
>  behavior of springs, or the effects when you don't actually do things
>  in a vacuum.  By your definition these lectures would be "wrong" ..
>  but they're not. They are attempting to introduce a topic painlessly.
>  And that's what I was doing.
That's not what you were doing. You were doing the equivalent of 'Hello, 
today we shall discuss gravity. I have 23 apples here, and I will throw 
them at 45 degrees - let's denote this by a vector. I throw them. Oh 
look, they fell! Well, that's gravity. Thank you for your attention.'

> > does it come from, why is it 'imaginary' etc.  Since we're being
> > arses here I'd hazard a guess you were educated in the USA where
> > doing without understanding has been mastered by teachers and
> > students alike. You're
>
>  I was initially educated in Europe where being rude was sometimes
>  encouraged to mask insecurity with a false sense of self-importance.
>  I was later educated in both Canada and the US wherein I learned both
>  they "why" and the how"
I'm sorry I made a generalisation from a tendency that I encountered. I 
shouldn't have done that. But the fact remains that your explanation, 
further statements and general mental sloppiness are excellent examples 
of what I said. Whether you learned how and why is debatable but you 
certainly don't know how to pass knowledge on.

GS.