So what exactly is a complex number?

[Tim Daneliuk]

Grzegorz Słodkowicz wrote:
> 
>> Here is a simple explanation (and it is not complete by a long shot).
>>
>> A number by itself is called a "scalar".  For example, when I say,
>> "I have 23 apples", the "23" is a scalar that just represents an
>> amount in this case.
>>
>> One of the most common uses for Complex Numbers is in what are
>> called "vectors".  In a vector, you have both an amount and
>> a *direction*.  For example, I can say, "I threw 23 apples in the air
>> at a 45 degree angle".  Complex Numbers let us encode both
>> the magnitude (23) and the direction (45 degrees) as a "number".
>>
>> There are actually two ways to represent Complex Numbers.
>> One is called the "rectangular" form, the other the "polar"
>> form, but both do the same thing - they encode a vector.
>>
>> Complex Numbers show up all over the place in engineering and
>> science problems.  Languages like Python that have Complex Numbers
>> as a first class data type allow you do to *arithmetic* on them
>> (add, subtract, etc.).  This makes Python very useful when solving
>> problems for engineering, science, navigation, and so forth.
>>
>>
>> HTH,
>>   
> You're mixing definition with application. You didn't say a word about 
> what complex numbers are, not a word about the imaginary unit, where 

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.   I am well aware of the
imaginary unit and from whence complex analysis springs.  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.

It is so hard to grasp that learning happens in layers, and that each
layer need not be complete or even precise?  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.  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.

> 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".



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Tim Daneliuk     tundra at tundraware.com
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