[OT] Re: So what exactly is a complex number?

Grzegorz Słodkowicz jergosh at wp.pl
Sun Sep 9 13:29:16 CEST 2007


>> Interesting. It appears that we are ran into a mathematical
>> cultural difference. Were I come from vectors *are* defined as
>> having four properties that I enumerated. After some research I
>> found that English sources (Wikipedia) indeed give the definition
>> you supplied. 
>>     
> Indeed, interesting.
>   
After some additional research I found that the most concise definition 
given here is 'A vector is an ordered pair of points'. From that follow 
the 4 properties:
if we describe a vector by 2 points A(x_1, y_1) and B(x_2, y_2) we can 
convert this representation into the form [x_2 - x_1, y_2 - y_1] but 
some information is then lost in the process - conversion in the 
opposite direction is not possible without knowing the starting point. 
The article on the English Wikipedia (again, 
http://en.wikipedia.org/wiki/Vector_%28spatial%29) says: "*vector*, is a 
concept characterized by a magnitude and a direction. A vector *can be 
thought of as* (emphasis mine - G.) an arrow in Euclidean space, drawn 
from an *initial point* /A/ pointing to a *terminal point* /B/." 
Apparently what you know as the definition is a property where I come 
from, and vice versa.

On a side note, I discovered more such differences (my studies are 
exclusively in English and we use mostly English textbooks even though 
I'm not in an English-speaking country). According to the article in the 
English Wikipedia a set is:
"Definition
(...)By a /set/ we understand any collection /S/ of definite, distinct 
objects /s/ of our perception or of our thought (which will be called 
the /elements/ of /S/) into a whole."

In 'Basic Concepts of Mathematics' (available for free at 
http://www.trillia.com/zakon1.html), the author (who was, BTW, a Russian 
that settled in Canada) states:
"A set is often described as a collection (“aggregate”, “class”, 
“totality”, “family”) of objects of any specified kind. However, such 
descriptions are no definitions, as they merely replace the term “set” 
by other undefined terms. Thus the term “set” must be accepted as a 
primitive notion, without definition."

To its credit, the Wikipedia article states that it describes the 'naive 
theory of sets'. Not a word about primary notions under 
http://en.wikipedia.org/wiki/Axiomatic_set_theory either, though.

Similarly, I was once given a definition of a number as 'the common 
feature of sets with the same size' (or something to that effect), one 
that I couldn't find in any English source. Another, trivial, example 
can be the Abel theorem which is known as Buniakovsky (Buniakowski) 
theorem in Eastern Europe. There are also some inconsistencies in 
mathematical notation that I couldn't settle with my university 
lecturers but I don't want to deviate from the subject any further. I 
suppose they're the kind of doubts you get after reading too much 
Dijkstra ;)

 > I didn't say that you must not know the point of application, but I
> said that it was not a property of the vector itself. It is true,
> however, that in physical calculations you should not "mix" many
> types of vectors (like force) that are, in the experiment, applied
> to different points of application.
>
>   
Here it would be an intrinsic property of a vector ;). I think we've 
nailed the crux of the argument now.
>> Again, I think we were given different definitions. Mine states
>> that direction is 'the line on which the vector lies', sense is
>> the 'arrow' and magnitude is the 'length' (thus non-negative). The
>> definition is separate from mathematical description (which can be
>> '[1 1] applied at (0, 0)' or 'sqrt(2) at 45 deg applied at (0, 0)'
>> or any other that is unambiguous).
>>     
>
> Oh, I thought we were talking about quite mathematical vectors? 
>   
But we were, I think?

>> No. In one-dimensional 'space' direction is a ± quantity (a
>> 'sense'). In 2-d it can be given as an angle.
>>     
>
> Indeed, you're right. So, those vectors have different properties
> depending on the used coordinate system? I myself prefer the
> concise definition ...
>   
I think it's similar to points in space: depending on the number of 
dimensions you need a corresponding number of coordinates to describe a 
point. Whether these are Cartesian or polar coordinates (complex plane 
also springs to mind) has no bearing on their number.

Regards,
Greg



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