Reply to Re: I gave up to the Computers National Olimpiad
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Complex numbers are logical if you are in the system in which there's no multiplication, only addition. In that case, you can see a+b*i as (a,b), which means "Go 'a' steps to the east, and 'b' to the north". That's a logical explanation of a coördinate, relative to where you're standing now, which I've just called (0,0).
(a,b) + (c,d) = (a+c, b+d) is the normal way of changing
"Go 'a' steps to the east, and 'b' steps to the north." "That's it?" "Euhm... no, wait I forgot! Then you should take 'c' steps to the east and 'd' steps to the north"
into
"Go 'a+c' steps to the east, and 'b+d' steps to the north".
Not surprisingly, (a+b*i) + (c+d*i) = (a+b)+(c+d)*i. The part without i is still the first coördinate, the part with i is still the second coördinate. All fits.
In this way, complex numbers are a completely logical way of describing 2D-coördinates.
So then they are logical in another system, being 2D-coördinates. In the post you replied to, you can leave anything away after "You don't even have to use i*i = -1". Why use all properties when you don't have need for all of them?
Now, you could ask why one would possibly choose complex numbers above 'normal' 2D-coördinates. You could say it's a matter of preference. They can have added value, if you have the need for it. Hence the preference, those who have the need probably prefer complex numbers, those who don't, probably prefer 2D-coördinates (or maybe even the "steps-taking" description).
Since you probably don't have need for that added value, I won't bother to explain it. It isn't part of this discussion, anyway, as I've already shown that there is a system in which they are logical, which satisfies "Evidence that it can be logical in nother system", in which 'it' refers to 'the use of complex numbers'.
#end of my 'argument' in return here. Some non-discussion related stuff below:
When you said that you "strugled alot with calculus and triginometry in high school. However, I excelled at algrebra and numbers (well normal ones anyway) and arithmetic." I thought 'oy'.
While complex numbers are a purely algebraïcal concept (they are an instance of a common kind of ring, which is an abstract algebraïcal concept... Too difficult to explain in this post), calc and trig are the areas in which they're used most by (to my standard) less mathy persons.
In any case, I won't mind teaching you the added value (of coördinate transformation). In fact I'd be really happy to share my mathematical knowledge with anyone who asks for it, and really shows interest. Then, you can do with that knowledge what you want. Reject, accept, ignore, print out and use as toilet paper, share further, point a shotgun at it... whatever you want. Really, I don't care. I'd prefer if you share it with other people who are interested, but if you choose to do otherwise, it's fine with me.
Also, I've found this link on Eulers formula (e^(i*pi) = -1, and in general what e^xi is supposed to be). It might contain some useful insights on the use of complex numbers, although the initial question doesn't seem like it, the last post describes why complex numbers are useful in real-life. If you have the need for it:
http://mathforum.org/library/drmath/view/52251.html.
LATE ADDITION:
More questions in this archive might be interesting to read. To pick a few:
Uses of Imaginary numbers
Imaginary Numbers in Real Life
Defining Complex Numbers
Imaginary Numbers - History and Commentary
There are more in the archives of people who wondered how complex numbers could be possible, how to visualise them, what they are, etc. Kinda like what I'm trying to explain here, but then by people who have the ability to explain.
(a,b) + (c,d) = (a+c, b+d) is the normal way of changing
"Go 'a' steps to the east, and 'b' steps to the north." "That's it?" "Euhm... no, wait I forgot! Then you should take 'c' steps to the east and 'd' steps to the north"
into
"Go 'a+c' steps to the east, and 'b+d' steps to the north".
Not surprisingly, (a+b*i) + (c+d*i) = (a+b)+(c+d)*i. The part without i is still the first coördinate, the part with i is still the second coördinate. All fits.
In this way, complex numbers are a completely logical way of describing 2D-coördinates.
So then they are logical in another system, being 2D-coördinates. In the post you replied to, you can leave anything away after "You don't even have to use i*i = -1". Why use all properties when you don't have need for all of them?
Now, you could ask why one would possibly choose complex numbers above 'normal' 2D-coördinates. You could say it's a matter of preference. They can have added value, if you have the need for it. Hence the preference, those who have the need probably prefer complex numbers, those who don't, probably prefer 2D-coördinates (or maybe even the "steps-taking" description).
Since you probably don't have need for that added value, I won't bother to explain it. It isn't part of this discussion, anyway, as I've already shown that there is a system in which they are logical, which satisfies "Evidence that it can be logical in nother system", in which 'it' refers to 'the use of complex numbers'.
#end of my 'argument' in return here. Some non-discussion related stuff below:
When you said that you "strugled alot with calculus and triginometry in high school. However, I excelled at algrebra and numbers (well normal ones anyway) and arithmetic." I thought 'oy'.
While complex numbers are a purely algebraïcal concept (they are an instance of a common kind of ring, which is an abstract algebraïcal concept... Too difficult to explain in this post), calc and trig are the areas in which they're used most by (to my standard) less mathy persons.
In any case, I won't mind teaching you the added value (of coördinate transformation). In fact I'd be really happy to share my mathematical knowledge with anyone who asks for it, and really shows interest. Then, you can do with that knowledge what you want. Reject, accept, ignore, print out and use as toilet paper, share further, point a shotgun at it... whatever you want. Really, I don't care. I'd prefer if you share it with other people who are interested, but if you choose to do otherwise, it's fine with me.
Also, I've found this link on Eulers formula (e^(i*pi) = -1, and in general what e^xi is supposed to be). It might contain some useful insights on the use of complex numbers, although the initial question doesn't seem like it, the last post describes why complex numbers are useful in real-life. If you have the need for it:
http://mathforum.org/library/drmath/view/52251.html.
LATE ADDITION:
More questions in this archive might be interesting to read. To pick a few:
Uses of Imaginary numbers
Imaginary Numbers in Real Life
Defining Complex Numbers
Imaginary Numbers - History and Commentary
There are more in the archives of people who wondered how complex numbers could be possible, how to visualise them, what they are, etc. Kinda like what I'm trying to explain here, but then by people who have the ability to explain.
