Complex numbers
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I am trying to understsand complex numbers, And I can imagine it is a vector, but why needs the y coordinate (imaginary part) to be multiplied by i. Why don''t one just use a normal vector for it.
Is there some paper on the net that explains how they got the idea to use that number
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http://www.clarku.edu/~djoyce/complex/
The first thing I came across on google. Certainly you can find more. A good math book also can help you out.
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The first thing I came across on google. Certainly you can find more. A good math book also can help you out.
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The idea probably came from equations like x² + 2 = 0. Then they found that complex numbers are useful for a lot of things, eg wave equations. Complex numbers are very different from 2D vectors though.
A complex number is the combination of a standard number and an imaginary number. Thus why you have the i component. If you didnt have the imaginary part, then it would just be any normal number.
Complex numbers are often represented as pairs or vectors, just as a means of storage: so 2 + 3i is often stored as (2, 3). But you cant forget the + sign, a complex number is truely single entity.
The question you should be asking is whats so magical about imaginary numbers and how they interact with real numbers. For example, go look up the source for calculating (x + iy)^n. Most implementations use the radian coordinate system and sin/cos to calculate complex pow().
On a related note, because complex numbers are stored as pairs, you can use the x,y coordinate field to generate fractals etc.
Complex numbers are often represented as pairs or vectors, just as a means of storage: so 2 + 3i is often stored as (2, 3). But you cant forget the + sign, a complex number is truely single entity.
The question you should be asking is whats so magical about imaginary numbers and how they interact with real numbers. For example, go look up the source for calculating (x + iy)^n. Most implementations use the radian coordinate system and sin/cos to calculate complex pow().
On a related note, because complex numbers are stored as pairs, you can use the x,y coordinate field to generate fractals etc.
You cannot write it as a normal vector, because that complex number is referring to a position on the Argand diagram, not on the XY plane. The Y axis of the Argand diagram is the imaginary number line, whereas the X is the real number line.
You can write complex numbers in numerous forms. There''s x + iy. Then ther is r(cos THETA + i sin THETA) which is also equivalent to re^i(THETA). r in the latter two cases is the modulus of the complex number, which is just the length of the complex number when drawn on the Argand diagram, and THETa is the angle that is made with the real number line.
Using these, you get the equation e^i(PI) + 1 = 0, which is beautiful, apparently :-)
You can write complex numbers in numerous forms. There''s x + iy. Then ther is r(cos THETA + i sin THETA) which is also equivalent to re^i(THETA). r in the latter two cases is the modulus of the complex number, which is just the length of the complex number when drawn on the Argand diagram, and THETa is the angle that is made with the real number line.
Using these, you get the equation e^i(PI) + 1 = 0, which is beautiful, apparently :-)
The idea of looking at complex numbers like you do a 2D vector is really just notation. I guess if you''re really hung up on the i, then try thinking of it like this:
complex number = a*r + b*i
where
r=real=1
i=imaginary=sqrt(-1)
and voila it''s in the same form as 2D vector notation of
a*i + b*j = (a,b)
complex number = a*r + b*i
where
r=real=1
i=imaginary=sqrt(-1)
and voila it''s in the same form as 2D vector notation of
a*i + b*j = (a,b)
Author
hm. I am getting to understanding how to work with them, but I still don't see how they came on the idea to make a number who's square is -1, and why they use it for complex numbers. Is there a reason for this, or should I just believe it.
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[edited by - Quasar3D on February 2, 2003 6:31:21 PM]
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[edited by - Quasar3D on February 2, 2003 6:31:21 PM]
it was not an idea to do like this.. it was automagically.. as everything by math, its just based on the basic rules...
they though about this:
2 = sqrt(4)
a*2 = sqrt(a*a*4)
now.. as they had more and more complex equations, and realised they need to find square roots of negative numbers:
i*2 = sqrt(-4)
i*2 = sqrt(-1*4)
i*2 = sqrt(-1)*2
i = sqrt(-1)
and voilà. the whole mis(t)ery is gone
(okay, there is much more than this.. but the basic idea on why i is what it is is now given..)
"take a look around" - limp bizkit
www.google.com
they though about this:
2 = sqrt(4)
a*2 = sqrt(a*a*4)
now.. as they had more and more complex equations, and realised they need to find square roots of negative numbers:
i*2 = sqrt(-4)
i*2 = sqrt(-1*4)
i*2 = sqrt(-1)*2
i = sqrt(-1)
and voilà. the whole mis(t)ery is gone
(okay, there is much more than this.. but the basic idea on why i is what it is is now given..)
"take a look around" - limp bizkit
www.google.com
quote:Original post by quasar3d
hm. I am getting to understanding how to work with them, but I still don''t see how they came on the idea to make a number who''s square is -1, and why they use it for complex numbers. Is there a reason for this, or should I just believe it.
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[edited by - Quasar3D on February 2, 2003 6:31:21 PM]
They came about because mathematicians found that there was a massive hole in their theories when it came to trying to solve x^2 + 2 = 0.
Try to solve it without complex numbers.
Well, complex numbers just make things easier to work with. You don''t have to keep track of the sign of numbers you''re dealing with. They always have a squareroot no matter if they are positive or negative so you have lass cases to deal with. Besides, every equation has a solution in complex numbers so you can always be sure that a solution exists.
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