where i(0,1) from??

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derarief الأحد, 30/05/2004 - 12:50ص

Complex space?? how they dicovered it??
one theory says that : they search about number keep the vector amplitude and rotrate it 90 degree and this number is "i" , if we rotrate it again with the same number "i" it will rotrate 180 degree but its amplitude the same and equal (- the first vector) so:
vector*i = newVector // vector with 90 dgree
(vector*i) * i = New(newVector) = - vector // vector with 180 dgree
So:
vector * i^2 = - vector
So:
i^2 = -1

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ammar_halaby الأحد, 30/05/2004 - 1:08ص #1

It is really one of the representations which helped to understand the complex i when it was first introduced, you know because it's idea was hard to accept untill the geometric representation was descovered.

*Complex numbers -i- was first introduced while finding roots for cubic and quadratic formulas, and later it appeared to be very useful in vector representation (because you can summarize the angle and the amplitude and the dest. in just one relation), and for the same reasons it became handy in signal representation and processing Wink

Nature uses only the longest threads to weave her patterns, so each small piece of her fabric reveals the organization of the entire tapestry
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أيمن الأحد, 30/05/2004 - 12:29م #2

Quaternions extend complex numbers, a quaternion has 4 dimensions, and 4 different imaginary numbers, something like:

xi + yj + zk + w

They are useful for rotating in 3D space.

More info here and here.

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nawwar الثلاثاء, 01/06/2004 - 3:58ص #3

Well.,Quaternions? I was just going to talk about them indeed, thanks for the links Ayman, here is another useful one
http://www.magic-software.com/Documentation/Quaternions.pdf
which explains its applications in 3D mathmatics which is to represent rotations,
there are many ways to represent rotations in 3D space, one is using Euler angles : phi and theta,
another way is using rotations matrices
but perhaps best way is using a quaternion, where 3 componets of the complex number represent a vector acting as a rotation axis, and the 4th component represents the angle of that rotation

note
talking about complex numbers with two comonents, means talking about a complex plane
whileas talking about quaternions "for instance", means talking about the complex space!!
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