with addition and multiplication operations defined as:

(a,b) * (c,d) = (a * c - b * d, a * d + b * c),

for any a,b,c,d belonging to R (R is Real numbers set).

let's note that addition's neutral element is (0,0).

let's note that multiplication's neutral element is (1,0).

let's note that opposite element to (a,b) is -(a,b) = (-a,-b).

let's note that inverse element to (a,b) ≠ (0,0) is:

^{(a,b)-1 = (a/(a2+b2),-b/(a2+b2)).}

let's define multiplication of complex number by real number as:

let's note that with this we have:

at last, identifying complex number (a,0) with real number a, and adding additional notation:

we get:

a = R

_{z}is

*real part*and b = I

_{z}imaginary part of complex number.

i itself we call imaginary unit.

let's note that:

^{2}= (-1,0) = -1.

Source: [1].

See also: Groups & Bodies.

Extension to complex numbers for 3D computer graphics (and more): http://en.wikipedia.org/wiki/Quaternion

ReplyDeletePerhaps they'll have use in Mindful Imaging (Stitie Machine 1.1).

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