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Complex Numbers Body.

def. Complex numbers body is ordered pairs set:

C := R x R = { (a,b) : a,b belongs to R }


with addition and multiplication operations defined as:

(a,b) + (c,d) = (a + c, b + d),
(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:

c * (a,b) = (a,b) * c = (c * a, c * b).


let's note that with this we have:

(a,b) = a * (1,0) + b * (0,1).


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

i := (0,1).


we get:

(a,b) = a + i*b.


a = Rz is real part and b = Iz imaginary part of complex number.

i itself we call imaginary unit.

let's note that:

i2 = (-1,0) = -1.



Source: [1].


See also: Groups & Bodies.

2 comments:

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

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    Replies
    1. Perhaps they'll have use in Mindful Imaging (Stitie Machine 1.1).

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