#header-inner {background-position: right !important; width: 100% !important;}

## 1/12/14

### 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].