Division of Complex Numbers
Except for 0, all complex numbers z have a reciprocal z^{1} = 1/z, such that
(1)  z·z^{1} = z^{1}·z = 1. 
How do we know? By explicitly finding it! Assume
(x + yi)(s + ti) = 1. 
This complex equation can be split into two linear equations with two real unknowns, s and t:
sx  ty = 1 and sy + tx = 0. 
Multiply the first by x, the second by y and add the results:
s(x^{2} + y^{2}) = x, 
from which we find s:
s = x / (x^{2} + y^{2}) = Re(z)/z^{2}. 
Similarly, we can get
t = Im(z)/z^{2}. 
Combining the two we see (and can also verify) that the number z'/z^{2} satisfies (1):
(2)  z·(z'/z^{2}) = 1 = (z'/z^{2})·z. 
We see now an easier way to obtain the formula for the reciprocal (and prove its existence to boot.) Indeed, (2) follows from the properties and the definition of the conjugate number. If
z^{2} = z·z' = z'^{2}. 
then certainly
1 = z·(z'/z^{2}). 
Armed with the formula for a reciprocal, we can divide two complex numbers, provided of course the second is not 0:
z/w = z·w^{1} = z·w'/w^{2}. 
As an example,

References
 T. Andreescu, D. Andrica, Complex Numbers From A to ... Z, Birkhäuser, 2006
 C. W. Dodge, Euclidean Geometry and Transformations, Dover, 2004 (reprint of 1972 edition)
 Liangshin Hahn, Complex Numbers & Geometry, MAA, 1994
 E. Landau, Foundations of Analisys, Chelsea Publ, 3^{rd} edition, 1966
Complex Numbers
 Algebraic Structure of Complex Numbers
 Division of Complex Numbers
 Useful Identities Among Complex Numbers
 Useful Inequalities Among Complex Numbers
 Trigonometric Form of Complex Numbers
 Real and Complex Products of Complex Numbers
 Complex Numbers and Geometry
 Plane Isometries As Complex Functions
 Remarks on the History of Complex Numbers
 Complex Numbers: an Interactive Gizmo
 Cartesian Coordinate System
 Fundamental Theorem of Algebra
 Complex Number To a Complex Power May Be Real
 One can't compare two complex numbers
 Riemann Sphere and Möbius Transformation
 Problems
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