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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 z = x + yi, |z| ≠ 0. We are looking for a complex number s + ti that satisfies

(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(x² + y²) = x,

from which we find s:

s = x / (x² + y²) = 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,

(2 - i)/(3 + 4i)	= (2 - i)(3 - 4i)/(3² + 4²)
	= (2 - 11i)/25
	= .08 - .44i.

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)
Liang-shin Hahn, Complex Numbers & Geometry, MAA, 1994
E. Landau, Foundations of Analisys, Chelsea Publ, 3^rd edition, 1966

Complex Numbers

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Copyright © 1996-2012 Alexander Bogomolny

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