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

  1. T. Andreescu, D. Andrica, Complex Numbers From A to ... Z, Birkhäuser, 2006
  2. C. W. Dodge, Euclidean Geometry and Transformations, Dover, 2004 (reprint of 1972 edition)
  3. Liang-shin Hahn, Complex Numbers & Geometry, MAA, 1994
  4. E. Landau, Foundations of Analisys, Chelsea Publ, 3rd edition, 1966

Complex Numbers

  1. Algebraic Structure of Complex Numbers
  2. Division of Complex Numbers
  3. Useful Identities Among Complex Numbers
  4. Useful Inequalities Among Complex Numbers
  5. Trigonometric Form of Complex Numbers
  6. Real and Complex Products of Complex Numbers
  7. Complex Numbers and Geometry
  8. Plane Isometries As Complex Functions
  9. Remarks on the History of Complex Numbers
  10. Complex Numbers: an Interactive Gizmo
  11. Cartesian Coordinate System
  12. Fundamental Theorem of Algebra
  13. Complex Number To a Complex Power May Be Real
  14. One can't compare two complex numbers
  15. Riemann Sphere and Möbius Transformation
  16. Problems

|Contact| |Front page| |Contents| |Algebra|

Copyright © 1996-2018 Alexander Bogomolny

 65122901