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
- Remarks on the History of Complex Numbers
- Complex Numbers: A Dynamic Tool
- Cartesian Coordinate System
- Fundamental Theorem of Algebra
- Complex Number To a Complex Power May Be Real
- One Can't Compare Two Complex Numbers
- Problems
Division of Complex Numbers
Except for 0, all complex numbers z have a reciprocal z-1 = 1/z, such that
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
This complex equation can be split into two linear equations with two real unknowns, s and t:
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sx - ty = 1 and
sy + tx = 0.
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Multiply the first by x, the second by y and add the results:
from which we find s:
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s = x / (x2 + y2) = Re(z)/|z|2.
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Similarly, we can get
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.
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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
then certainly
Armed with the formula for a reciprocal, we can divide two complex numbers, provided of course the second is not 0:
As an example,
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| (2 - i)/(3 + 4i) | = (2 - i)(3 - 4i)/(32 + 42) |
| | = (2 - 11i)/25 |
| | = .08 - .44i. |
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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, 3rd edition, 1966
Copyright © 1996-2008 Alexander Bogomolny
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