Stanley Rabinowitz published the article
how to find the square roots of complex numbers
in the journal Mathematics and Informatics Quarterly, 3(1993)54-56 . By following the same approach as
Rabinowitz I am showing how to obtain the general equations for numbers containing imaginary radicals of the type
Let us start with the complex number where k is positive
Let us assume that a square root of c is p + q√−k where p and q are real
Equating the real and radical parts gives us the two equations
We must have p ≠ 0 since b ≠ 0. Solving equation (4) for q gives
and we can substitute this value for q into equation (4) to get
This is a quadratic in p2, so we can solve for p2 using the quadratic formula.
and after factoring out
At this point r which contains k is set to the following:
Giving p and consequently x:
The value of b from (11) is
Substituting p from equation (12) into equation (5)
Multiplying the top and bottom of equation (14) by √a ∓ r and substituting the value for b from (13) gives the following:
This means that whenever the radical in equation (12) is √a + r the value of the radical in (15) would be √a − r and vice versa. Moreover, substituting x in (14) gives (17).
1321 + 7392√−2
Find rr = √13212 + 2×73922 = 10537
Find x then y (note that √(a − r )/2 gives a negative value).
√1321 + 10537 = 77
Giving the answer
(77 + 48√−2)2
7078 − 1190√−3
Find rr = √70782 + 3×11902 = 7372
Find x then y (note √(a − r )/2 gives an negative value).
±√7078 + 7372 = 85, −85
Giving the answers
(85 − 7√ −3)2 and ( − 85 + 7√ −3)2
This concludes Part II.
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Copyright © 2011 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com