General Equations for Square Roots of Complex Numbers

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 a + b√k where k is any number greater than 0. In addition, k is square free, i.e.,any squares are removed from under the square root sign. For example if k = −18 the square 9 is removed and its root placed outside the radical, √ , to give k = −2. This article follows Part I for General equation for the square roots of non complex numbers

Construction of the Equations

Let us start with the complex number where k is positive
c= a + b√−k (1)
Let us assume that a square root of c is p + q√−k where p and q are real
(p + q√−k)² = a + b√−k (2)
Equating the real and radical parts gives us the two equations
p² − kq² = a (3)
2pq = b (4)
We must have p ≠ 0 since b ≠ 0. Solving equation (4) for q gives
q = b (5)
2p
and we can substitute this value for q into equation (4) to get
p² − k( b²) = a (6)
(4p²)
or
4p⁴ − 4ap² − kb² = 0 (7)
This is a quadratic in p², so we can solve for p² using the quadratic formula.
p² = 4a ±√16a2 + 16kb2 (8)
8
and after factoring out
p² = a ±√a2 + kb2 (9)
2
so that
p = √a ±√a2 + kb2 (10)
√ 2
At this point r which contains k is set to the following:
r = ±√a2 + kb2 (11)
Giving p and consequently x:
p = x = √a ± r (12)
√ 2
The value of b from (11) is
b = ±√r2 − a2 (13)
√ k
Substituting p from equation (12) into equation (5)
q = b (14)
2√(a ± r)/2
Multiplying the top and bottom of equation (14) by √a ∓ r and substituting the value for b from (13) gives the following:
q = √a ∓ r ±√r2 − a2 (15)
√ 2k ±√a2 − r2
q = √r − a (16)
√ 2k
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).
y = b (17)
2x

Two Examples

Example I:

1321 + 7392√−2

Find r

r = √13212 + 2×73922 = 10537

Find x then y (note that √(a − r )/2 gives a negative value).

x = √1321 + 10537 = 77
√ 2

y = 7392 = 48
2 × 77

Giving the answer

(77 + 48√−2)²

Example II:

7078 − 1190√−3

Find r

r = √70782 + 3×11902 = 7372

Find x then y (note √(a − r )/2 gives an negative value).

x = ±√7078 + 7372 = 85, −85
√ 2

y = −1190 = −7, 7
2 × ±85

Giving the answers

(85 − 7√ −3)² and ( − 85 + 7√ −3)²

SQUARE ROOTS OF COMPLEX NUMBERS

GENERAL EQUATION FOR NUMBERS CONTAINING IMAGINARY RADICALS (Part II)

Construction of the Equations

Two Examples