SQUARE ROOTS OF NON-COMPLEX NUMBERS

GENERAL EQUATIONS FOR NUMBERS CONTAINING RADICALS (Part I)

Picture of an equation

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 non 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.

Construction of the Equations

Three Examples

Example I:

4036 + 2322√3

Find r

r = √40362 − 3×23222  = 338

Find x then y  (note √(a + r  )/2 does not give an integer value).

x = 4036 − 338     = 43
2

y =     2322    = 27
2 × 43

Giving the answer

(43 + 27√3)2


Example II:

229 − 132√3

Find r

r = √2292 − 3×1322  = 13

Find x then y  (note √(a − r  )/2 does not give an integer value).

x = 229 + 13     = 11
2

y =     −132    = −6
2 × 11

Giving the answer

(11 − 6√3) 2


Example III:

15675 + 11050√2

Find r

r = √156752 − 2×110502  = 1225

Find x then y  (note √(a + r  )/2 does not give an integer value).

x = 15675 − 1225     = 85
2

y =     11050    = 65
2 × 85

Giving the answer

(85 + 65√2) 2

This concludes Part I. To continue to Part II, which contructs general equations of the type a + b√−k.
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Copyright © 2011 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com