Editing Square root (section)

===Algebraic formula===
[[Image:Imaginary2Root.svg|right|thumb|The square roots of {{mvar|i}}]]
When the number is expressed using its real and imaginary parts, the following formula can be used for the principal square root:<ref>{{cite book
 |title       = Handbook of mathematical functions with formulas, graphs, and mathematical tables
 |first1      = Milton
 |last1       = Abramowitz
 |first2      = Irene A.
 |last2       = Stegun
 |publisher   = Courier Dover Publications
 |year        = 1964
 |isbn        = 0-486-61272-4
 |page        = 17
 |url         = https://books.google.com/books?id=MtU8uP7XMvoC
 |url-status     = live
 |archive-url  = https://web.archive.org/web/20160423180235/https://books.google.com/books?id=MtU8uP7XMvoC
 |archive-date = 2016-04-23
}}, [http://www.math.sfu.ca/~cbm/aands/page_17.htm Section 3.7.27, p. 17] {{webarchive|url=https://web.archive.org/web/20090910094533/http://www.math.sfu.ca/~cbm/aands/page_17.htm |date=2009-09-10 }}
</ref><ref>{{cite book
 |title       = Classical algebra: its nature, origins, and uses
 |first1      = Roger
 |last1       = Cooke
 |publisher   = John Wiley and Sons
 |year        = 2008
 |isbn        = 978-0-470-25952-8
 |page        = 59
 |url         = https://books.google.com/books?id=lUcTsYopfhkC&pg=PA59
 |url-status     = live
 |archive-url  = https://web.archive.org/web/20160423183239/https://books.google.com/books?id=lUcTsYopfhkC&pg=PA59
 |archive-date = 2016-04-23
}}
</ref>

<math display=block>
\sqrt{x+iy} = \sqrt{\tfrac12\bigl(\sqrt{\textstyle x^2+y^2} + x\bigr)} + 
              i\sgn(y) \sqrt{\tfrac12\bigl(\sqrt{\textstyle x^2+y^2} - x\bigr)},
</math>

where {{math|1=sgn(''y'') = 1}} if {{math|''y'' ≥ 0}} and {{math|1=sgn(''y'') =  −1}} otherwise.<ref>This sign function differs from the usual [[sign function]] by its value at {{math|0}}.</ref> In particular, the imaginary parts of the original number and the principal value of its square root have the same sign. The real part of the principal value of the square root is always nonnegative.

For example, the principal square roots of {{math|±''i''}} are given by:

<math display=block>
   \sqrt{i} = \frac{1+i}{\sqrt2}, \qquad
   \sqrt{-i} = \frac{1 - i}{\sqrt2}.</math>