Editing Imaginary unit (section)

==Proper use==
The imaginary unit was historically written <math display=inline>\sqrt{-1},</math> and still is in some modern works. However, great care needs to be taken when manipulating formulas involving [[Nth root|radicals]]. The radical sign notation <math display=inline>\sqrt{x}</math> is reserved either for the principal square root function, which is defined for ''only'' real {{math|''x'' ≥ 0,}} or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function can produce false results:<ref>{{cite book |first=Bryan |last=Bunch |year=2012 |title=Mathematical Fallacies and Paradoxes |edition=illustrated |publisher=Courier Corporation |page=[https://books.google.com/books?id=jUTCAgAAQBAJ&pg=PA31 31]-34 |isbn=978-0-486-13793-3 |via=Google Books |url=https://books.google.com/books?id=jUTCAgAAQBAJ}}</ref>
<math display=block>-1 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} \mathrel{\stackrel{\mathrm{fallacy}}{=}} {\textstyle \sqrt{(-1) \cdot (-1)}} = \sqrt{1} = 1 \qquad \text{(incorrect).}</math>

Generally, the calculation rules
<math display=inline>\sqrt{x\vphantom{ty}} \cdot\! \sqrt{y\vphantom{ty}} = \sqrt{x \cdot y\vphantom{ty}}</math>
and
<math display=inline>\sqrt{x\vphantom{ty}}\big/\!\sqrt{y\vphantom{ty}} = \sqrt{x/y}</math>
are guaranteed to be valid only for real, positive values of {{mvar|x}} and {{mvar|y}}.<ref>{{cite book |first=Arthur |last=Kramer |publisher=Cengage Learning |year=2012 |edition=4th |title=Math for Electricity & Electronics |isbn=978-1-133-70753-0 |page=[https://books.google.com/books?id=gdAJAAAAQBAJ&pg=PA81 81] |via=Google Books |url=https://books.google.com/books?id=gdAJAAAAQBAJ}}</ref><ref>{{cite book |last1=Picciotto |first1=Henri |last2=Wah |first2=Anita |year=1994 |title=Algebra: Themes, tools, concepts |edition=Teachers' |publisher=Henri Picciotto |isbn=978-1-56107-252-1 |page=[https://books.google.com/books?id=_cOhDl3J3ZMC&pg=PA424 424] |via=Google Books |url=https://books.google.com/books?id=_cOhDl3J3ZMC}}</ref><ref>{{cite book |first=Paul J. |last=Nahin |year=2010 |title=An Imaginary Tale: The story of "{{mvar|i}}" [the square root of minus one] |publisher=Princeton University Press |isbn=978-1-4008-3029-9 |page=[https://books.google.com/books?id=PflwJdPhBlEC&pg=PA12 12] |via=Google Books |url=https://books.google.com/books?id=PflwJdPhBlEC}}</ref>

When {{mvar|x}} or {{mvar|y}} is real but negative, these problems can be avoided by writing and manipulating expressions like <math display=inline>i \sqrt{7}</math>, rather than <math display=inline>\sqrt{-7}</math>. For a more thorough discussion, see the articles [[Square root]] and [[Branch point]].