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{{Short description|Square root of a
{{Redirect|Imaginary Numbers|the 2013 EP by The Maine|Imaginary Numbers (EP)}}
{{pp-pc1}}
{| class="wikitable" style="float: right; margin-left: 1em; text-align: center;"
! The powers of {{mvar|i}}<br/> are cyclic:
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|<math>\ \vdots</math>
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|<math>\ i^{-2} = -1\phantom{i}</math>
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|style="background:#e1edfd;" | <math>\ \ i^{0}\ = \phantom-1\phantom{i}</math>
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|style="background:#
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|style="background:#
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|style="background:#
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|<math>\ \ i^{4}\ = \phantom-1\phantom{i}</math>
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|<math>\ \ i^{5}\ = \phantom-i\phantom1</math>
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|<math>\ \vdots</math>
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|}
An '''imaginary number''' is the product of a [[real number]]
{{cite book
|chapter-url=https://books.google.com/books?id=SGVfGIewvxkC&pg=PA38
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==Square roots of negative numbers==
Care must be used when working with imaginary numbers that are expressed as the [[principal value]]s of the [[square root]]s of [[negative number]]s.<ref>{{cite book |title=An Imaginary Tale: The Story of "i" [the square root of minus one] |first1=Paul J. |last1=Nahin |publisher=Princeton University Press |year=2010 |isbn=978-1-4008-3029-9 |page=12 |url=https://books.google.com/books?id=PflwJdPhBlEC}} [https://books.google.com/books?id=PflwJdPhBlEC&pg=PA12 Extract of page 12]</ref> For example, if {{mvar|x}} and {{mvar|y}} are both positive real numbers, the following chain of equalities appears reasonable at first glance:
: <math>\textstyle
\sqrt{x \cdot y \vphantom{t}}
=\sqrt{(-x) \cdot (-y)}
\mathrel{\stackrel{\text{ (fallacy) }}{=}} \sqrt{-x\vphantom{ty}} \cdot \sqrt{-y\vphantom{ty}}
= i\sqrt{x\vphantom{ty}} \cdot i\sqrt{y\vphantom{ty}}
= -\sqrt{x \cdot y \vphantom{ty}}\,.
</math>
But the result is clearly nonsense. The step where the square root was broken apart was illegitimate. (See [[Mathematical fallacy]].)
==See also==
* [[−1]]
{{Classification of numbers}}
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* [https://www.math.toronto.edu/mathnet/answers/imagexist.html How can one show that imaginary numbers really do exist?] – an article that discusses the existence of imaginary numbers.
* [https://www.bbc.co.uk/radio4/science/5numbers4.shtml 5Numbers programme 4] BBC Radio 4 programme
* [http://www2.dsu.nodak.edu/users/mberg/Imaginary/imaginary.htm Why Use Imaginary Numbers?] {{Webarchive|url=https://web.archive.org/web/20190825172656/http://www2.dsu.nodak.edu/users/mberg/Imaginary/imaginary.htm |date=2019-08-25 }} Basic Explanation and Uses of Imaginary Numbers
{{Complex numbers}}
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