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{{Short description|Mathematical symbol denoting a root}}
{{Redirect|Root symbol|the root symbols for gene and protein families|Gene nomenclature#Symbol and name}}
{{about|the mathematical symbol denoting a root|other uses}}
{{redirect|√|the check symbol (✓)|Check mark}}

In mathematics, the '''radical symbol''', '''radical sign''', '''root symbol''', or '''surd''' is a [[Mathematical notation|symbol]] for the [[square root]] or [[Nth root|higher-order root]] of a number. The square root of a number {{math|''x''}} is written as
:<math>\sqrt{x},</math>
while the {{math|''n''}}th root of {{math|''x''}} is written as
:<math>\sqrt[n]{x}.</math>
It is also used for other meanings in more advanced mathematics, such as the [[radical of an ideal]].

In [[linguistics]], the symbol is used to denote a [[root word]].

==Principal square root==
Each positive real number has two square roots, one positive and the other negative. The radical symbol refers to the [[principal value]] of the square root function called the principal square root, which is the positive one. The two square roots of a negative number are both [[imaginary number]]s, and the square root symbol refers to the principal square root, the one with a positive imaginary part. For the definition of the principal square root of other [[complex number]]s, see {{slink|Square root|Principal square root of a complex number}}.

==Origin==

The origin of the root symbol √ is largely speculative.  Some sources imply that the symbol was first used by Arab mathematicians. One of those mathematicians was [[Abū al-Hasan ibn Alī al-Qalasādī]] (1421–1486). Legend has it that it was taken from the [[Arabic]] letter "{{lang|tg-Arab|ج}}" (''[[Gimel#Arabic ǧīm|ǧīm]]''), which is the first letter in the Arabic word "{{lang|tg-Arab|جذر}}" (''jadhir'', meaning "root").<ref>{{cite web|url=http://itre.cis.upenn.edu/~myl/languagelog/archives/002662.html |title=Language Log: ''Ab surd'' |access-date=22 June 2012}}</ref>  However, [[Leonhard Euler]]<ref>{{cite book|title=Institutiones calculi differentialis|url=https://archive.org/details/bub_gb_sYE_AAAAcAAJ|author=Leonhard Euler|year=1755|publisher=Petropolis |language=la}}</ref> believed it originated from the letter "r", the first letter of the [[Latin]] word "[[radix]]" (meaning "root"), referring to the same [[Operation (mathematics)|mathematical operation]].

The symbol was first seen in print without the [[vinculum (symbol)|vinculum]] (the horizontal "bar" over the numbers inside the radical symbol) in the year 1525 in ''Die Coss'' by [[Christoff Rudolff]], a [[German people|German]] mathematician. In 1637 [[Renee Descartes|Descartes]] was the first to unite the German radical sign √ with the vinculum to create the radical symbol in common use today.<ref>{{cite book|first=Florian|last=Cajori|title=A History of Mathematical Notations|volume=I|year=2012|orig-year=1928|publisher=Dover|isbn=978-0-486-67766-8|page=[https://archive.org/details/historyofmathema00cajo_0/page/208 208]|url-access=registration|url=https://archive.org/details/historyofmathema00cajo_0/page/208}}</ref>

==Encoding==
The Unicode and HTML character codes for the radical symbols are:
* {{unichar|221A|html=}}
* {{unichar|221B|html=}}
* {{unichar|221C|html=}}
* {{unichar|23B7|html=}}

However, these characters differ in appearance from most mathematical typesetting by omitting the [[overline]] connected to the radical symbol, which surrounds the argument of the square root function. The [[OpenType]] math table allows adding this overline following the radical symbol.

The Symbol font displays the character without any vinculum whatsoever; the overline may be a separate character at 0x60.<ref>{{citation|mode=cs1 |archive-url=https://web.archive.org/web/20150708120330/http://www-03.ibm.com/systems/resources/systems_i_software_globalization_pdf_cp01038z.pdf |archive-date=2015-07-08 |url=http://www-03.ibm.com/systems/resources/systems_i_software_globalization_pdf_cp01038z.pdf |title=Code Page 01038 |author=IBM |author-link=IBM}}</ref> The JIS,<ref>{{cite iso-ir |title=Japanese Graphic Character Set for Information Interchange |sponsor=ISO/IEC JTC 1/SC 2 |sponsor-link=ISO/IEC JTC 1/SC 2 |date=1992-07-13 |number=168}}</ref> Wansung<ref>{{cite iso-ir |title=Korean Graphic Character Set for Information Interchange |sponsor=Korea Bureau of Standards |date=1988-10-01 |number=149}}</ref> and CNS 11643<ref name="cnstwradic" /><ref>{{cite iso-ir |title=Chinese Standard Interchange Code (CSIC) - Set 1 |sponsor=ECMA |sponsor-link=Ecma International |date=1994 |orig-date=1993 |number=171}}</ref> code charts include a short overline attached to the radical symbol, whereas the GB 2312<ref>{{cite iso-ir |title=Coded Chinese Graphic Character Set for Information Interchange |sponsor=China Association for Standardization |date=1980 |number=58}}</ref> and [[GB 18030]] charts do not.<ref>{{citation|mode=cs1 |url=https://archive.org/details/GB18030-2005 |author=Standardization Administration of China |title=Information Technology—Chinese coded character set |id=GB 18030-2005 |date=2005 |page=8}}</ref>

Additionally a "Radical Symbol Bottom" (U+23B7, ⎷) is available in the [[Miscellaneous Technical]] block.<ref>{{cite web |url=https://www.unicode.org/charts/PDF/U2300.pdf |title=Miscellaneous Technical |work=The Unicode Standard |edition=15.0 |access-date=2023-07-16 |author=Unicode Consortium |date=2022-09-16}}</ref> This was used in contexts where [[box-drawing character]]s are used, such as in the [[DEC Technical Character Set|technical character set]] of [[Digital Equipment Corporation|DEC]] terminals, to join up with box drawing characters on the line above to create the vinculum.<ref>{{cite web |url=https://vt100.net/charsets/technical.html |title=DEC Technical Character Set (TCS) |last=Williams |first=Paul Flo |date=2002 |work=VT100.net |access-date=2023-07-16}}</ref>

In [[LaTeX]] the square root symbol may be generated by the <code>\sqrt</code> macro,<ref>{{cite web |url=http://mirrors.ctan.org/macros/latex/base/source2e.pdf |title=The LATEX 2ε Sources |edition=2023-06-01 Patch Level 1 |at=§ ltmath.dtx: Math Environments |access-date=2023-07-16 |date=2023-06-01 |last1=Braams |first1=Johannes |last2=Carlisle |first2=David |last3=Jeffrey |first3=Alan |last4=Lamport |first4=Leslie |last5=Mittelbach |first5=Frank |last6=Rowley |first6=Chris |last7=Schöpf |first7=Rainer |display-authors=1}}</ref> and the square root symbol without the overline may be generated by the <code>\surd</code> macro.<ref>{{cite book|title=Practical LaTeX|first=George|last=Grätzer|publisher=Springer|year=2014|isbn=9783319064253|contribution=Table B.5: Miscellaneous symbols|contribution-url=https://books.google.com/books?id=BxhRBAAAQBAJ&pg=PA172|page=172}}</ref>

===Legacy encodings===
Legacy encodings of the square root character U+221A include:
* 0xC3 in [[Mac OS Roman]] and [[Mac OS Cyrillic encoding|Mac OS Cyrillic]]
* 0xFB ({{keypress|Alt}}+{{keypress|2}}{{keypress|5}}{{keypress|1}}) in [[Code page 437]] and [[Code page 866]] (but not [[Code page 850]]) on DOS and the Windows console
* 0xD6 in the [[Symbol (typeface)|Symbol font encoding]]<ref>{{citation|mode=cs1 |url=https://unicode.org/Public/MAPPINGS/VENDORS/APPLE/SYMBOL.TXT |title=Map (external version) from Mac OS Symbol character set to Unicode 4.0 and later |id=SYMBOL.TXT |author=Apple Computer |author-link=Apple, Inc |date=2005-04-05 |orig-year=1995-04-15 |publisher=[[Unicode Consortium]]}}</ref>
* 02-69 (7-bit 0x2265, [[Shift JIS|SJIS]] 0x81E3, [[EUC-JP|EUC]] 0xA2E5) in Japanese [[JIS X 0208]]<ref>{{citation|mode=cs1 |title=JIS X 0208 (1990) to Unicode |id=JIS0208.TXT |date=2015-12-02 |orig-year=1994-03-08 |author=Unicode Consortium |author-link=Unicode Consortium |url=https://www.unicode.org/Public/MAPPINGS/OBSOLETE/EASTASIA/JIS/JIS0208.TXT}}</ref>
* 01-78 ([[EUC-KR|EUC]]/[[Unified Hangul Code|UHC]] 0xA1EE) in Korean [[KS X 1001|Wansung code]]<ref>{{citation|mode=cs1 |title=Unified Hangeul(KSC5601-1992) to Unicode table |id=KSC5601.TXT |date=2011-10-14 |orig-year=1995-07-24 |author=Unicode Consortium |author-link=Unicode Consortium |url=https://www.unicode.org/Public/MAPPINGS/OBSOLETE/EASTASIA/KSC/KSC5601.TXT}}</ref>
* 01-44 ([[EUC-CN|EUC]] 0xA1CC) in Mainland Chinese [[GB 2312]] or [[GBK (character encoding)|GBK]]<ref>{{cite web |url=https://opensource.apple.com/source/ICU/ICU-551.51/icuSources/data/mappings/windows-936-2000.ucm |author=IBM |author-link=IBM |title=windows-936-2000 |work=[[International Components for Unicode]] |date=2002}}</ref>
* Traditional Chinese: 0xA1D4 in [[Big5]]<ref>{{citation|mode=cs1 |title=BIG5 to Unicode table (complete) |id=BIG5.TXT |date=2015-12-02 |orig-year=1994-02-11 |author=Unicode Consortium |author-link=Unicode Consortium |url=https://www.unicode.org/Public/MAPPINGS/OBSOLETE/EASTASIA/OTHER/BIG5.TXT}}</ref><ref name="cnstwradic" /> or 1-2235 (kuten 01-02-21, [[EUC-TW|EUC]] 0xA2B5 or 0x8EA1A2B5) in [[CNS 11643]]<ref name="cnstwradic">{{cite web |url=https://www.cns11643.gov.tw/wordView.jsp?ID=74293&SN=&lang=en |title=[√] 1-2235 |work=Word Information |publisher=[[National Development Council (Taiwan)|National Development Council]]}}</ref><ref>{{cite web |url=https://opensource.apple.com/source/ICU/ICU-551.51/icuSources/data/mappings/euc-tw-2014.ucm |author=IBM |author-link=IBM |title=euc-tw-2014 |work=[[International Components for Unicode]] |date=2014}}</ref>

==See also==
*[[Solution in radicals]]
==References==
{{Reflist}}

{{Navbox punctuation}}

[[Category:Mathematical symbols]]