In mathematics, the radical sign, radical symbol, root symbol, radix, or surd is a
symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
for the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
or
higher-order root of a number. The square root of a number
is written as
:
while the
th root of
is written as
:
It is also used for other meanings in more advanced mathematics, such as the
radical of an ideal
In ring theory, a branch of mathematics, the radical of an ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in I. Taking the radical of an ideal is called ' ...
.
In
linguistics
Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...
, the symbol is used to denote a
root word
A root (or root word) is the core of a word that is irreducible into more meaningful elements. In morphology, a root is a morphologically simple unit which can be left bare or to which a prefix or a suffix can attach. The root word is the prima ...
.
Principal square root
Each positive real number has two square roots, one positive and the other negative. The square root symbol refers to the principal square root, which is the positive one. The two square roots of a negative number are both
imaginary number
An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...
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 numbers, see
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
Abū'l-Ḥasan ibn ʿAlī ibn Muḥammad ibn ʿAlī al-Qurashī al-Qalaṣādī ( ar, أبو الحسن علي بن محمد بن علي القرشي البسطي; 1412–1486) was a Muslim Arab mathematician from Al-Andalus specializing in Is ...
(1421–1486). Legend has it that it was taken from the
Arabic
Arabic (, ' ; , ' or ) is a Semitic languages, Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C ...
letter "" (''
ǧīm
Gimel is the third letter of the Semitic abjads, including Phoenician Gīml , Hebrew Gimel , Aramaic Gāmal , Syriac Gāmal , and Arabic (in alphabetical order; fifth in spelling order). Its sound value in the original Phoenician and in all d ...
''), which is the first letter in the Arabic word "" (''jadhir'', meaning "root"). However,
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
believed it originated from the letter "r", the first letter of the
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
word "
radix
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
" (meaning "root"), referring to the same
mathematical operation
In mathematics, an operation is a Function (mathematics), function which takes zero or more input values (also called "''operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation.
The most c ...
.
The symbol was first seen in print without the
vinculum (the horizontal "bar" over the numbers inside the radical symbol) in the year 1525 in ''Die Coss'' by
Christoff Rudolff, a
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
mathematician. In 1637
Descartes was the first to unite the German radical sign √ with the vinculum to create the radical symbol in common use today.
Encoding
The Unicode and HTML character codes for the radical symbols are:
However, these characters differ in appearance from most mathematical typesetting by omitting the
overline
An overline, overscore, or overbar, is a typographical feature of a horizontal line drawn immediately above the text. In old mathematical notation, an overline was called a '' vinculum'', a notation for grouping symbols which is expressed in m ...
connected to the radical symbol, which surrounds the argument of the square root function. The
OpenType
OpenType is a format for scalable computer fonts. It was built on its predecessor TrueType, retaining TrueType's basic structure and adding many intricate data structures for prescribing typographic behavior. OpenType is a registered trademark o ...
math table allows adding this overline following the radical symbol.
Legacy encodings of the square root character U+221A include:
* 0xC3 in
Mac OS Roman
Mac OS Roman is a character encoding created by Apple Computer, Inc. for use by Macintosh computers. It is suitable for representing text in English and several other Western languages. Mac OS Roman encodes 256 characters, the first 128 of which ...
and
Mac OS Cyrillic
* 0xFB (+) in
Code page 437
Code page 437 (CCSID 437) is the character set of the original IBM PC (personal computer). It is also known as CP437, OEM-US, OEM 437, PC-8, or DOS Latin US. The set includes all printable ASCII characters as well as some accented letters (diacri ...
and
Code page 866
Code page 866 (CCSID 866) (CP 866, "DOS Cyrillic Russian") is a code page used under DOS and OS/2 in Russia to write Cyrillic script. It is based on the "alternative code page" (russian: Альтернативная кодировка) develope ...
(but not
Code page 850
Code page 850 (CCSID 850) (also known as CP 850, IBM 00850, OEM 850, DOS Latin 1) is a code page used under DOS and Psion's EPOC16 operating systems in Western Europe. Depending on the country setting and system configuration, code page 850 i ...
) on DOS and the Windows console
* 0xD6 in the
Symbol font encoding
* 02-69 (7-bit 0x2265,
SJIS 0x81E3,
EUC 0xA2E5) in Japanese
JIS X 0208
JIS X 0208 is a 2-byte character set specified as a Japanese Industrial Standard, containing 6879 graphic characters suitable for writing text, place names, personal names, and so forth in the Japanese language. The official title of the current ...
* 01-78 (
EUC/
UHC 0xA1EE) in Korean
Wansung code
* 01-44 (
EUC 0xA1CC) in Mainland Chinese
GB 2312
is a key official character set of the People's Republic of China, used for Simplified Chinese characters. GB2312 is the registered internet name for EUC-CN, which is its usual encoded form. ''GB'' refers to the Guobiao standards (国家标准 ...
or
GBK
* Traditional Chinese: 0xA1D4 in
Big5
Big-5 or Big5 is a Chinese character encoding method used in Taiwan, Hong Kong, and Macau for traditional Chinese characters.
The People's Republic of China (PRC), which uses simplified Chinese characters, uses the GB 18030 character set inst ...
or 1-2235 (kuten 01-02-21,
EUC 0xA2B5 or 0x8EA1A2B5) in
CNS 11643
The CNS 11643 character set (Chinese National Standard 11643), also officially known as the Chinese Standard Interchange Code or CSIC ( zh, tr=, t=中文標準交換碼), is officially the standard character set of Taiwan (Republic of China). In p ...
The Symbol font displays the character without any vinculum whatsoever; the overline may be a separate character at 0x60. The JIS, Wansung and CNS 11643
code charts include a short overline attached to the radical symbol, whereas the GB 2312 and
GB 18030
GB 18030 is a Chinese government standard, described as ''Information Technology — Chinese coded character set'' and defines the required language and character support necessary for software in China. GB18030 is the registered Internet ...
charts do not.
Additionally a "Radical Symbol Bottom" (U+23B7, ⎷) is available in the
Miscellaneous Technical
Miscellaneous Technical is a Unicode block ranging from U+2300 to U+23FF, which contains various common symbols which are related to and used in the various technical, programming language, and academic professions. For example:
* Symbol ⌂ (HT ...
block. This was used in contexts where
box-drawing character
Box-drawing characters, also known as line-drawing characters, are a form of semigraphics widely used in text user interfaces to draw various geometric frames and boxes. Box-drawing characters typically only work well with monospaced fonts. In ...
s are used, such as in the
technical character set of
DEC terminals, to join up with box drawing characters on the line above to create the vinculum.
In
LaTeX
Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well.
In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...
the square root symbol may be generated by the
\sqrt
macro,
and the square root symbol without the overline may be generated by the
\surd
macro.
References
{{Reflist
Mathematical symbols