In mathematics, theradical symbol,radical sign,root symbol, orsurd is asymbol for thesquare root orhigher-order root of a number. The square root of a number${\displaystyle x}$ is written as

${\displaystyle \sqrt{x},}$

while the${\displaystyle n}$th root of${\displaystyle x}$ is written as

${\displaystyle \sqrt[n]{x}.}$

It is also used for other meanings in more advanced mathematics, such as theradical of an ideal.

Inlinguistics, the symbol is used to denote aroot word.

Each positive real number has two square roots, one positive and the other negative. The radical symbol refers to theprincipal 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 bothimaginary numbers, 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 othercomplex numbers, seeSquare root § Principal square root of a complex number.

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 wasAbū al-Hasan ibn Alī al-Qalasādī (1421–1486). Legend has it that it was taken from theArabic letter "ج" (ǧīm), which is the first letter in the Arabic word "جذر" (jadhir, meaning "root").^[1] However,Leonhard Euler^[2] believed it originated from the letter "r", the first letter of theLatin word "radix" (meaning "root"), referring to the samemathematical operation.

The symbol was first seen in print without thevinculum (the horizontal "bar" over the numbers inside the radical symbol) in the year 1525 inDie Coss byChristoff Rudolff, aGerman mathematician. In 1637Descartes was the first to unite the German radical sign √ with the vinculum to create the radical symbol in common use today.^[3]

The Unicode and HTML character codes for the radical symbols are:

• U+221A √SQUARE ROOT (√, √)
• U+221B ∛CUBE ROOT
• U+221C ∜FOURTH ROOT
• U+23B7 ⎷RADICAL SYMBOL BOTTOM

However, these characters differ in appearance from most mathematical typesetting by omitting theoverline connected to the radical symbol, which surrounds the argument of the square root function. TheOpenType 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.^[4] The JIS,^[5] Wansung^[6] and CNS 11643^[7][8] code charts include a short overline attached to the radical symbol, whereas the GB 2312^[9] andGB 18030 charts do not.^[10]

Additionally a "Radical Symbol Bottom" (U+23B7, ⎷) is available in theMiscellaneous Technical block.^[11] This was used in contexts wherebox-drawing characters are used, such as in thetechnical character set ofDEC terminals, to join up with box drawing characters on the line above to create the vinculum.^[12]

InLaTeX the square root symbol may be generated by the\sqrt macro,^[13] and the square root symbol without the overline may be generated by the\surd macro.^[14]

Legacy encodings of the square root character U+221A include:

• 0xC3 inMac OS Roman andMac OS Cyrillic
• 0xFB (Alt+251) inCode page 437 andCode page 866 (but notCode page 850) on DOS and the Windows console
• 0xD6 in theSymbol font encoding^[15]
• 02-69 (7-bit 0x2265,SJIS 0x81E3,EUC 0xA2E5) in JapaneseJIS X 0208^[16]
• 01-78 (EUC/UHC 0xA1EE) in KoreanWansung code^[17]
• 01-44 (EUC 0xA1CC) in Mainland ChineseGB 2312 orGBK^[18]
• Traditional Chinese: 0xA1D4 inBig5^[19][7] or 1-2235 (kuten 01-02-21,EUC 0xA2B5 or 0x8EA1A2B5) inCNS 11643^[7][20]

• Solution in radicals

• Mathematical symbols

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