section_path: Definition and notation
section_name: Definition and notation
paragraph_text: Definition and notation
  
  An n th root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose n th power is x:
  
  r n = x. {\displaystyle r^{n}=x.}
  
  Every positive real number x has a single positive n th root, called the principal n th root, which is written x n {\displaystyle {\sqrt{x}}}. For n equal to 2 this is called the principal square root and the n is omitted. The n th root can also be represented using exponentiation as x.
  
  For even values of n, positive numbers also have a negative n th root, while negative numbers do not have a real n th root. For odd values of n, every negative number x has a real negative n th root. For example, −2 has a real 5th root, − 2 5 = − 1.148698354 … {\displaystyle {\sqrt{-2}}=-1.148698354\ldots } but −2 does not have any real 6th roots.
  
  Every non-zero number x, real or complex, has n different complex number n th roots. (In the case x is real, this count includes any real n th roots.) The only complex root of 0 is 0.
  
  The n th roots of almost all numbers (all integers except the n th powers, and all rationals except the quotients of two n th powers) are irrational. For example,
  
  2 = 1.414213562 … {\displaystyle {\sqrt {2}}=1.414213562\ldots }
  
  All n th roots of rational numbers are algebraic numbers, and all n th roots of integers are algebraic integers.
  
  The term "surd" traces back to Al-Khwarizmi (c. 825), who referred to rational and irrational numbers as audible and inaudible, respectively. This later led to the Arabic word أصم (asamm, meaning "deaf" or "dumb") for irrational number being translated into Latin as surdus (meaning "deaf" or "mute"). Gerard of Cremona (c. 1150), Fibonacci (1202), and then Robert Recorde (1551) all used the term to refer to unresolved irrational roots, that is, expressions of the form r n {\displaystyle {\sqrt{r}}}, in which n {\displaystyle n} and r {\displaystyle r} are integer numerals and the whole expression denotes an irrational number. Irrational numbers of the form ± a, {\displaystyle \pm {\sqrt {a}},} where a {\displaystyle a} is rational, are called pure quadratic surds; irrational numbers of the form a ± b {\displaystyle a\pm {\sqrt {b}}}, where a {\displaystyle a} and b {\displaystyle b} are rational, are called mixed quadratic surds.