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An th complex root (root of degree ) is one of the complex solutions of

Real roots

An th real root (root of degree ) is one of the real solutions of

If is an even positive integer, then the two real roots are

while if is an odd positive integer, then the single real root is

where is the root index and a is the radicand.


A surd is an algebraic irrational root, e.g. is a cubic surd. The quadratic surd is a mixed surd (i.e. a rational number multiplied by a surd).

See also

Hierarchical list of operations pertaining to numbers [1] [2]

0th iteration
1st iteration
  • Addition:
    S(S( "a times" (S(n))))
    , the sum
    n  +  a
    , where
    is the augend and
    is the addend. (When addition is commutative both are simply called terms.)
  • Subtraction:
    P(P( "s times" (P(n))))
    , the difference
    n  −  s
    , where
    is the minuend and
    is the subtrahend.
2nd iteration
3rd iteration
4th iteration
5th iteration
6th iteration
7th iteration
8th iteration
  • Octation (
    as "degree",
    as "base",
    as "variable").
    • Octa-powers:
      n ^^^^^ (n ^^^^^ ( "d times" (n ^^^^^ (n))))
      , written
      n ^^^^^^ d or n ↑↑↑↑↑↑ d
    • Octa-exponentials:
      b ^^^^^ (b ^^^^^ ( "n times" (b ^^^^^ (b))))
      , written
      b ^^^^^^ n or b ↑↑↑↑↑↑ n
  • Octation inverses


  2. Grzegorczyk
  3. There is a lack of consensus on which comes first. Having the multiplier come second makes it consistent with the definitions for exponentiation and higher operations. This is also the convention used with transfinite ordinals:
    ω  ×  2 := ω  +  ω