A square root of is a number such that . This is written ( to the 1/2 Power) or
. The square root function is the Inverse Function of . Square roots are also
called Radicals or Surds. A general Complex Number has *two* square
roots. For example, for the real Positive number , the two square roots are
, since
. Similarly, for the real Negative number , the two square roots are
, where
*i* is the Imaginary Number defined by . In common usage, unless otherwise specified, ``the'' square
root is generally taken to mean the Positive square root.

The square root of 2 is the Irrational Number (Sloane's A002193), which has the simple periodic Continued Fraction 1, 2, 2, 2, 2, 2, .... The square root of 3 is the Irrational Number (Sloane's A002194), which has the simple periodic Continued Fraction 1, 1, 2, 1, 2, 1, 2, .... In general, the Continued Fractions of the square roots of all Positive integers are periodic.

The square roots of a Complex Number are given by

(1) |

A Nested Radical of the form
can sometimes be simplified into a simple square root
by equating

(2) |

(3) |

(4) | |||

(5) |

Solving for and gives

(6) |

A sequence of approximations to can be derived by factoring

(7) |

(8) |

(9) |

(10) | |||

(11) | |||

(12) |

Therefore, and are given by the Recurrence Relations

(13) | |||

(14) |

with . The error obtained using this method is

(15) |

(16) |

Another general technique for deriving this sequence, known as Newton's Iteration, is obtained by letting .
Then , so the Sequence

(17) |

(18) |

**References**

Sloane, N. J. A. Sequences
A002193/M3195
and A002194/M4326
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Spanier, J. and Oldham, K. B. ``The Square-Root Function and Its Reciprocal,''
``The
Function and Its Reciprocal,'' and ``The Function.''
Chs. 12, 14, and 15 in *An Atlas of Functions.*
Washington, DC: Hemisphere, pp. 91-99, 107-115, and 115-122, 1987.

Williams, H. C. ``A Numerical Investigation into the Length of the Period of the Continued Fraction Expansion of .''
*Math. Comp.* **36**, 593-601, 1981.

© 1996-9

1999-05-26