“Square Root”的概念、定义、习题解题方法及翻译-数学辞典

单词

Square Root

释义

Square Root

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 alsocalled Radicals or Surds. A general Complex Number has two squareroots. For example, for the real Positive number , the two square roots are , since. Similarly, for the real Negative number , the two square roots are , wherei is the Imaginary Number defined by . In common usage, unless otherwise specified, ``the'' squareroot 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 periodicContinued 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)

As can be seen in the above figure, the Imaginary Part of the complex square root function has a BranchCut along the Negative real axis.

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

(2)

Squaring gives

(3)

			(4)
			(5)

Solving for

and

gives

(6)

A sequence of approximations to can be derived by factoring

(7)

(where

is possible only if

is a Quadratic Residue of

). Then

(8)

(9)

and

			(10)
			(11)
			(12)

Therefore,

and

are given by the Recurrence Relations

			(13)
			(14)

with

. The error obtained using this method is

(15)

The first few approximants to

are therefore given by

(16)

This Algorithm is sometimes known as the Bhaskara-BroucknerAlgorithm. For the case

, this gives the convergents to

as 1, 3/2, 7/5, 17/12, 41/29, 99/70, ....

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

(17)

converges quadratically to the root. The first few approximants to

are therefore given by

(18)

For

, this gives the convergents 1, 3/2, 17/12, 577/408, 665857/470832, ....

See also Continued Square Root, Cube Root, Nested Radical, Newton's Iteration, QuadraticSurd, Root of Unity, Square Number, Square Triangular Number, Surd

References

Sloane, N. J. A. SequencesA002193/M3195and A002194/M4326in ``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.

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