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Gaussian Brackets

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Gaussian brackets are notation published by Gauss in Disquisitiones Arithmeticae and defined by

[ ]=1
(1)
[a_1]=a_1
(2)
[a_1,a_2]=[a_1]a_2+[ ]
(3)
[a_1,a_2,...,a_n]=[a_1,a_2,...,a_(n-1)]a_n+[a_1,a_2,...,a_(n-2)].
(4)

Gaussian brackets are useful for computing simple continued fractions because

K_(k=1)^(n)1/(a_k)=1/(a_1+1/(a_2+1/(a_3+...+1/(a_n))))
(5)
=([a_2,...,a_n])/([a_1,...,a_n]).
(6)

Note that the Gaussian bracket notation [a_1,a_2,...] corresponds to a different quantity than that denoted by the more established simple continued fraction notation

[b_0;b_1,b_2,...,b_n]=b_0+1/(b_1+1/(b_2+1/(b_3+...+1/(b_n)))).
(7)

See also

Continued Fraction, Regular Continued Fraction, Simple Continued Fraction

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References

Fowler, D. H. The Mathematics of Plato's Academy: A New Reconstruction, 2nd ed. Oxford, England: Oxford University Press, 1999.Herzberger, M. Modern Geometrical Optics. New York: Interscience Publishers, pp. 457-462, 1958.

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Gaussian Brackets

Cite this as:

Weisstein, Eric W. "Gaussian Brackets." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GaussianBrackets.html

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