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Gaussian Brackets

Published by Gauß in Disquisitiones Arithmeticae. They are defined as follows.

\begin{displaymath}[\ ]= 1
\end{displaymath} (1)

\begin{displaymath}[a_1]= a_1
\end{displaymath} (2)

\begin{displaymath}[a_1,a_2]=[a_1]a_2+[\ ]
\end{displaymath} (3)

\begin{displaymath}[a_1,a_2,\ldots,a_n]= [a_1,a_2,\ldots,a_{n-1}]a_n+[a_1,a_2,\ldots,a_{n-2}].
\end{displaymath} (4)

Gaussian brackets are useful for treating Continued Fractions because
{1\over a_1+{\strut\displaystyle 1\over\strut\displaystyle a...
...\over\strut\displaystyle a_n}}}} = {[a_2,a_n]\over [a_1,a_n]}.
\end{displaymath} (5)

The Notation $[x]$ conflicts with that of Gaussian Polynomials and the Nint function.


Herzberger, M. Modern Geometrical Optics. New York: Interscience Publishers, pp. 457-462, 1958.

© 1996-9 Eric W. Weisstein