Gaussian Brackets

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

$$[\ ]= 1$$ (1)

$$[a_1]= a_1$$ (2)

$$[a_1,a_2]=[a_1]a_2+[\ ]$$ (3)

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

Gaussian brackets are useful for treating Continued Fractions because

$${1\over a_1+{\strut\displaystyle 1\over\strut\displaystyle a... ...\over\strut\displaystyle a_n}}}} = {[a_2,a_n]\over [a_1,a_n]}.$$ (5)

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

References

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