In mathematics, Gaussian brackets are a special notation invented by Carl Friedrich Gauss to represent the convergents of a simple continued fraction in the form of a simple fraction.
Gauss used this notation in the context of finding solutions of the indeterminate equations of the form
[1] This notation should not be confused with the widely prevalent use of square brackets to denote the greatest integer function:
This notation was also invented by Gauss and was used in the third proof of the quadratic reciprocity law.
[2] The Gaussian brackets notation is defined as follows:[3][4] The expanded form of the expression
"[4] With this notation, one can easily verify that[3] The Gaussian brackets have been used extensively by optical designers as a time-saving device in computing the effects of changes in surface power, thickness, and separation of focal length, magnification, and object and image distances.
[4][5] The following papers give additional details regarding the applications of Gaussian brackets in optics.