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: denotes the greatest integer less than or equal to . This notation was also invented by Gauss and was used in the third proof of the quadratic reciprocity law. The notation , denoting the floor function, is now more commonly used to denote the greatest integer less than or equal to . [2]
The Gaussian brackets notation is defined as follows: [3] [4]
The expanded form of the expression can be described thus: "The first term is the product of all n members; after it come all possible products of (n -2) members in which the numbers have alternately odd and even indices in ascending order, each starting with an odd index; then all possible products of (n-4) members likewise have successively higher alternating odd and even indices, each starting with an odd index; and so on. If the bracket has an odd number of members, it ends with the sum of all members of odd index; if it has an even number, it ends with unity." [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.
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: CS1 maint: multiple names: authors list (
link)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: denotes the greatest integer less than or equal to . This notation was also invented by Gauss and was used in the third proof of the quadratic reciprocity law. The notation , denoting the floor function, is now more commonly used to denote the greatest integer less than or equal to . [2]
The Gaussian brackets notation is defined as follows: [3] [4]
The expanded form of the expression can be described thus: "The first term is the product of all n members; after it come all possible products of (n -2) members in which the numbers have alternately odd and even indices in ascending order, each starting with an odd index; then all possible products of (n-4) members likewise have successively higher alternating odd and even indices, each starting with an odd index; and so on. If the bracket has an odd number of members, it ends with the sum of all members of odd index; if it has an even number, it ends with unity." [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.
{{
cite journal}}
: CS1 maint: multiple names: authors list (
link)