Gaussian brackets

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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 notation

Summarize
Perspective

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]

Properties

  1. The bracket notation can also be defined by the recursion relation:
  2. The notation is symmetric or reversible in the arguments:
  3. The Gaussian brackets expression can be written by means of a determinant:
  4. The notation satisfies the determinant formula (for use the convention that ):
  5. Let the elements in the Gaussian bracket expression be alternatively 0. Then

Applications

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]

References

Additional reading

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