Gaussian brackets explained

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

ax=by\pm1

.[1]

This notation should not be confused with the widely prevalent use of square brackets to denote the greatest integer function:

[x]

denotes the greatest integer less than or equal to

x

. This notation was also invented by Gauss and was used in the third proof of the quadratic reciprocity law. The notation

\lfloorx\rfloor

, denoting the floor function, is now more commonly used to denote the greatest integer less than or equal to

x

.[2]

The notation

The Gaussian brackets notation is defined as follows:[3]

\begin{align} []&=1\\[1mm] [a1]&=a1\\[1mm] [a1,a2]&=[a1]a2+[]\\[1mm] &=a1a2+1\\[1mm][a1,a2,a3]&=[a1,a2]a3+[a1]\\[1mm] &=a1a2a3+a1+a3\\[1mm] [a1,a2,a3,a4]&=[a1,a2,a3]a4+[a1,a2]\\[1mm] &=a1a2a3a4+a1a2+a1a4+a3a4+1\\[1mm] [a1,a2,a3,a4,a5]&=[a1,a2,a3,a4]a5+[a1,a2,a3]\\[1mm] &=a1a2a3a4a5+a1a2a3+a1a2a5+a1a4a5+a3a4a5+a1+a3+a5\\[1mm] \vdots&\\[1mm] [a1,a2,\ldots,an]&=[a1,a2,\ldots,an-1]an+[a1,a2,\ldots,an-2] \end{align}

The expanded form of the expression

[a1,a2,\ldots,an]

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]

\cfrac{1}{a1+\cfrac{1}{a2+\cfrac{1}{a3+

\ddots
\cfrac{1

{an-1+

1
an
} } }}} = \frac

Properties

  1. The bracket notation can also be defined by the recursion relation:

[a1,a2,a3,\ldots,an]=a1[a2,a3,\ldots,an]+[a3,\ldots,an]

  1. The notation is symmetric or reversible in the arguments:

[a1,a2,\ldots,an-1,an]=[an,an-1,\ldots,a2,a1]

  1. The Gaussian brackets expression can be written by means of a determinant:

[a1,a2,\ldots,an]=\begin{vmatrix}a1&-1&0&0&&0&0&0\\[1mm]1&a2&-1&0&&0&0&0\\[1mm] 0&1&a3&-1&&0&0&0\\[1mm] \vdots&&&&&&&\\[1mm] 0&0&0&0&&1&an-1&-1\\[1mm] 0&0&0&0&&0&1&an \end{vmatrix}

  1. The notation satisfies the determinant formula (for

n=1

use the convention that

[a2,\ldots,a0]=0

):

\begin{vmatrix}[a1,\ldots,an]&[a1,\ldots,an-1]\\[1mm][a2,\ldots,an]&[a2,\ldots,an-1]\end{vmatrix}=(-1)n

[-a1,-a2,\ldots,-an]=

n[a
(-1)
1,a

2,\ldots,an]

  1. Let the elements in the Gaussian bracket expression be alternatively 0. Then

\begin{align} [a1,0,a3,0,\ldots,a2m+1]&=a1+a3+ … +a2m+1\\[1mm] [a1,0,a3,0,\ldots,a2m+1,0]&=1\\[1mm] [0,a2,0,a4,\ldots,a2m]&=1\\[1mm] [0,a2,0,a4,\ldots,a2m,0]&=0\end{align}

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]

Additional reading

The following papers give additional details regarding the applications of Gaussian brackets in optics.

Notes and References

  1. Book: Carl Friedrich Gauss (English translation by Arthur A. Clarke and revised by William C. Waterhouse) . Disquisitiones Arithmeticae . 1986 . Springer-Verlag . New York . 0-387-96254-9 . 10–11.
  2. Web site: Weisstein, Eric W. . Floor Function . MathWorld--A Wolfram Web Resource. . 25 January 2023.
  3. Web site: Weisstein, Eric W. . Gaussian Brackets . MathWorld - A Wolfram Web Resource . 24 January 2023.
  4. M. Herzberger . Gaussian Optics and Gaussian Brackets . Journal of the Optical Society of America . December 1943 . 33 . 12 . 10.1364/JOSA.33.000651.
  5. Book: Kazuo Tanaka . Paraxial theory in optical design in terms of Gaussian brackets . Progess in Optics . Progress in Optics . 1986 . XXIII . 63–111 . 10.1016/S0079-6638(08)70031-3. 1986PrOpt..23...63T . 9780444869821 .