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In [[Mathematics|mathematics]], '''Gaussian brackets''' are a special notation invented by [[Biography:Carl Friedrich Gauss|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 equation|indeterminate equation]]s of the form <math>ax=by\pm 1 </math>.<ref>{{cite book |last1=Carl Friedrich Gauss (English translation by Arthur A. Clarke and revised by William C. Waterhouse) |title=Disquisitiones Arithmeticae |date=1986 |publisher=Springer-Verlag |location=New York |isbn=0-387-96254-9 |pages=10–11}}</ref>
This notation should not be confused with the widely prevalent use of square brackets to denote the greatest integer function: <math>[x]</math> denotes the greatest integer less than or equal to <math>x</math>. This notation was also invented by Gauss and was used in the third proof of the quadratic reciprocity law. The notation <math>\lfloor x \rfloor </math>, denoting the floor function,  is now more commonly used to denote the greatest integer less than or equal to <math>x</math>.<ref>{{cite web |last1=Weisstein, Eric W. |title=Floor Function |url=https://mathworld.wolfram.com/FloorFunction.html |website=MathWorld--A Wolfram Web Resource. |access-date=25 January 2023}}</ref>

==The notation==
The Gaussian brackets notation is defined as follows:<ref name=Wolfram>{{cite web |last1=Weisstein, Eric W. |title=Gaussian Brackets |url=https://mathworld.wolfram.com/GaussianBrackets.html |website=MathWorld - A Wolfram Web Resource |access-date=24 January 2023}}</ref><ref name=Herzberger/>

:<math>\begin{align}
\quad[\,\,] & = 1\\[1mm]
[a_1] & = a_1\\[1mm]
[a_1, a_2] & = [a_1]a_2 + [\,\,]\\[1mm]
           & = a_1a_2+1\\[1mm] 
[a_1, a_2, a_3] & = [a_1, a_2]a_3 + [a_1] \\[1mm]
                & = a_1a_2a_3  + a_1 + a_3 \\[1mm]
[a_1,a_2,a_3,a_4] & = [a_1,a_2,a_3]a_4 + [a_1,a_2]\\[1mm]
                  & = a_1a_2a_3a_4 + a_1a_2 + a_1a_4 + a_3a_4 + 1\\[1mm]
[a_1,a_2,a_3,a_4,a_5] & = [a_1,a_2,a_3,a_4]a_5 + [a_1, a_2,a_3]\\[1mm]
                      & = a_1a_2a_3a_4a_5 + a_1a_2a_3 + a_1a_2a_5 + a_1a_4a_5 + a_3a_4a_5 + a_1+a_3+a_5\\[1mm]
\vdots & \\[1mm]
[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}]
\end{align}
</math>

The expanded form of the expression <math>[a_1,a_2,\ldots, a_n]</math> 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."<ref name=Herzberger>{{cite journal |last1=M. Herzberger |title=Gaussian Optics and Gaussian Brackets |journal=Journal of the Optical Society of America |date=December 1943 |volume=33 |issue=12 |doi=10.1364/JOSA.33.000651}}</ref>

With this notation, one can easily verify that<ref name=Wolfram/> 
 
::<math> \cfrac{1}{a_1  + \cfrac{1}{ a_2 + \cfrac{1}{a_3 + \cdots \frac{\ddots}{    \cfrac{1}{a_{n-1} +\frac{1}{a_n}} } }}} = \frac{[a_2,\ldots,a_n]}{[a_1,a_2,\ldots,a_n]}</math>

==Properties==

# The bracket notation can also be defined by the [[Recursion|recursion]] relation: <math>\,\,[a_1,a_2, a_3, \ldots, a_n]=a_1[a_2,a_3, \ldots,a_n] + [a_3,\ldots,a_n]</math>
# The notation is symmetric or reversible in the arguments: <math>\,\,[a_1,a_2, \ldots,a_{n-1},a_n]=[a_n,a_{n-1},\ldots, a_2,a_1]</math>
# The Gaussian brackets expression can be written by means of a determinant:  <math>\,\,[a_1,a_2,\ldots,a_n] = 
\begin{vmatrix} 
a_1 &  -1  & 0   & 0  & \cdots & 0 & 0 & 0 \\[1mm] 
1   &  a_2 & -1  & 0  & \cdots & 0 & 0 & 0 \\[1mm]
0   &  1   & a_3 & -1 & \cdots & 0 & 0 & 0 \\[1mm]
\vdots &   &     &    &        &   &   &   \\[1mm]
0   &  0   & 0   & 0  & \cdots & 1 & a_{n-1} & -1 \\[1mm]
0   &  0   & 0   & 0  & \cdots & 0 &  1  & a_n
\end{vmatrix}
</math>
# The notation satisfies the [[Determinant|determinant]] formula (for <math>n=1</math> use the convention that <math>[a_2,\ldots,a_0]=0</math>): <math>\,\, \begin{vmatrix} [a_1,\ldots,a_n] & [a_1,\ldots,a_{n-1}]\\[1mm] [a_2, \ldots, a_{n}] &  [a_2,\ldots, a_{n-1}]\end{vmatrix}=(-1)^n</math>
# <math>[-a_1, -a_2, \ldots, -a_n] = (-1)^n[a_1,a_2, \ldots,a_n]</math>
# Let the elements in the Gaussian bracket expression be alternatively 0. Then 

:::<math>
\begin{align}
\,\,\quad[a_1,0,a_3,0,\ldots,a_{2m+1}] & = a_1+a_3+\cdots + a_{2m+1}\\[1mm]
[a_1,0,a_3,0,\ldots,a_{2m+1}, 0] & = 1\\[1mm]
[0, a_2, 0, a_4, \ldots, a_{2m}] & = 1 \\[1mm]
[0, a_2, 0, a_4, \ldots, a_{2m}, 0] & = 0 
\end{align}
</math>

==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.<ref name=Herzberger/><ref>{{cite journal |last1=Kazuo Tanaka |title=Paraxial theory in optical design in terms of Gaussian brackets |journal=Progress in Optics |date=1986 |volume=XXIII |pages=63–111 |doi=10.1016/S0079-6638(08)70031-3|bibcode=1986PrOpt..23...63T |isbn=9780444869821 }}</ref>

==References==
{{reflist}}

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

* {{cite journal |last1=Chen Ma, Dewen Cheng, Q. Wang and Chen Xu|title=Optical System Design of a Liquid Tunable Fundus Camera Based on Gaussian Brackets Method |journal=Acta Optica Sinica |date=November 2014 |volume=34 |issue=11 |doi=10.3788/AOS201434.1122001}}
* {{cite journal |last1=Yi Zhong, Herbert Gross |title=Initial system design method for non-rotationally symmetric systems based on Gaussian brackets and Nodal aberration theory |journal=Opt Express |date=May 2017 |volume=25 |issue=9 |pages=10016–10030 |doi=10.1364/OE.25.010016 |pmid=28468369 |bibcode=2017OExpr..2510016Z |url=https://opg.optica.org/abstract.cfm?URI=oe-25-9-10016 |access-date=24 January 2023|doi-access=free }}
*{{cite journal |last1=Xiangyu Yuan and Xuemin Cheng |editor-first1=Yongtian |editor-first2=Chunlei |editor-first3=José |editor-first4=Kimio |editor-last1=Wang |editor-last2=Du |editor-last3=Sasián |editor-last4=Tatsuno |title=Lens design based on lens form parameters using Gaussian brackets |journal=Proc. SPIE 9272, Optical Design and Testing VI, 92721L |series=Optical Design and Testing VI |date=November 2014 |volume=9272 |pages=92721L |doi=10.1117/12.2073422|bibcode=2014SPIE.9272E..1LY |s2cid=121201008 }}


[[Category:Continued fractions]]

{{Sourceattribution|Gaussian brackets}}