Complex Continued Fractions and The Gaussian Integers

Continued fractions off er a useful method and representation of expressing numbers and functions. One of interesting ideas of this method is that any number can be represented by a point on the real line and that falls between two integers.

Several attempts have been made during the last 100 years to develop continued fraction algorithm for complex numbers[1]; such algorithm having properties that are known to possess on the regular continued fraction algorithm in the real numbers case.

In the real case, for example, let s be any real number, and if t is an integer, then

t\leq s<t+1.

There is one and only one such integer t for any given real s. The integer t is sometimes called the floor of s,

t= \left [ s \right ]

For example : 3= \left [ \pi \right ], -2= \left [-1.5 \right ].

For any real s with t= \left [ s \right ], there is a unique decomposition s= t+u where t is an integer and u is in the unit interval. So, u = 0 if and only if s is an integer.

The process of finding the continued fraction expansion of a real number is a recursive process that procedes one step at a time. Given any real number s, after k steps, the process goes like this one :

s= t_1+\frac{1}{t_2+\frac{1}{t_3+\frac{1}{...+\frac{1}{t_k+u_k}}}}

Which has integers t_{1}, t_{2}, . . . , t_{k} and real numbers u_{1}, u_{2}, ..., u_{k} that are members of the unit interval I with u_{1}, u_{2}, ..., u_{k-1} all positive.  We can write another representation :

s= \left [t_1,t_2,t_3,...,t_k+u_k \right].

If u_{k}=0, the process ends after k steps. Otherwise, the process continues at least one more step with

\frac{1}{u_k}= t_{k+1}+u_{k+1}.

In this way one associates with any real number s a sequence, which could be either finite or infinite, t_{1}, t_{2}, . . . of integers. This sequence is called the continued fraction expansion of real number s. For example, \pi can be represented as a continued fraction

\pi = 3.14159265... = 3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{3+\frac{1}{1+\frac{1}{14+\frac{1}{2+\frac{1}{1}}}}}}}}}}}}}}

or \pi = \left [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3 ...\right].

Continued fractions have become more common in various other areas since the beginning of the twentieth century. For example, Robert M. Corless [2] examines the connection between theory of chaotic dynamical systems and continued fractions. It has also been used in computer algorithms for computing rational approximations to real numbers, as well as for solving Diophantine approximation of complex numbers [1] .

It turns out that complex numbers can also be represented as continued fractions, and, as such, falls between two Gaussian integers, and the unit is any divisor of z whose norm is 1.

The best method at the moment was devised by Asmus Schmidt. [1]

Let

\mathbf{Z}\left [ i \right ] = \left\{a + bi : a, b \in \mathbf{Z},b \geq 0 \right\} \cup \left\{ \infty\right\},

\mathbf{Z^{*}}\left [ i \right ]= \left \{ a+bi : a,b \in \mathbf{Z}, 0\leq a\leq 1, b\geq \left ( a-a^{2} \right )^{1/2} \right \}\cup \left \{ \infty \right \}.

The set \mathbf{Z}[i] , \mathbf{Z}^{*}[i] and their subdivisions into

\mathbf{Z}[i] = \mathbf{V_1}\cup \mathbf{V_2}\cup \mathbf{V_3}\cup \mathbf{E_1}\cup \mathbf{E_2}\cup \mathbf{E_3}\cup \mathbf{C}, and \mathbf{Z}^{*}[i] = \mathbf{V^{*}_1}\cup \mathbf{V^{*}_2}\cup \mathbf{V^{*}_3}\cup\mathbf{C^{*}}.

are shown in Fig. 1, Fig. 1*, respectively.

Figure 1, Figure 1*. Every boundary point occurring in Fig. 1 is properly equivalent to a real number, and that every boundary point occurring in Fig. 1* is improperly equivalent to a real number. [1]

(Note : all regions are bounded by straight lines or circles with radii 1/2 (or part thereof), and are supposed to be closed; \infty \in \mathbf{V_1},\mathbf{E_2},\mathbf{E_3},\mathbf{V^{*}_1}.)

The 8 matrices v_1, v_2, v_3, e_1, e_2, e_3, c, s are defined  as follows :

v_1= \begin{pmatrix}  1 & i\\  0 & 1  \end{pmatrix},  v_2= \begin{pmatrix}  1 & 0\\  -i & 1  \end{pmatrix},  v_3= \begin{pmatrix}  1-i & i\\  -i & 1+i  \end{pmatrix},

e_1= \begin{pmatrix}1 & 0\\ 1-i & i\end{pmatrix}, e_2= \begin{pmatrix}1 & -1+i\\ 0 & i \end{pmatrix}, e_3= \begin{pmatrix}i & 0\\ 0 & 1\end{pmatrix},

c = \begin{pmatrix}1 & -1+i\\ 1-i & i\end{pmatrix}, s = \begin{pmatrix}0 & -1\\ 1 & -1\end{pmatrix}.

For any invertible matrix M,

M= \begin{pmatrix}p & q\\ r & s\end{pmatrix}

with elements in C, let m denote the corresponding homographic map :

m:z \mapsto \frac{pz+q}{rz+s}

And \chi :z \mapsto \bar{z} denotes complex conjugation of C.

A homographic map m:z \mapsto (pz+q)/(rz+s) is called unimodular if p,q,r,s\in \mathbf{Z}\left [ i \right ] (the ring of Gaussian integers) and \mathrm{det}~m= ps-qr \in \mathfrak{U}= \left \{ \pm 1,\pm i \right \}. Notice that the determinant of a unimodular map is defined only as an element in the quotient group \mathfrak{U}/\mathfrak{U}^{2} consisting of the two elements \left \{ \pm 1 \right \}, \left \{ \pm i \right \}. A unimodular map m is called properly unimoduIar if  \mathrm{det}~m= \left \{ \pm 1 \right \} and improperly unimodular if \mathrm{det}~m= \left \{ \pm i \right \}.

An immediate consequence of these definitions is that every boundary point occurring in Fig. 1 is properly equivalent to a real number, and that every boundary point occurring in Fig. 1* is improperly equivalent to a real number.

A complex number, for example \mathrm{e}^i can be represented as a series of these matrices: c ~c ~v_3 ~c ~c ~v_3 ~v_3 ~v_3 ~c ~c ~v_3 ~v_3 ~v_3 ~v_3 ~v_3 ~c ~c and so on.  Multiplying them together would yield :

\begin{pmatrix}  1541-1037i & -40+1857i\\  -40-1857i & 1541+1037i  \end{pmatrix}

If the top and bottom are added up, the fraction \left(1501 + 820 i \right)/\left(1501 - 820 i\right) can be made, which approximately equals 0.5403023380 + 0.8414709642 i. e^i starts off 0.5403023059 + 0.8414709848 i.

The Asmus Schmidt method uses partitions to divide the complex plane into regions, then subregions, and so on, and in this way, each point is encoded by the sequence of regions to which it belongs. In the first generation :

Figure 2.  The first generation of Asmus Schmidt’s complex continued fraction method. (picture credite to Ed Pegg Jr., courtesy of MAA)

After that, each triangular region is divided into 4 parts, by adding a circle. In addition, each circle is divided into 8 regions, by adding 3 tangent circles.

Figure 3.  The third generation of Asmus Schmidt’s complex continued fraction method. (picture credite to Ed Pegg Jr., courtesy of MAA)

Here is a portion of fourth generation :

Figure 4.  The fourth generation of Asmus Schmidt’s complex continued fraction method. (picture credite to Ed Pegg Jr., courtesy of MAA)

Another interesting visualization on further successive generation was done by Mathematica software[7] in which seven matrices are used. Each use of these matrices divides the plane into seven parts as shown in the Figure 5.

Figure 5.  Sucessive generations of Asmus Schmidt’s complex continued fraction method (picture credite to Doug Hensley, at Wolfram Library Archive).

References :

[1] Asmus L. Schmidt, Diophantine Approximation of Complex Numbers, University of Copenhagen, Denmark.

[2] Robert M. Corless, Continued Fractions and Chaos, Dept. Applied Math, University of Western Ontario, Canada.

[3] Ed Pegg Jr., Gaussian Numbers, editorial Math Games, March 15, 2004, MAA.

[4] MathWorld, Continued Fractions, Pi Continued Fraction.

[5] W. F. Hammond, Continued Fractions and the Euclidean Algorithm, Department of Mathematics and Statistics, University at Albany.

[6] Wikipedia, Continued Fraction, Generalized continued fraction, Gaussian integer.

[7] Wolfram Library Archive, Complex Continued Fractions, picture download : PDF document.

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