View source for Diagonal continued fraction

A [[Continued fraction|continued fraction]]

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in which the sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314702.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314703.png" /> must satisfy the following conditions:

1) the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314705.png" /> are integers; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314706.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314707.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314708.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d0314709.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147010.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147011.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147012.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147014.png" /> for an infinite set of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147015.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147016.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147017.png" />;

4) the partial fractions of the continued fraction are all irreducible fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147021.png" /> is value of the continued fraction.

For each real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147022.png" /> there exists one and only one diagonal continued fraction with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147023.png" /> as its value; this fraction is periodic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147024.png" /> is a [[Quadratic irrationality|quadratic irrationality]].



====Comments====
After truncation and evaluation one obtains

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with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147028.png" />. These are the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147029.png" /> alluded to in condition . The continued fraction as described above for a real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147030.png" /> can be obtained by the nearest integer algorithm, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147035.png" />, etc., where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147036.png" /> denotes the nearest integer to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147037.png" />. It is also possible to use the entier function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147038.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147039.png" />. One then has the continued fraction algorithm which is more commonly used.

The adjective  "diagonal"  stems from the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147040.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031470/d03147041.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,   E.M. Wright,   "An introduction to the theory of numbers" , Oxford Univ. Press  (1979)  pp. Chapts. 5; 7; 8</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Perron,   "Die Lehre von den Kettenbrüchen" , '''I''' , Teubner  (1977)</TD></TR></table>