Editing Golden ratio base (section)

==Representing rational numbers as golden ratio base numbers==

Every non-negative rational number can be represented as a recurring base-φ expansion, as can any non-negative element of the [[field (mathematics)|field]] '''Q'''[{{sqrt|5}}] = '''Q''' + {{sqrt|5}}'''Q''', the field generated by the [[rational number]]s and [[square root of 5|<math display=inline>\sqrt{5}</math>]]. Conversely any recurring (or terminating) base-φ expansion is a non-negative element of '''Q'''[{{sqrt|5}}].  For recurring decimals, the recurring part has been overlined:

*{{sfrac|1|2}} = 0.<span style="text-decoration:overline;>010</span><sub>φ</sub>
*{{sfrac|1|3}} = 0.<span style="text-decoration:overline;>00101000</span><sub>φ</sub>
*{{sfrac|1|4}} = 0.<span style="text-decoration:overline;>001000</span><sub>φ</sub>
*{{sfrac|1|5}} = 0.<span style="text-decoration:overline;>001001010100100100</span><sub>φ</sub>
*{{sfrac|1|10}} = 0.<span style="text-decoration:overline;>000010000100010100001010001010101000100101000001001000100000</span><sub>φ</sub>

The justification that a rational gives a recurring expansion is analogous to the equivalent proof for a base-''n'' numeration system (''n'' = 2,3,4,...).  Essentially in base-φ [[long division]] there are only a finite number of possible remainders, and so once there must be a recurring pattern.  For example, with {{sfrac|1|2}} = {{sfrac|1|10.01<sub>φ</sub>}} = {{sfrac|100<sub>φ</sub>|1001<sub>φ</sub>}} long division looks like this (note that base-φ subtraction may be hard to follow at first):
<pre>
                .0 1 0 0 1
         ________________________
 1 0 0 1 ) 1 0 0.0 0 0 0 0 0 0 0
             1 0 0 1                        trade: 10000 = 1100 = 1011
             -------                            so 10000 − 1001 = 1011 − 1001 = 10
                 1 0 0 0 0
                   1 0 0 1
                   -------
                       etc.
</pre>
The converse is also true, in that a number with a recurring base-φ; representation is an element of the field '''Q'''[{{sqrt|5}}].  This follows from the observation that a recurring representation with period k involves a [[geometric series]] with ratio φ<sup>−k</sup>, which will sum to an element of '''Q'''[{{sqrt|5}}].