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Quaternary numeral system
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==Fractions== Due to having only factors of two, many quaternary fractions have repeating digits, although these tend to be fairly simple: {|class="wikitable" | colspan="3" align="center" | Decimal base<br><SMALL>Prime factors of the base: <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span></SMALL><br><SMALL>Prime factors of one below the base: <span style="color:Blue">'''3'''</span></SMALL><br><SMALL>Prime factors of one above the base: <span style="color:Magenta">'''11'''</span></SMALL><br><SMALL>Other prime factors: <span style="color:Red">'''7 13 17 19 23 29 31'''</span></SMALL> | colspan="3" align="center" | '''Quaternary base'''<br><SMALL>Prime factors of the base: <span style="color:Green">'''2'''</span></SMALL><br><SMALL>Prime factors of one below the base: <span style="color:Blue">'''3'''</span></SMALL><br><SMALL>Prime factors of one above the base: <span style="color:Magenta">'''5 (=11<sub>4</sub>)'''</span></SMALL><br><SMALL>Other prime factors: <span style="color:Red">'''13 23 31 101 103 113 131 133'''</span></SMALL> |- | align="center" | Fraction | align="center" | <SMALL>Prime factors of<br>the denominator</SMALL> | align="center" | Positional<br>representation | align="center" | Positional<br>representation | align="center" | <SMALL>Prime factors of<br>the denominator</SMALL> | align="center" | Fraction |- | align="center" | {{sfrac|1|2}} | align="center" | <span style="color:Green">'''2'''</span> | '''0.5''' | '''0.2''' | align="center" | <span style="color:Green">'''2'''</span> | align="center" | {{sfrac|1|2}} |- | align="center" | {{sfrac|1|3}} | align="center" | <span style="color:Blue">'''3'''</span> | bgcolor=#c0c0c0 | '''0.'''3333... = '''0.'''{{overline|3}} | bgcolor=#c0c0c0 | '''0.'''1111... = '''0.'''{{overline|1}} | align="center" | <span style="color:Blue">'''3'''</span> | align="center" | {{sfrac|1|3}} |- | align="center" | {{sfrac|1|4}} | align="center" | <span style="color:Green">'''2'''</span> | '''0.25''' | '''0.1''' | align="center" | <span style="color:Green">'''2'''</span> | align="center" | {{sfrac|1|10}} |- | align="center" | {{sfrac|1|5}} | align="center" | <span style="color:Green">'''5'''</span> | '''0.2''' | bgcolor=#c0c0c0 | '''0.'''{{overline|03}} | align="center" | <span style="color:Magenta">'''11'''</span> | align="center" | {{sfrac|1|11}} |- | align="center" | {{sfrac|1|6}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span> | bgcolor=#c0c0c0 | '''0.1'''{{overline|6}} | bgcolor=#c0c0c0 | '''0.0'''{{overline|2}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span> | align="center" | {{sfrac|1|12}} |- | align="center" | {{sfrac|1|7}} | align="center" | <span style="color:Red">'''7'''</span> | bgcolor=#c0c0c0 | '''0.'''{{overline|142857}} | bgcolor=#c0c0c0 | '''0.'''{{overline|021}} | align="center" | <span style="color:Red">'''13'''</span> | align="center" | {{sfrac|1|13}} |- | align="center" | {{sfrac|1|8}} | align="center" | <span style="color:Green">'''2'''</span> | '''0.125''' | '''0.02''' | align="center" | <span style="color:Green">'''2'''</span> | align="center" | {{sfrac|1|20}} |- | align="center" | {{sfrac|1|9}} | align="center" | <span style="color:Blue">'''3'''</span> | bgcolor=#c0c0c0 | '''0.'''{{overline|1}} | bgcolor=#c0c0c0 | '''0.'''{{overline|013}} | align="center" | <span style="color:Blue">'''3'''</span> | align="center" | {{sfrac|1|21}} |- | align="center" | {{sfrac|1|10}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span> | '''0.1''' | bgcolor=#c0c0c0 | '''0.0'''{{overline|12}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''11'''</span> | align="center" | {{sfrac|1|22}} |- | align="center" | {{sfrac|1|11}} | align="center" | <span style="color:Magenta">'''11'''</span> | bgcolor=#c0c0c0 | '''0.'''{{overline|09}} | bgcolor=#c0c0c0 | '''0.'''{{overline|01131}} | align="center" | <span style="color:Red">'''23'''</span> | align="center" | {{sfrac|1|23}} |- | align="center" | {{sfrac|1|12}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span> | bgcolor=#c0c0c0 | '''0.08'''{{overline|3}} | bgcolor=#c0c0c0 | '''0.0'''{{overline|1}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span> | align="center" | {{sfrac|1|30}} |- | align="center" | {{sfrac|1|13}} | align="center" | <span style="color:Red">'''13'''</span> | bgcolor=#c0c0c0 | '''0.'''{{overline|076923}} | bgcolor=#c0c0c0 | '''0.'''{{overline|010323}} | align="center" | <span style="color:Red">'''31'''</span> | align="center" | {{sfrac|1|31}} |- | align="center" | {{sfrac|1|14}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span> | bgcolor=#c0c0c0 | '''0.0'''{{overline|714285}} | bgcolor=#c0c0c0 | '''0.0'''{{overline|102}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''13'''</span> | align="center" | {{sfrac|1|32}} |- | align="center" | {{sfrac|1|15}} | align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Green">'''5'''</span> | bgcolor=#c0c0c0 | '''0.0'''{{overline|6}} | bgcolor=#c0c0c0 | '''0.'''{{overline|01}} | align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Magenta">'''11'''</span> | align="center" | {{sfrac|1|33}} |- | align="center" | {{sfrac|1|16}} | align="center" | <span style="color:Green">'''2'''</span> | '''0.0625''' | '''0.01''' | align="center" | <span style="color:Green">'''2'''</span> | align="center" | {{sfrac|1|100}} |- | align="center" | {{sfrac|1|17}} | align="center" | <span style="color:Red">'''17'''</span> | bgcolor=#c0c0c0 | '''0.'''{{overline|0588235294117647}} | bgcolor=#c0c0c0 | '''0.'''{{overline|0033}} | align="center" | <span style="color:Red">'''101'''</span> | align="center" | {{sfrac|1|101}} |- | align="center" | {{sfrac|1|18}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span> | bgcolor=#c0c0c0 | '''0.0'''{{overline|5}} | bgcolor=#c0c0c0 | '''0.0'''{{overline|032}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span> | align="center" | {{sfrac|1|102}} |- | align="center" | {{sfrac|1|19}} | align="center" | <span style="color:Red">'''19'''</span> | bgcolor=#c0c0c0 | '''0.'''{{overline|052631578947368421}} | bgcolor=#c0c0c0 | '''0.'''{{overline|003113211}} | align="center" | <span style="color:Red">'''103'''</span> | align="center" | {{sfrac|1|103}} |- | align="center" | {{sfrac|1|20}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span> |'''0.05''' | bgcolor=#c0c0c0 | '''0.0'''{{overline|03}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''11'''</span> | align="center" | {{sfrac|1|110}} |- | align="center" | {{sfrac|1|21}} | align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Red">'''7'''</span> | bgcolor=#c0c0c0 | '''0.'''{{overline|047619}} | bgcolor=#c0c0c0 | '''0.'''{{overline|003}} | align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Red">'''13'''</span> | align="center" | {{sfrac|1|111}} |- | align="center" | {{sfrac|1|22}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''11'''</span> | bgcolor=#c0c0c0 | '''0.0'''{{overline|45}} | bgcolor=#c0c0c0 | '''0.0'''{{overline|02322}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''23'''</span> | align="center" | {{sfrac|1|112}} |- | align="center" | {{sfrac|1|23}} | align="center" | <span style="color:Red">'''23'''</span> | bgcolor=#c0c0c0 | '''0.'''{{overline|0434782608695652173913}} | bgcolor=#c0c0c0 | '''0.'''{{overline|00230201121}} | align="center" | <span style="color:Red">'''113'''</span> | align="center" | {{sfrac|1|113}} |- | align="center" | {{sfrac|1|24}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span> | bgcolor=#c0c0c0 | '''0.041'''{{overline|6}} | bgcolor=#c0c0c0 | '''0.00'''{{overline|2}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span> | align="center" | {{sfrac|1|120}} |- | align="center" | {{sfrac|1|25}} | align="center" | <span style="color:Green">'''5'''</span> |'''0.04''' | bgcolor=#c0c0c0 | '''0.'''{{overline|0022033113}} | align="center" | <span style="color:Magenta">'''11'''</span> | align="center" | {{sfrac|1|121}} |- | align="center" | {{sfrac|1|26}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''13'''</span> | bgcolor=#c0c0c0 | '''0.0'''{{overline|384615}} | bgcolor=#c0c0c0 | '''0.0'''{{overline|021312}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''31'''</span> | align="center" | {{sfrac|1|122}} |- | align="center" | {{sfrac|1|27}} | align="center" | <span style="color:Blue">'''3'''</span> | bgcolor=#c0c0c0 | '''0.'''{{overline|037}} | bgcolor=#c0c0c0 | '''0.'''{{overline|002113231}} | align="center" | <span style="color:Blue">'''3'''</span> | align="center" | {{sfrac|1|123}} |- | align="center" | {{sfrac|1|28}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span> | bgcolor=#c0c0c0 | '''0.03'''{{overline|571428}} | bgcolor=#c0c0c0 | '''0.0'''{{overline|021}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''13'''</span> | align="center" | {{sfrac|1|130}} |- | align="center" | {{sfrac|1|29}} | align="center" | <span style="color:Red">'''29'''</span> | bgcolor=#c0c0c0 | '''0.'''{{overline|0344827586206896551724137931}} | bgcolor=#c0c0c0 | '''0.'''{{overline|00203103313023}} | align="center" | <span style="color:Red">'''131'''</span> | align="center" | {{sfrac|1|131}} |- | align="center" | {{sfrac|1|30}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>, <span style="color:Green">'''5'''</span> | bgcolor=#c0c0c0 | '''0.0'''{{overline|3}} | bgcolor=#c0c0c0 | '''0.0'''{{overline|02}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>, <span style="color:Magenta">'''11'''</span> | align="center" | {{sfrac|1|132}} |- | align="center" | {{sfrac|1|31}} | align="center" | <span style="color:Red">'''31'''</span> | bgcolor=#c0c0c0 | '''0.'''{{overline|032258064516129}} | bgcolor=#c0c0c0 | '''0.'''{{overline|00201}} | align="center" | <span style="color:Red">'''133'''</span> | align="center" | {{sfrac|1|133}} |- | align="center" | {{sfrac|1|32}} | align="center" | <span style="color:Green">'''2'''</span> |'''0.03125''' |'''0.002''' | align="center" | <span style="color:Green">'''2'''</span> | align="center" | {{sfrac|1|200}} |- | align="center" | {{sfrac|1|33}} | align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Magenta">'''11'''</span> | bgcolor=#c0c0c0 | '''0.'''{{overline|03}} | bgcolor=#c0c0c0 | '''0.'''{{overline|00133}} | align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Red">'''23'''</span> | align="center" | {{sfrac|1|201}} |- | align="center" | {{sfrac|1|34}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''17'''</span> | bgcolor=#c0c0c0 | '''0.0'''{{overline|2941176470588235}} | bgcolor=#c0c0c0 | '''0.0'''{{overline|0132}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''101'''</span> | align="center" | {{sfrac|1|202}} |- | align="center" | {{sfrac|1|35}} | align="center" | <span style="color:Green">'''5'''</span>, <span style="color:Red">'''7'''</span> | bgcolor=#c0c0c0 | '''0.0'''{{overline|285714}} | bgcolor=#c0c0c0 | '''0.'''{{overline|001311}} | align="center" | <span style="color:Magenta">'''11'''</span>, <span style="color:Red">'''13'''</span> | align="center" | {{sfrac|1|203}} |- | align="center" | {{sfrac|1|36}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span> | bgcolor=#c0c0c0 | '''0.02'''{{overline|7}} | bgcolor=#c0c0c0 | '''0.0'''{{overline|013}} | align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span> | align="center" | {{sfrac|1|210}} |}
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