Editing Octal (section)

==Real numbers==
===Fractions===
Due to having only factors of two, many octal 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" | '''Octal 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">'''7'''</span></SMALL><br><SMALL>Prime factors of one above the base: <span style="color:Magenta">'''3'''</span></SMALL><br><SMALL>Other Prime factors: <span style="color:Red">'''5 13 15 21 23 27 35 37'''</span></SMALL>
|-
| align="center" | Fraction
| align="center" | <SMALL>Prime factors<br>of the denominator</SMALL>
| align="center" | Positional representation
| align="center" | Positional representation
| align="center" | <SMALL>Prime factors<br>of the denominator</SMALL>
| align="center" | Fraction
|-
| align="center" | 1/2
| align="center" | <span style="color:Green">'''2'''</span>
| '''0.5'''
| '''0.4'''
| align="center" | <span style="color:Green">'''2'''</span>
| align="center" | 1/2
|-
| align="center" | 1/3
| align="center" | <span style="color:Blue">'''3'''</span>
| bgcolor=#c0c0c0 | '''0.'''3333... = '''0.'''{{overline|3}}
| bgcolor=#c0c0c0 | '''0.'''2525... = '''0.'''{{overline|25}}
| align="center" | <span style="color:Magenta">'''3'''</span>
| align="center" | 1/3
|-
| align="center" | 1/4
| align="center" | <span style="color:Green">'''2'''</span>
| '''0.25'''
| '''0.2'''
| align="center" | <span style="color:Green">'''2'''</span>
| align="center" | 1/4
|-
| align="center" | 1/5
| align="center" | <span style="color:Green">'''5'''</span>
| '''0.2'''
| bgcolor=#c0c0c0 | '''0.'''{{overline|1463}}
| align="center" | <span style="color:Red">'''5'''</span>
| align="center" | 1/5
|-
| align="center" | 1/6
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| bgcolor=#c0c0c0 | '''0.1'''{{overline|6}}
| bgcolor=#c0c0c0 | '''0.1'''{{overline|25}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''3'''</span>
| align="center" | 1/6
|-
| align="center" | 1/7
| align="center" | <span style="color:Red">'''7'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|142857}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|1}}
| align="center" | <span style="color:Blue">'''7'''</span>
| align="center" | 1/7
|-
| align="center" | 1/8
| align="center" | <span style="color:Green">'''2'''</span>
| '''0.125'''
| '''0.1'''
| align="center" | <span style="color:Green">'''2'''</span>
| align="center" | 1/10
|-
| align="center" | 1/9
| align="center" | <span style="color:Blue">'''3'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|1}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|07}}
| align="center" | <span style="color:Magenta">'''3'''</span>
| align="center" | 1/11
|-
| align="center" | 1/10
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>
| '''0.1'''
| bgcolor=#c0c0c0 | '''0.0'''{{overline|6314}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''5'''</span>
| align="center" | 1/12
|-
| align="center" | 1/11
| align="center" | <span style="color:Magenta">'''11'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|09}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|0564272135}}
| align="center" | <span style="color:Red">'''13'''</span>
| align="center" | 1/13
|-
| align="center" | 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|52}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''3'''</span>
| align="center" | 1/14
|-
| align="center" | 1/13
| align="center" | <span style="color:Red">'''13'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|076923}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|0473}}
| align="center" | <span style="color:Red">'''15'''</span>
| align="center" | 1/15
|-
| align="center" | 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|4}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''7'''</span>
| align="center" | 1/16
|-
| align="center" | 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|0421}}
| align="center" | <span style="color:Magenta">'''3'''</span>, <span style="color:Red">'''5'''</span>
| align="center" | 1/17
|-
| align="center" | 1/16
| align="center" | <span style="color:Green">'''2'''</span>
| '''0.0625'''
| '''0.04'''
| align="center" | <span style="color:Green">'''2'''</span>
| align="center" | 1/20
|-
| align="center" | 1/17
| align="center" | <span style="color:Red">'''17'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|0588235294117647}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|03607417}}
| align="center" | <span style="color:Red">'''21'''</span>
| align="center" | 1/21
|-
| align="center" | 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|34}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''3'''</span>
| align="center" | 1/22
|-
| align="center" | 1/19
| align="center" | <span style="color:Red">'''19'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|052631578947368421}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|032745}}
| align="center" | <span style="color:Red">'''23'''</span>
| align="center" | 1/23
|-
| align="center" | 1/20
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>
| '''0.05'''
| bgcolor=#c0c0c0 | '''0.0'''{{overline|3146}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''5'''</span>
| align="center" | 1/24
|-
| align="center" | 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|03}}
| align="center" | <span style="color:Magenta">'''3'''</span>, <span style="color:Blue">'''7'''</span>
| align="center" | 1/25
|-
| align="center" | 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|2721350564}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''13'''</span>
| align="center" | 1/26
|-
| align="center" | 1/23
| align="center" | <span style="color:Red">'''23'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|0434782608695652173913}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|02620544131}}
| align="center" | <span style="color:Red">'''27'''</span>
| align="center" | 1/27
|-
| align="center" | 1/24
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| bgcolor=#c0c0c0 | '''0.041'''{{overline|6}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|25}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''3'''</span>
| align="center" | 1/30
|-
| align="center" | 1/25
| align="center" | <span style="color:Green">'''5'''</span>
| '''0.04'''
| bgcolor=#c0c0c0 | '''0.'''{{overline|02436560507534121727}}
| align="center" | <span style="color:Red">'''5'''</span>
| align="center" | 1/31
|-
| align="center" | 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|2354}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''15'''</span>
| align="center" | 1/32
|-
| align="center" | 1/27
| align="center" | <span style="color:Blue">'''3'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|037}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|022755}}
| align="center" | <span style="color:Magenta">'''3'''</span>
| align="center" | 1/33
|-
| align="center" | 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|2}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''7'''</span>
| align="center" | 1/34
|-
| align="center" | 1/29
| align="center" | <span style="color:Red">'''29'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|0344827586206896551724137931}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|0215173454106475626043236713}}
| align="center" | <span style="color:Red">'''35'''</span>
| align="center" | 1/35
|-
| align="center" | 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|2104}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''3'''</span>, <span style="color:Red">'''5'''</span>
| align="center" | 1/36
|-
| align="center" | 1/31
| align="center" | <span style="color:Red">'''31'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|032258064516129}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|02041}}
| align="center" | <span style="color:Red">'''37'''</span>
| align="center" | 1/37
|-
| align="center" | 1/32
| align="center" | <span style="color:Green">'''2'''</span>
| '''0.03125'''
| '''0.02'''
| align="center" | <span style="color:Green">'''2'''</span>
| align="center" | 1/40
|}

===Irrational numbers===
The table below gives the expansions of some common [[irrational number]]s in decimal and octal.
{| class="wikitable"
! rowspan=2 | Number
! colspan=2 | Positional representation
|-
! Decimal
! Octal
|-
| [[Square root of 2|{{sqrt|2}}]] {{small|(the length of the [[diagonal]] of a unit [[Square (geometry)|square]])}}
| {{val|1.414213562373095048}}...
| 1.3240 4746 3177 1674...
|-
| [[Square root of 3|{{sqrt|3}}]] {{small|(the length of the diagonal of a unit [[cube]])}}
| {{val|1.732050807568877293}}...
| 1.5666 3656 4130 2312...
|-
| [[Square root of 5|{{sqrt|5}}]] {{small|(the length of the [[diagonal]] of a 1×2 [[rectangle]])}}
| {{val|2.236067977499789696}}...
| 2.1706 7363 3457 7224...
|-
| {{mvar|[[Golden ratio|φ]]}} {{small|1=(phi, the [[golden ratio]] = {{math|(1+{{radical|5}})/2}})}}
| {{val|1.618033988749894848}}...
| 1.4743 3571 5627 7512...
|-
| {{mvar|[[Pi|π]]}} {{small|(pi, the ratio of [[circumference]] to [[diameter]] of a circle)}}
| {{val|3.141592653589793238462643}}<br>{{val|383279502884197169399375105}}...
| 3.1103 7552 4210 2643...
|-
| {{mvar|[[E (mathematical constant)|e]]}} {{small|(the base of the [[natural logarithm]])}}
| {{val|2.718281828459045235}}...
| 2.5576 0521 3050 5355...
|}