Editing Hexadecimal (section)

==Real numbers==

=== Rational numbers ===
As with other numeral systems, the hexadecimal system can be used to represent [[rational number]]s, although [[repeating decimal|repeating expansions]] are common since sixteen (10<sub>16</sub>) has only a single prime factor: two.

For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system. Thus, whether dividing one by two for [[binary numeral system|binary]] or dividing one by sixteen for hexadecimal, both of these fractions are written as <code>0.1</code>. Because the radix 16 is a [[square number|perfect square]] (4<sup>2</sup>), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no [[cyclic number]]s (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a [[prime factor]] not found in the radix; thus, when using hexadecimal notation, all fractions with denominators that are not a [[power of two]] result in an infinite string of recurring digits (such as thirds and fifths). This makes hexadecimal (and binary) less convenient than [[decimal]] for representing rational numbers since a larger proportion lies outside its range of finite representation.

All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, [[duodecimal]] and [[sexagesimal]]: that is, any hexadecimal number with a finite number of digits also has a finite number of digits when expressed in those other bases. Conversely, only a fraction of those finitely representable in the latter bases are finitely representable in hexadecimal. For example, decimal 0.1 corresponds to the infinite recurring representation 0.1{{overline|9}} in hexadecimal. However, hexadecimal is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator. For example, 0.0625<sub>10</sub> (one-sixteenth) is equivalent to 0.1<sub>16</sub>, 0.09<sub>12</sub>, and 0;3,45<sub>60</sub>.

{|class="wikitable"
! rowspan=2 style="vertical-align:bottom;" | n
! colspan="3" | Decimal<br />Prime factors of: base, b = 10: {{color|#920000|2}}, {{color|#920000|5}};<br />b − 1 = 9: {{color|#000092|3}};<br />b + 1 = 11: {{color|#004900|11}}
! colspan="3" | Hexadecimal<br />Prime factors of: base, b = 16{{sub|10}} = 10: {{color|#920000|2}}; b − 1 = 15{{sub|10}} = F: {{color|#000092|3, 5}}; b + 1 = 17{{sub|10}} = 11: {{color|#004900|11}}
|-
! Reciprocal
! Prime factors
! Positional representation<br />(decimal)
! Positional representation<br />(hexadecimal)
! Prime factors
! Reciprocal
|-
| 2
| align="center" | 1/2
| align="center" | {{color|#920000|'''2'''}}
| '''0.5'''
| '''0.8'''
| align="center" | {{color|#920000|'''2'''}}
| align="center" | 1/2
|-
| 3
| align="center" | 1/3
| align="center" | {{color|#000092|'''3'''}}
| bgcolor=#e4e4e4 | '''0.'''3333... = '''0.'''{{overline|3}}
| bgcolor="#e4e4e4" | '''0.'''5555... = '''0.'''{{overline|5}}
| align="center" | {{color|#000092|'''3'''}}
| align="center" | 1/3
|-
| 4
| align="center" | 1/4
| align="center" | {{color|#920000|'''2'''}}
| '''0.25'''
| '''0.4'''
| align="center" | {{color|#920000|'''2'''}}
| align="center" | 1/4
|-
| 5
| align="center" | 1/5
| align="center" | {{color|#920000|'''5'''}}
| '''0.2'''
| bgcolor="#e4e4e4" | '''0.'''{{overline|3}}
| align="center" | {{color|#000092|'''5'''}}
| align="center" | 1/5
|-
| 6
| align="center" | 1/6
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| bgcolor=#e4e4e4 | '''0.1'''{{overline|6}}
| bgcolor="#e4e4e4" | '''0.2'''{{overline|A}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| align="center" | 1/6
|-
| 7
| align="center" | 1/7
| align="center" | '''7'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|142857}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|249}}
| align="center" | '''7'''
| align="center" | 1/7
|-
| 8
| align="center" | 1/8
| align="center" | {{color|#920000|'''2'''}}
| '''0.125'''
| '''0.2'''
| align="center" | {{color|#920000|'''2'''}}
| align="center" | 1/8
|-
| 9
| align="center" | 1/9
| align="center" | {{color|#000092|'''3'''}}
| bgcolor=#e4e4e4 | '''0.'''{{overline|1}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|1C7}}
| align="center" | {{color|#000092|'''3'''}}
| align="center" | 1/9
|-
| 10
| align="center" | 1/10
| align="center" | {{color|#920000|'''2'''}}, {{color|#920000|'''5'''}}
| '''0.1'''
| bgcolor="#e4e4e4" | '''0.1'''{{overline|9}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''5'''}}
| align="center" | 1/A
|-
| 11
| align="center" | 1/11
| align="center" | {{color|#004900|'''11'''}}
| bgcolor=#e4e4e4 | '''0.'''{{overline|09}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|1745D}}
| align="center" | '''B'''
| align="center" | 1/B
|-
| 12
| align="center" | 1/12
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| bgcolor=#e4e4e4 | '''0.08'''{{overline|3}}
| bgcolor="#e4e4e4" | '''0.1'''{{overline|5}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| align="center" | 1/C
|-
| 13
| align="center" | 1/13
| align="center" | '''13'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|076923}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|13B}}
| align="center" | '''D'''
| align="center" | 1/D
|-
| 14
| align="center" | 1/14
| align="center" | {{color|#920000|'''2'''}}, '''7'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|714285}}
| bgcolor="#e4e4e4" | '''0.1'''{{overline|249}}
| align="center" | {{color|#920000|'''2'''}}, '''7'''
| align="center" | 1/E
|-
| 15
| align="center" | 1/15
| align="center" | {{color|#000092|'''3'''}}, {{color|#920000|'''5'''}}
| bgcolor=#e4e4e4 | '''0.0'''{{overline|6}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|1}}
| align="center" | {{color|#000092|'''3'''}}, {{color|#000092|'''5'''}}
| align="center" | 1/F
|-
| 16
| align="center" | 1/16
| align="center" | {{color|#920000|'''2'''}}
| '''0.0625'''
| '''0.1'''
| align="center" | {{color|#920000|'''2'''}}
| align="center" | 1/10
|-
| 17
| align="center" | 1/17
| align="center" | '''17'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|0588235294117647}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|0F}}
| align="center" | {{color|#004900|'''11'''}}
| align="center" | 1/11
|-
| 18
| align="center" | 1/18
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| bgcolor=#e4e4e4 | '''0.0'''{{overline|5}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|E38}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| align="center" | 1/12
|-
| 19
| align="center" | 1/19
| align="center" | '''19'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|052631578947368421}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|0D79435E5}}
| align="center" | '''13'''
| align="center" | 1/13
|-
| 20
| align="center" | 1/20
| align="center" | {{color|#920000|'''2'''}}, {{color|#920000|'''5'''}}
| '''0.05'''
| bgcolor="#e4e4e4" | '''0.0'''{{overline|C}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''5'''}}
| align="center" | 1/14
|-
| 21
| align="center" | 1/21
| align="center" | {{color|#000092|'''3'''}}, '''7'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|047619}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|0C3}}
| align="center" | {{color|#000092|'''3'''}}, '''7'''
| align="center" | 1/15
|-
| 22
| align="center" | 1/22
| align="center" | {{color|#920000|'''2'''}}, {{color|#004900|'''11'''}}
| bgcolor=#e4e4e4 | '''0.0'''{{overline|45}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|BA2E8}}
| align="center" | {{color|#920000|'''2'''}}, '''B'''
| align="center" | 1/16
|-
| 23
| align="center" | 1/23
| align="center" | '''23'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|0434782608695652173913}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|0B21642C859}}
| align="center" | '''17'''
| align="center" | 1/17
|-
| 24
| align="center" | 1/24
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| bgcolor=#e4e4e4 | '''0.041'''{{overline|6}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|A}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| align="center" | 1/18
|-
| 25
| align="center" | 1/25
| align="center" | {{color|#920000|'''5'''}}
| '''0.04'''
| bgcolor="#e4e4e4" | '''0.'''{{overline|0A3D7}}
| align="center" | {{color|#000092|'''5'''}}
| align="center" | 1/19
|-
| 26
| align="center" | 1/26
| align="center" | {{color|#920000|'''2'''}}, '''13'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|384615}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|9D8}}
| align="center" | {{color|#920000|'''2'''}}, '''D'''
| align="center" | 1/1A
|-
| 27
| align="center" | 1/27
| align="center" | {{color|#000092|'''3'''}}
| bgcolor=#e4e4e4 | '''0.'''{{overline|037}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|097B425ED}}
| align="center" | {{color|#000092|'''3'''}}
| align="center" | 1/1B
|-
| 28
| align="center" | 1/28
| align="center" | {{color|#920000|'''2'''}}, '''7'''
| bgcolor=#e4e4e4 | '''0.03'''{{overline|571428}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|924}}
| align="center" | {{color|#920000|'''2'''}}, '''7'''
| align="center" | 1/1C
|-
| 29
| align="center" | 1/29
| align="center" | '''29'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|0344827586206896551724137931}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|08D3DCB}}
| align="center" | '''1D'''
| align="center" | 1/1D
|-
| 30
| align="center" | 1/30
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}, {{color|#920000|'''5'''}}
| bgcolor=#e4e4e4 | '''0.0'''{{overline|3}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|8}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}, {{color|#000092|'''5'''}}
| align="center" | 1/1E
|-
| 31
| align="center" | 1/31
| align="center" | '''31'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|032258064516129}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|08421}}
| align="center" | '''1F'''
| align="center" | 1/1F
|-
| 32
| align="center" | 1/32
| align="center" | {{color|#920000|'''2'''}}
| '''0.03125'''
| '''0.08'''
| align="center" | {{color|#920000|'''2'''}}
| align="center" | 1/20
|-
| 33
| align="center" | 1/33
| align="center" | {{color|#000092|'''3'''}}, {{color|#004900|'''11'''}}
| bgcolor=#e4e4e4 | '''0.'''{{overline|03}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|07C1F}}
| align="center" | {{color|#000092|'''3'''}}, '''B'''
| align="center" | 1/21
|-
| 34
| align="center" | 1/34
| align="center" | {{color|#920000|'''2'''}}, '''17'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|2941176470588235}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|78}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#004900|'''11'''}}
| align="center" | 1/22
|-
| 35
| align="center" | 1/35
| align="center" | {{color|#920000|'''5'''}}, '''7'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|285714}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|075}}
| align="center" | {{color|#000092|'''5'''}}, '''7'''
| align="center" | 1/23
|-
| 36
| align="center" | 1/36
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| bgcolor=#e4e4e4 | '''0.02'''{{overline|7}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|71C}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| align="center" | 1/24
|-
| 37
| align="center" | 1/37
| align="center" | '''37'''
| bgcolor=#e4e4e4 | '''0.'''{{overline|027}}
| bgcolor="#e4e4e4" | '''0.'''{{overline|06EB3E453}}
| align="center" | '''25'''
| align="center" | 1/25
|-
| 38
| align="center" | 1/38
| align="center" | {{color|#920000|'''2'''}}, '''19'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|2631578947368421}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|435E50D79435E4AC62B4}}
| align="center" | {{color|#920000|'''2'''}}, '''13'''
| align="center" | 1/26
|-
| 39
| align="center" | 1/39
| align="center" | {{color|#000092|'''3'''}}, '''13'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|256410}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|69}}
| align="center" | {{color|#000092|'''3'''}}, '''D'''
| align="center" | 1/27
|-
| 40
| align="center" | 1/40
| align="center" | {{color|#920000|'''2'''}}, {{color|#920000|'''5'''}}
| '''0.025'''
| bgcolor="#e4e4e4" | '''0.0'''{{overline|6}}
| align="center" | {{color|#000092|'''2'''}}, {{color|#000092|'''5'''}}
| align="center" | 1/28
|-
| 41
| align="center" | 1/41
| align="center" | '''41'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|2439}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|63E9538D283B5B62FB8}}
| align="center" | '''29'''
| align="center" | 1/29
|-
| 42
| align="center" | 1/42
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}, '''7'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|238095}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|618}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}, '''7'''
| align="center" | 1/2A
|-
| 43
| align="center" | 1/43
| align="center" | '''43'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|23255813953488372093}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|5F417D}}
| align="center" | '''2B'''
| align="center" | 1/2B
|-
| 44
| align="center" | 1/44
| align="center" | {{color|#920000|'''2'''}}, {{color|#004900|'''11'''}}
| bgcolor=#e4e4e4 | '''0.02'''{{overline|27}}
| bgcolor="#e4e4e4" | '''0.05'''{{overline|D1745}}
| align="center" | {{color|#920000|'''2'''}}, '''B'''
| align="center" | 1/2C
|-
| 45
| align="center" | 1/45
| align="center" | {{color|#000092|'''3'''}}, {{color|#920000|'''5'''}}
| bgcolor=#e4e4e4 | '''0.0'''{{overline|2}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|5B}}
| align="center" | {{color|#000092|'''3'''}}, '''5'''
| align="center" | 1/2D
|-
| 46
| align="center" | 1/46
| align="center" | {{color|#920000|'''2'''}}, '''23'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|2173913043478765869567}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|590B21642C9C4EF44A9}}
| align="center" | {{color|#920000|'''2'''}}, '''17'''
| align="center" | 1/2E
|-
| 47
| align="center" | 1/47
| align="center" | '''47'''
| bgcolor=#e4e4e4 | '''0.0'''{{overline|212765957446808510638297872340425531914893617}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|572620AE4C415C9882B931}}
| align="center" | '''2F'''
| align="center" | 1/2F
|-
| 48
| align="center" | 1/48
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| bgcolor=#e4e4e4 | '''0.0208'''{{overline|3}}
| bgcolor="#e4e4e4" | '''0.0'''{{overline|5}}
| align="center" | {{color|#920000|'''2'''}}, {{color|#000092|'''3'''}}
| align="center" | 1/30
|}

===Irrational numbers===
The table below gives the expansions of some common [[irrational number]]s in decimal and hexadecimal.
{| class="wikitable"
! rowspan=2 | Number
! colspan=2 | Positional representation
|-
! Decimal
! Hexadecimal
|-
| [[Square root of 2|{{sqrt|2}}]] (the length of the [[diagonal]] of a unit [[Square (geometry)|square]])
| {{val|1.414213562373095048}}...
| 1.6A09E667F3BCD...
|-
| [[Square root of 3|{{sqrt|3}}]] (the length of the diagonal of a unit [[cube]])
| {{val|1.732050807568877293}}...
| 1.BB67AE8584CAA...
|-
| [[Square root of 5|{{sqrt|5}}]] (the length of the [[diagonal]] of a 1×2 [[rectangle]])
| {{val|2.236067977499789696}}...
| 2.3C6EF372FE95...
|-
| {{mvar|[[Golden ratio|φ]]}} (phi, the [[golden ratio]] = {{math|(1+{{radical|5}})/2}})
| {{val|1.618033988749894848}}...
| 1.9E3779B97F4A...
|-
| {{mvar|[[Pi|π]]}} (pi, the ratio of [[circumference]] to [[diameter]] of a circle)
| {{val|3.141592653589793238462643}}<br />{{val|383279502884197169399375105}}...
| 3.243F6A8885A308D313198A2E0<br />3707344A4093822299F31D008...
|-
| {{mvar|[[E (mathematical constant)|e]]}} (the base of the [[natural logarithm]])
| {{val|2.718281828459045235}}...
| 2.B7E151628AED2A6B...
|-
| {{mvar|[[Thue–Morse constant|τ]]}} (the [[Thue–Morse constant]])
| {{val|0.412454033640107597}}...
| 0.6996 9669 9669 6996...
|-
| {{mvar|[[Euler-Mascheroni constant|γ]]}} (the limiting difference between the [[harmonic series (mathematics)|harmonic series]] and the natural logarithm)
| {{val|0.577215664901532860}}...
| 0.93C467E37DB0C7A4D1B...
|}

===Powers===
Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below.
{| class="wikitable"
! 2<sup>''x''</sup> !! Value !! Value (Decimal)
|-
| 2<sup>0</sup> || style="text-align:right;" | 1 || style="text-align:right;" | 1
|-
| 2<sup>1</sup> || style="text-align:right;" | 2 || style="text-align:right;" | 2
|-
| 2<sup>2</sup> || style="text-align:right;" | 4 || style="text-align:right;" | 4
|-
| 2<sup>3</sup> || style="text-align:right;" | 8 || style="text-align:right;" | 8
|-
| 2<sup>4</sup> || style="text-align:right;" | 10<sub>hex</sub> || style="text-align:right;" | 16<sub>dec</sub>
|-
| 2<sup>5</sup> || style="text-align:right;" | 20<sub>hex</sub> || style="text-align:right;" | 32<sub>dec</sub>
|-
| 2<sup>6</sup> || style="text-align:right;" | 40<sub>hex</sub> || style="text-align:right;" | 64<sub>dec</sub>
|-
| 2<sup>7</sup> || style="text-align:right;" | 80<sub>hex</sub> || style="text-align:right;" | 128<sub>dec</sub>
|-
| 2<sup>8</sup> || style="text-align:right;" | 100<sub>hex</sub> || style="text-align:right;" | 256<sub>dec</sub>
|-
| 2<sup>9</sup> || style="text-align:right;" | 200<sub>hex</sub> || style="text-align:right;" | 512<sub>dec</sub>
|-
| 2<sup>A</sup> (2{{sup|10{{sub|dec}}}}) || style="text-align:right;" | 400<sub>hex</sub> || style="text-align:right;" | 1,024<sub>dec</sub>
|-
| 2<sup>B</sup> (2{{sup|11{{sub|dec}}}}) || style="text-align:right;" | 800<sub>hex</sub> || style="text-align:right;" | 2,048<sub>dec</sub>
|-
| 2<sup>C</sup> (2{{sup|12{{sub|dec}}}}) || style="text-align:right;" | 1,000<sub>hex</sub> || style="text-align:right;" | 4,096<sub>dec</sub>
|-
| 2<sup>D</sup> (2{{sup|13{{sub|dec}}}}) || style="text-align:right;" | 2,000<sub>hex</sub> || style="text-align:right;" | 8,192<sub>dec</sub>
|-
| 2<sup>E</sup> (2{{sup|14{{sub|dec}}}}) || style="text-align:right;" | 4,000<sub>hex</sub> || style="text-align:right;" | 16,384<sub>dec</sub>
|-
| 2<sup>F</sup> (2{{sup|15{{sub|dec}}}}) || style="text-align:right;" | 8,000<sub>hex</sub> || style="text-align:right;" | 32,768<sub>dec</sub>
|-
| 2<sup>10</sup> (2{{sup|16{{sub|dec}}}}) || style="text-align:right;" | 10,000<sub>hex</sub> || style="text-align:right;" | 65,536<sub>dec</sub>
|}