Editing Duodecimal (section)

== Comparison to other number systems ==
:''In this section, numerals are in decimal. For example, "10" means 9+1, and "12" means 9+3.''

The Dozenal Society of America argues that if a base is too small, significantly longer expansions are needed for numbers; if a base is too large, one must memorise a large multiplication table to perform arithmetic. Thus, it presumes that "a number base will need to be between about 7 or 8 through about 16, possibly including 18 and 20".<ref name="dsafaq" />

The number 12 has six factors, which are 1, 2, 3, 4, 6, and 12, of which 2 and 3 are [[prime number|prime]]. It is the smallest number to have six factors, the largest number to have at least half of the numbers below it as divisors, and is only slightly larger than 10. (The numbers 18 and 20 also have six factors but are much larger.) Ten, in contrast, only has four factors, which are 1, 2, 5, and 10, of which 2 and 5 are prime.<ref name="dsafaq" /> Six shares the prime factors 2 and 3 with twelve; however, like ten, six only has four factors (1, 2, 3, and 6) instead of six. Its corresponding base, [[senary]], is below the DSA's stated threshold.

Eight and sixteen only have 2 as a prime factor. Therefore, in [[octal]] and [[hexadecimal]], the only [[Repeating decimal|terminating fractions]] are those whose [[denominator]] is a [[power of two]].

Thirty is the smallest number that has three different prime factors (2, 3, and 5, the first three primes), and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30). [[Sexagesimal]] was actually used by the ancient [[Sumer]]ians and [[Babylonia]]ns, among others; its base, sixty, adds the four convenient factors 4, 12, 20, and 60 to 30 but no new prime factors. The smallest number that has four different prime factors is 210; the pattern follows the [[primorial]]s. However, these numbers are quite large to use as bases, and are far beyond the DSA's stated threshold.

In all base systems, there are similarities to the representation of multiples of numbers that are one less than or one more than the base.{{Clear}}''In the following multiplication table, numerals are written in duodecimal. For example, "10" means twelve, and "12" means fourteen.''<!-- the {{-}} template keeps the multiplication table from squeezing the heading for the next section-->

{| class="wikitable" style="text-align: right;"
|+ Duodecimal multiplication table
!style="width:7.69%"|×
!style="text-align: right; width:7.69%"|1
!style="text-align: right; width:7.69%"|2
!style="text-align: right; width:7.69%"|3
!style="text-align: right; width:7.69%"|4
!style="text-align: right; width:7.69%"|5
!style="text-align: right; width:7.69%"|6
!style="text-align: right; width:7.69%"|7
!style="text-align: right; width:7.69%"|8
!style="text-align: right; width:7.69%"|9
!style="text-align: right; width:7.69%"|{{d2}}
!style="text-align: right; width:7.69%"|{{d3}}
!style="text-align: right; width:7.69%"|10
|-
!style="text-align: right; width:7.69%"|1
|1||2||3||4||5||6||7||8||9||{{d2}}||{{d3}}||10
|-
!style="text-align: right;"|2
|2||4||6||8||{{d2}}||10||12||14||16||18||1{{d2}}||20
|-
!style="text-align: right;|3
|3||6||9||10||13||16||19||20||23||26||29||30
|-
!style="text-align: right;"|4
|4||8||10||14||18||20||24||28||30||34||38||40
|-
!style="text-align: right;"|5
|5||{{d2}}||13||18||21||26||2{{d3}}||34||39||42||47||50
|-
!style="text-align: right;"|6
|6||10||16||20||26||30||36||40||46||50||56||60
|-
!style="text-align: right;"|7
|7||12||19||24||2{{d3}}||36||41||48||53||5{{d2}}||65||70
|-
!style="text-align: right;"|8
|8||14||20||28||34||40||48||54||60||68||74||80
|-
!style="text-align: right;"|9
|9||16||23||30||39||46||53||60||69||76||83||90
|-
!style="text-align: right;"|{{d2}}
|{{d2}}||18||26||34||42||50||5{{d2}}||68||76||84||92||A0
|-
!style="text-align: right;"|{{d3}}
|{{d3}}||1{{d2}}||29||38||47||56||65||74||83||92||{{d2}}1||B0
|-
!style="text-align: right;"|10
|10||20||30||40||50||60||70||80||90||A0||B0||100
|}