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What Makes a Good Base?
The following chart is an organization of fractions of 1/n, increasing n by 1 each time until n = 18 decimally. Note that I've only chosen even bases, since odd bases cannot be divisible by 2 without recurrence (ie. 0.2222). Choosing even bases is arguably the most important property in finding our base.
Furthermore, to represent our higher values, I've also chosen to use lower-case letters instead of upper-case, because:
- 8 & B
- D & 0
- G & 6 (although this one isn't so bad in most fonts)
- I and 1
...they all look similar.
Another problem arises when you reach letter L. We can't use lower-case L because it looks like 1 and the letter 'I' as in "igloo".
At base 18, we use the letter H, but not I; it becomes a problem beyond base 18.
It is for these reasons that I only show the bases from 6 - 18.
Here's a chart I made on Google sheets so you can see the elegance primes and composite numbers within the dozenal system.
I highly recommend you check it out if you want to see how primes appear in different bases:
docs.google.com/spreadsheets/d/1gC2DNbtCgM6PAfQWc0xcXWrtvZfPMob-uOdXPl4UqYw/edit?usp=sharing
This sheet below shows the fractions shown in base form:
Furthermore, to represent our higher values, I've also chosen to use lower-case letters instead of upper-case, because:
- 8 & B
- D & 0
- G & 6 (although this one isn't so bad in most fonts)
- I and 1
...they all look similar.
Another problem arises when you reach letter L. We can't use lower-case L because it looks like 1 and the letter 'I' as in "igloo".
At base 18, we use the letter H, but not I; it becomes a problem beyond base 18.
It is for these reasons that I only show the bases from 6 - 18.
Here's a chart I made on Google sheets so you can see the elegance primes and composite numbers within the dozenal system.
I highly recommend you check it out if you want to see how primes appear in different bases:
docs.google.com/spreadsheets/d/1gC2DNbtCgM6PAfQWc0xcXWrtvZfPMob-uOdXPl4UqYw/edit?usp=sharing
This sheet below shows the fractions shown in base form:
Senary
|
Octal
|
Decimal
|
Dozenal
|
1/2
1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/a (10) 1/b (11) 1/c (12) 1/d (13) 1/e (14) 1/f (15) 1/g (16) 1/h (17) 1/i (18) |
.3
.2 .13 .1--- .1 .05--- .043 .04 .0 3--- .0313452421--- .03 .024340531215--- .0 23--- .0 2--- .0213 .020412245351433... .02 |
.4
.25--- .2 .1463--- .125 25--- .1--- .1 .07--- .06314--- .0564272135--- .052--- .0473--- .0 4--- .0421--- .04 .03607417--- .0 34--- |
.5
.3--- .25 .2 .16--- .142857--- .125 .1--- .1 .09--- .083--- .076923--- .07 142857--- .0 6--- .0625 .0588235294117647--- .0 5--- |
.6
.4 .3 .2497--- .2 .186a35--- .16 .14 .1 2497--- .1--- .1 .0b--- .0 a35186--- .0 9724--- .09 .0857924b3643... .08 |
1/2
1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/a (10) 1/b (11) 1/c (12) 1/d (13) 1/e (14) 1/f (15) 1/g (16) 1/h (17) 1/i (18) |
Symbols used above:
--- recurrance of digits to immediate left ( .07 142857--- is .07 142857 142857 142857... )
... recurrance, like above, but sequence is too long to fit in chart.
--- recurrance of digits to immediate left ( .07 142857--- is .07 142857 142857 142857... )
... recurrance, like above, but sequence is too long to fit in chart.
Bases and their compatibility with the two most important divisors:
Intervals of 2:
Best -Octal -Dozenal -Senary -Decimal Worst |
Intervals of 3:
Best -Senary -Dozenal -Octal -Decimal Worst |
Tetradecimal
|
Hexadecimal
|
Octadecimal
|
1/2
1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/a (10) 1/b (11) 1/c (12) 1/d (13) 1/e (14) 1/f (15) 1/g (16) 1/h (17) 1/i (18) |
.7
.49--- .37 .2b--- .2 49--- .2 .1a7 .17ac63--- .1 58--- .13b65--- .12 49--- .1--- .1 .0d--- .0c37 .0b75a9c4d268... .0 ac6317--- |
.8
.5--- .4 .3--- .2 a--- .249--- .2 .1c7--- .1 9--- .1745d--- .1 5--- .13b--- .1 249--- .1--- .1 .0f--- .0 e38--- |
.9
.6 .49 .3ae7--- .3 .2a5--- .249 .2 .1 e73a--- .1b834g69ed... .19 .16gb--- .1 52a--- .13 ae73--- .1249 .1--- .1 |
1/2
1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/a (10) 1/b (11) 1/c (12) 1/d (13) 1/e (14) 1/f (15) 1/g (16) 1/h (17) 1/i (18) |
Bases and their compatibility with the two most important divisors:
Intervals of 2:
Best -Hexadecimal -Octadecimal -Tetradecimal Worst |
Intervals of 3:
Best -Octadecimal -Hexadecimal -Tetradecimal Worst |
Chart Conclusion
As you can see, the dozenal and octadecimal systems seem to be the clear victors. With around half of the divisions coming out as short terminations, these bases prove to be very useful in most mathematical calculations.
However, the dozenal system has the edge because of 3 factors:
1: You only need to learn 2 extra symbols in dozenal instead of the 8 new symbols required for octadecimal. Imagine all the names you'd have to remember for each new digit.
2: Dozenal has the advantage of having a good relationship with 2 and 3, whereas octadecimal only has advantages over multiples of 3. Since the multiples of 2 are far more useful, dozenal also wins this.
3: There are less occurrences of recurring decimals and more occurrences of short termination.
Compare them side by side and you'll see:
However, the dozenal system has the edge because of 3 factors:
1: You only need to learn 2 extra symbols in dozenal instead of the 8 new symbols required for octadecimal. Imagine all the names you'd have to remember for each new digit.
2: Dozenal has the advantage of having a good relationship with 2 and 3, whereas octadecimal only has advantages over multiples of 3. Since the multiples of 2 are far more useful, dozenal also wins this.
3: There are less occurrences of recurring decimals and more occurrences of short termination.
Compare them side by side and you'll see:
1/2
1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/a (10) 1/b (11) 1/c (12) 1/d (13) 1/e (14) 1/f (15) 1/g (16) 1/h (17) 1/i (18) |
Dozenal
|
Balanced
Balanced <----Dozenal Wins (multiple of 2) Balanced Balanced Octadecimal Wins (prime)----> <----Dozenal Wins (multiple of 2) Octadecimal Wins (multiple of 3)----> Balanced <----Dozenal Wins (prime) <----Dozenal Wins (multiple of 2 and 3) <----Dozenal Wins (prime) Octadecimal Wins (multiple of 2)----> <----Dozenal Wins (multiple of 3) <----Dozenal Wins (multiple of 2) Octadecimal Wins (prime)----> Octadecimal Wins (multiple of 2 and 3)----> |
Octadecimal
|
|
Oh, and if you're still not convinced, consider that in octadecimal, 10 / 2 = 9; This is an odd number.
In dozenal, 10 / 2 = 6; This is divisible by both 2 and 3.
There is absolutely no better base than dozenal.
In dozenal, 10 / 2 = 6; This is divisible by both 2 and 3.
There is absolutely no better base than dozenal.