Random Bases

anirudhkCasual Member
 Joined: Aug 8 2013, 11:58 AM
I have this idea. Why don't we each write about some random base? It can be any base, even an imaginary base. Write the pros and cons of that base. Also, rate that base at the end of the review out of 6 stars (5 is too decimal).
I'll start with 48
I'll start with 48

anirudhkCasual Member
 Joined: Aug 8 2013, 11:58 AM
Base 48 (40 in dozenal)
Base 48 is a pretty decent base.However, it is overshadowed by base 60, which is pretty closeby. It's a 3smooth base with a prime factorization of 2^4 * 3.
Pros
It's 3smooth and has a totient ratio of 1/3, which is pretty low.
It has many factors and produces many regular numbers.
It's not a prime flank and has a nice neighbor in 49, which gives better sevenths.
It's divisible by 16.
It's a highly composite number.
Cons
5 is maximally recurrent.
Overshadowed by 60.
It's good neighbor is alpha and not omega
I would give base 48
5.1/6 stars
Base 48 is a pretty decent base.However, it is overshadowed by base 60, which is pretty closeby. It's a 3smooth base with a prime factorization of 2^4 * 3.
Pros
It's 3smooth and has a totient ratio of 1/3, which is pretty low.
It has many factors and produces many regular numbers.
It's not a prime flank and has a nice neighbor in 49, which gives better sevenths.
It's divisible by 16.
It's a highly composite number.
Cons
5 is maximally recurrent.
Overshadowed by 60.
It's good neighbor is alpha and not omega
I would give base 48
5.1/6 stars

icarusDozens Demigod
 Joined: Apr 11 2006, 12:29 PM
Here's what I wrote about base 48: 48: You Got Your 3 In My Hexadecimal!.
I'll chat about base 21 (see Unvigesimal Light and Magic). Note, this isn't exhaustive.
Pros: "Intuitive" divisibility tests for the smallest 5 primes {2, 3, 5, 7, 11} and the "SPD" for 13.
Ideal if {3, 7} are particularly important.
Cons: there are more highly divisible numbers in this neighborhood {18, 20, 24} so we aren't getting a bargain. Here's a bit about tuning up your number base.
21 isn't even. Here's a bit about why even bases are important.
I'll chat about base 21 (see Unvigesimal Light and Magic). Note, this isn't exhaustive.
Pros: "Intuitive" divisibility tests for the smallest 5 primes {2, 3, 5, 7, 11} and the "SPD" for 13.
Ideal if {3, 7} are particularly important.
Cons: there are more highly divisible numbers in this neighborhood {18, 20, 24} so we aren't getting a bargain. Here's a bit about tuning up your number base.
21 isn't even. Here's a bit about why even bases are important.

wendy.krieger
One should remember that arithmetic can be got for bases of pretty any size, if you really want it. Twelfty is pretty much out of the decimal/dozenal style tables, and digits, but you can break it down into littler steps, and it works well.
The stonetable calculations in europe tend to favour four to six stones in a row, so 'optimal bases' there are things like 28 and 30.
Just the way the cookie crumbles, i suppose.
The stonetable calculations in europe tend to favour four to six stones in a row, so 'optimal bases' there are things like 28 and 30.
Just the way the cookie crumbles, i suppose.

anirudhkCasual Member
 Joined: Aug 8 2013, 11:58 AM
Icarus, actually I was sort of hoping you would write about bases not covered in your tour yet, because I thought it would be a brief description, but you can write about base 21 if you want to. I'm actually interested about base 21 because it is a really lucky base, receiving great divisibility tests. Strangely,it's also a lucky number (not the numerological, fortune telling type).

Stella of the SapphireCasual Member
 Joined: Apr 22 2013, 08:08 PM
Base six: 2*3
PRO
CON
3/6 Standard, 3/6 Balanced
Base eight: 2^3
PRO
CON
3/6 Standard, 4/6 Balanced
Base dozen and four: 2^4
PRO
CON
3/6 Standard, 4/6 Balanced
Base two dozen: 2^3*3
PRO
CON
4/6 Standard, 5/6 Balanced
Base five dozen: 2^2*3*5
PRO
CON
3/6 Standard, 4/6 Balanced
Yes, I did take the magnitude of the base into account because I think that's important enough to consider for evaluation.
Shall I do more?
PRO
 Halves, thirds, sixths are all succinct (+5)
 Multiplication table very short (+4)
 **Balanced signedfigure notation allows symmetry between addition and subtraction, positive and negative numbers (+3)
 **Balanced notation allows shorter multiplication table (+1)
CON
 Fifths are not handled well (1)
 Fourths require two significant figures (3)
 Smaller base than decimal and dozenal requires more figures per number (4)
 **Balanced notation magnifies figure count disadvantage of smaller base (2)
3/6 Standard, 3/6 Balanced
Base eight: 2^3
PRO
 Halves, fourths, and eighths are all succinct (+5)
 Multiplication table very short (+2)
 **Balanced notation allows symmetry between addition and subtraction, positive and negative numbers (+3)
 **Balanced notation allows shorter multiplication table (+2)
CON
 Smaller base than decimal and dozenal requires more figures per number (2)
 Thirds and fifths are not handled well (3)
 **Balanced notation magnifies figure count disadvantage of smaller base (2)
3/6 Standard, 4/6 Balanced
Base dozen and four: 2^4
PRO
 Halves, fourths, and eighths are all succinct (+5)
 Larger base than decimal and dozenal allows fewer figures per number (+2)
 **Balanced notation allows shorter multiplication table (+3)
 **Balanced notation allows symmetry between addition and subtraction, positive and negative numbers (+3)
CON
 Thirds and fifths are not handled well (3)
 Large base requires more symbols and more complicated multiplication table (3)
 **Balanced notation limits figure count advantage of larger base (2)
3/6 Standard, 4/6 Balanced
Base two dozen: 2^3*3
PRO
 Halves, thirds, fourths, sixths and eighths are all succinct (+7)
 Larger base than decimal and dozenal allows fewer figures per number (+3)
 **Balanced notation allows shorter multiplication table (+4)
 **Balanced notation allows symmetry between addition and subtraction, positive and negative numbers (+4)
CON
 Fifths are not handled well (1)
 Large base requires more symbols and more complicated multiplication table (6)
 **Balanced notation limits figure count advantage of larger base (2)
4/6 Standard, 5/6 Balanced
Base five dozen: 2^2*3*5
PRO
 Halves, thirds, fourths, fifths and sixths are all succinct (+7)
 Larger base than decimal and dozenal allows fewer figures per number (+6)
 **Balanced notation allows shorter multiplication table (+6)
 **Balanced notation allows symmetry between addition and subtraction, positive and negative numbers (+4)
CON
 Large base requires more symbols and more complicated multiplication table (11;)
 **Balanced notation limits figure count advantage of larger base (2)
3/6 Standard, 4/6 Balanced
Yes, I did take the magnitude of the base into account because I think that's important enough to consider for evaluation.
Shall I do more?

wendy.krieger
I'm still not convinced about rating bases with single primes highly, because they never expand their regulars. For example, 16 has more divisors than 14, (1,2,4,8 vs 1,2,7), but 14 continues to add more over powers (ie 1,2,3.7,4,7 vs 1,2,4,8).
I don't think size is an issue, since the arithmetic follows the base, rather than the other way around. Most people seem to only know decimal digitstyle arithmetic, which is why bases like 10 and 12 win over larger, more able numbers like 18 and 30.
But seriously, once you have used some base that has three prime divisors, the twodivisor bases look like primes. You enumerate the various say 4stick regulars in 12, and you get something like 8 or 9, and you say "That's it? Isn't there any more?" In twelfty they pretty much rain out of the roof tops.
The same could be said of 120 compared with bases with more divisors, like 210 or 840, but the counterance here is that you have to accomodate all of these primes in every layer of the measurement system: you just can't have one 7. You got to sprinke 7's everywhere.
Some things make 18 better than 12. For example, 12 has no convenient sevenite: eighteen is spoilt for choices here: 5, 7, 37, 331, ... 68 does it better.
Twelve has tight opposition: that is, for some small ranges, the numbers that divide 100 but not 10, are 8, 9, 14, 16, the ratio of smallest to largest is 2;3. For 1000 but not 100, we get 23, 28, 46, 54, all together a range of 2;454. Only 56 does it better, where you get sevenplace numbers having a total range of 4.2 from bottom to top.
When you look at the logrithmic wheel, for example, where 360 = base, and then plot the regulars by order (ie 9 is second order because it divides 10^2 but not 10), bases like 8 and 16 get pretty much stuck.
This thread deals with the division of regulars by size and recriprocal (ie what power of b it divides), for a number of bases, of several different divisors. So a number like decimal 8, which is one place long, still has a threeplace dual (ie 8*125).
But raw numbers from these sort of tests do not equate to onground evaluations. Bases of the style 237 do better at covering the full range of firstdigits than 235 ones, because it is harder to approximate the logs of 237 together than it is for 235. So something like 120, takes a six place number to find one that starts with '95', while 126 fills all 126 possible starting positions at four places.
I don't think size is an issue, since the arithmetic follows the base, rather than the other way around. Most people seem to only know decimal digitstyle arithmetic, which is why bases like 10 and 12 win over larger, more able numbers like 18 and 30.
But seriously, once you have used some base that has three prime divisors, the twodivisor bases look like primes. You enumerate the various say 4stick regulars in 12, and you get something like 8 or 9, and you say "That's it? Isn't there any more?" In twelfty they pretty much rain out of the roof tops.
The same could be said of 120 compared with bases with more divisors, like 210 or 840, but the counterance here is that you have to accomodate all of these primes in every layer of the measurement system: you just can't have one 7. You got to sprinke 7's everywhere.
Some things make 18 better than 12. For example, 12 has no convenient sevenite: eighteen is spoilt for choices here: 5, 7, 37, 331, ... 68 does it better.
Twelve has tight opposition: that is, for some small ranges, the numbers that divide 100 but not 10, are 8, 9, 14, 16, the ratio of smallest to largest is 2;3. For 1000 but not 100, we get 23, 28, 46, 54, all together a range of 2;454. Only 56 does it better, where you get sevenplace numbers having a total range of 4.2 from bottom to top.
When you look at the logrithmic wheel, for example, where 360 = base, and then plot the regulars by order (ie 9 is second order because it divides 10^2 but not 10), bases like 8 and 16 get pretty much stuck.
This thread deals with the division of regulars by size and recriprocal (ie what power of b it divides), for a number of bases, of several different divisors. So a number like decimal 8, which is one place long, still has a threeplace dual (ie 8*125).
But raw numbers from these sort of tests do not equate to onground evaluations. Bases of the style 237 do better at covering the full range of firstdigits than 235 ones, because it is harder to approximate the logs of 237 together than it is for 235. So something like 120, takes a six place number to find one that starts with '95', while 126 fills all 126 possible starting positions at four places.

Stella of the SapphireCasual Member
 Joined: Apr 22 2013, 08:08 PM
@wendy.krieger
Sure, prime power bases don't have regular numbers beyond powers of the prime, but they have the advantage that repeated division by the prime does not continually increase the count of significant figures, as in base 2^4 3>6>S>18>30 where halving a number four times is equivalent to a rightshift by one place, so no more significant figures! There's no shortage of applications where repeated doubling and halving takes place. I do it all the time in the kitchen and in the marketplace!
Indeed, adding primes to the base number means that one has to use them commonly enough that nonuse of the prime would lead to too many cumbersome representations. Therein lies the advantage of avoiding a prime that isn't relevant, and if only one ternary power is needed and general trisection doesn't occur such as with circular arcs, then even dozenal falls short of binary power bases because with dozenal, or any base that isn't a binary power, repeated bisection will increase the count of significant figures... 0;6, 0;3, 0;16, 0;09, 0;046, 0;023, 0;0116, 0;0069 and so on, it's good in the first few powers but then it becomes a mess later on.
You say (decimal)18 is better than dozenal in some ways, noting a list of 'sevenites' What exactly is a 'sevenite' and what sort of practical application would that concept be useful aside from number theory?
Sure, prime power bases don't have regular numbers beyond powers of the prime, but they have the advantage that repeated division by the prime does not continually increase the count of significant figures, as in base 2^4 3>6>S>18>30 where halving a number four times is equivalent to a rightshift by one place, so no more significant figures! There's no shortage of applications where repeated doubling and halving takes place. I do it all the time in the kitchen and in the marketplace!
Indeed, adding primes to the base number means that one has to use them commonly enough that nonuse of the prime would lead to too many cumbersome representations. Therein lies the advantage of avoiding a prime that isn't relevant, and if only one ternary power is needed and general trisection doesn't occur such as with circular arcs, then even dozenal falls short of binary power bases because with dozenal, or any base that isn't a binary power, repeated bisection will increase the count of significant figures... 0;6, 0;3, 0;16, 0;09, 0;046, 0;023, 0;0116, 0;0069 and so on, it's good in the first few powers but then it becomes a mess later on.
You say (decimal)18 is better than dozenal in some ways, noting a list of 'sevenites' What exactly is a 'sevenite' and what sort of practical application would that concept be useful aside from number theory?

wendy.krieger
@stella
Logrelated Multiples
One often sees in proposals for weights and measures, units rounded to the nearest power of 2. In part, this is because there are already a good selection of such units, but also that it's relatively easy to calculate with these numbers.
My experience with systems like TIOF and COF (bases 120, 12), suggest that such benefit is largely illusionary, and that bases like 120 and 12 do much better jobs of it than binary.
In essence, if you keep the range of units to a relatively tight range, you don't get very large numbers to calculate, outside the base. None the same, you can have things like 1 lb = 14; oz in dozenal.
COF is a cross between c.g.s and t.g.m., in that the units of length and mass are about the size of the centimetre and gram, but the same principles that are used in t.g.m. are used. What is different, is that instead of stopping at this base relation, there is a series of classical units built on it, so one is in effect using an ordinary dozenal version of f.p.s. or metric, but each has a coherent number which is easy to multiply by.
Take for example, weight. The COF unit of weight is an obol, or 5/9 gram. You can build a fairly classical system on this, in the manner of Jeffison, like
1 pound = 14; ounces = 600 obols = 480 g,
1 ounce = 9 ; dram = 46 obols = 30 g
1 dram = 3; scruples = 6 obols = 3 1/3 g
1 scruple = 2 obols = 10/9 gm
1 obol = 10; grains. = 5/9 gm = 540 mg
1 carat = 4 grains; (as it should) = 5/27 gm =
1 grain = 10; mites = 5/108 g or about 50 mgm.
So you already have a system that divides the pound into ounces, and a system that is suited for apothecaries (drugs), for example.
If you live in metric countries, then 1000 obols is pretty close to a kg.
Because water has pretty much the same density in both systems,
1 cup = 300 obols , say 240 mls, divided into 1/2, 1/3, 1/4, 2/3 and 3/4
1 tablespoon = 1/14; cup, = gives 23 obols or 1 cu inch = 15 ml
1 teaspoon = 1/3 tablespoon = 5 ml = 9 obols = 1/3 cu in.
So have ordinary cooking measure in there as well.
Since we have many numbers that come from the same series (negative powers of 2), none of the ratios are really large. 3/23;, for example, gives 14;
The system can be made to cover a great number of units like Btuinch / ft^2 hr deg, because all of these have very simple numbers, like 6D6 * 3 / (30^2 * 1d5 * 1d4), which is fairly straight forward to reduce to (16/9)D(6254) = 2D5.
Unlike TGM, the temperature scale is directly derived as well. We set the specific heat of water to 10000, which makes the difference between icepoint and boiling point as 251 = 191;. But this can be remedied because if you set 0F = 0; F' then 100 d F = 100; F', and you read an ordinary fahrenheit temperature as 100 = 100. Freezing is 40, and 68F becomes 80.
It's a little more complex, but the system as a whole gives both the binary series and its intermediate (ie 3 as 2 sqrt(2)).
Sevenites
Sevenites are instances where a prime divides its own period: for example, in base 18, 1/5 = 0:(3,10,14,7) while 1/5^2 = 0;(0,12,17,5). It's even better with sevenths, where 1/7 gives 0;2,(10,5) and 1/7^2 gives 0;(0,6,11) and 1/7^3 = 0:(0,0,17).
Ten has three sevenites, but only '3' is useful: 1/3 = 0.(3) and 1/9 = 0.(1). This makes things like 1/3 and 1/9 more useful in decimal then they are in hexadecimal. For example
1/12 = 0.08(3) = 0.1,(5) ie: 3 vs 2
1/144 = 0.0069(4) = 0;0,(1,12,7) ie 5 vs 4
1/1728 = 0.000 578 (703) = 0;0,0,(2,5,14,13,0,9,7,11,4) ie 9 vs 11.
The decimal period for powers of 3 are 1/3 of the hexadecimal, for the reason that in decimal 3 divides its own period (ie is a decimal sevenite).
Large sevenites like 18's 37 are not much use, because there is not much call for either fractions in 1/37 or its square, or for 36digit places.
It should be noted that the recriprocal of 19! is shorter in base 18, than in bases like 120, 80. Even 30, i think, is longer. But you have to go to 210 (which is 7smooth), to find a shorter period.
The general utility of sevenites depends on whether you want to include powers of 5 and 7 into the measurement system. For example, having a number like 1.3 b18 in a system, allows even its cube to have a simple period ie 0;0,0,(0,11,6). Otherwise, they're pretty much like curios, rather like having divisors of the base if they're never used.
Logrelated Multiples
One often sees in proposals for weights and measures, units rounded to the nearest power of 2. In part, this is because there are already a good selection of such units, but also that it's relatively easy to calculate with these numbers.
My experience with systems like TIOF and COF (bases 120, 12), suggest that such benefit is largely illusionary, and that bases like 120 and 12 do much better jobs of it than binary.
In essence, if you keep the range of units to a relatively tight range, you don't get very large numbers to calculate, outside the base. None the same, you can have things like 1 lb = 14; oz in dozenal.
COF is a cross between c.g.s and t.g.m., in that the units of length and mass are about the size of the centimetre and gram, but the same principles that are used in t.g.m. are used. What is different, is that instead of stopping at this base relation, there is a series of classical units built on it, so one is in effect using an ordinary dozenal version of f.p.s. or metric, but each has a coherent number which is easy to multiply by.
Take for example, weight. The COF unit of weight is an obol, or 5/9 gram. You can build a fairly classical system on this, in the manner of Jeffison, like
1 pound = 14; ounces = 600 obols = 480 g,
1 ounce = 9 ; dram = 46 obols = 30 g
1 dram = 3; scruples = 6 obols = 3 1/3 g
1 scruple = 2 obols = 10/9 gm
1 obol = 10; grains. = 5/9 gm = 540 mg
1 carat = 4 grains; (as it should) = 5/27 gm =
1 grain = 10; mites = 5/108 g or about 50 mgm.
So you already have a system that divides the pound into ounces, and a system that is suited for apothecaries (drugs), for example.
If you live in metric countries, then 1000 obols is pretty close to a kg.
Because water has pretty much the same density in both systems,
1 cup = 300 obols , say 240 mls, divided into 1/2, 1/3, 1/4, 2/3 and 3/4
1 tablespoon = 1/14; cup, = gives 23 obols or 1 cu inch = 15 ml
1 teaspoon = 1/3 tablespoon = 5 ml = 9 obols = 1/3 cu in.
So have ordinary cooking measure in there as well.
Since we have many numbers that come from the same series (negative powers of 2), none of the ratios are really large. 3/23;, for example, gives 14;
The system can be made to cover a great number of units like Btuinch / ft^2 hr deg, because all of these have very simple numbers, like 6D6 * 3 / (30^2 * 1d5 * 1d4), which is fairly straight forward to reduce to (16/9)D(6254) = 2D5.
Unlike TGM, the temperature scale is directly derived as well. We set the specific heat of water to 10000, which makes the difference between icepoint and boiling point as 251 = 191;. But this can be remedied because if you set 0F = 0; F' then 100 d F = 100; F', and you read an ordinary fahrenheit temperature as 100 = 100. Freezing is 40, and 68F becomes 80.
It's a little more complex, but the system as a whole gives both the binary series and its intermediate (ie 3 as 2 sqrt(2)).
Sevenites
Sevenites are instances where a prime divides its own period: for example, in base 18, 1/5 = 0:(3,10,14,7) while 1/5^2 = 0;(0,12,17,5). It's even better with sevenths, where 1/7 gives 0;2,(10,5) and 1/7^2 gives 0;(0,6,11) and 1/7^3 = 0:(0,0,17).
Ten has three sevenites, but only '3' is useful: 1/3 = 0.(3) and 1/9 = 0.(1). This makes things like 1/3 and 1/9 more useful in decimal then they are in hexadecimal. For example
1/12 = 0.08(3) = 0.1,(5) ie: 3 vs 2
1/144 = 0.0069(4) = 0;0,(1,12,7) ie 5 vs 4
1/1728 = 0.000 578 (703) = 0;0,0,(2,5,14,13,0,9,7,11,4) ie 9 vs 11.
The decimal period for powers of 3 are 1/3 of the hexadecimal, for the reason that in decimal 3 divides its own period (ie is a decimal sevenite).
Large sevenites like 18's 37 are not much use, because there is not much call for either fractions in 1/37 or its square, or for 36digit places.
It should be noted that the recriprocal of 19! is shorter in base 18, than in bases like 120, 80. Even 30, i think, is longer. But you have to go to 210 (which is 7smooth), to find a shorter period.
The general utility of sevenites depends on whether you want to include powers of 5 and 7 into the measurement system. For example, having a number like 1.3 b18 in a system, allows even its cube to have a simple period ie 0;0,0,(0,11,6). Otherwise, they're pretty much like curios, rather like having divisors of the base if they're never used.

Stella of the SapphireCasual Member
 Joined: Apr 22 2013, 08:08 PM
So it looks like this whole concept of 'sevenite' is based on infinitely repeating representations, so in practice, it's utility is minimal. I don't care if there are factors of five or seven in the system, if they're not divisors of the base then they don't have a finite representation in that base. Plain and simple, however short or long the period of recurrence, it makes no difference to a layman who cares about exact, finite and succinct representations.
Here are some more evaluations:
Base 16; 2*3^2
PRO
CON
2~3/6 Standard, 3/6 Balanced
Base 18; 2^2*5
PRO
CON
2~3/6 Standard, 3/6 Balanced
Base 26; 2*3*5
PRO
CON
3/6 Standard, 3~4/6 Balanced
Base 54; 2^6
PRO
CON
2/6 Standard, 4/6 Balanced
Base X0; 2^3*3*5
PRO
CON
3~4/6 Standard, 5/6 Balanced
Here are some more evaluations:
Base 16; 2*3^2
PRO
 Halves, thirds, sixths are all succinct (+4)
 **Balanced notation halves size of multiplication table (+3)
 **Balanced notation allows symmetry between addition and subtraction, positive and negative numbers (+3)
CON
 Fifths do not have finite representation (1)
 Fourths require two significant figures (4)
 **Balanced notation halves thresholds for figure counts (2)
2~3/6 Standard, 3/6 Balanced
Base 18; 2^2*5
PRO
 Halves, fourths, fifths are all succinct (+4)
 **Balanced notation halves size of multiplication table (+3)
 **Balanced notation allows symmetry between addition and subtraction, positive and negative numbers (+3)
CON
 Thirds do not have finite representation (4)
 **Balanced notation halves thresholds for figure counts (2)
2~3/6 Standard, 3/6 Balanced
Base 26; 2*3*5
PRO
 Halves, thirds, fifths and sixths are all succinct (+5)
 **Balanced notation halves size of multiplication table (+4)
 **Balanced notation allows symmetry between addition and subtraction, positive and negative numbers (+3)
CON
 Fourths require two significant figures (5)
 **Balanced notation halves thresholds for figure counts (2)
3/6 Standard, 3~4/6 Balanced
Base 54; 2^6
PRO
 Halves, fourths and eighths are all succinct (+4)
 Base is a square and a cube (+3)
 Repeated halving does not increase count of significant figures (+3)
 **Balanced notation halves size of multiplication table (+5)
 **Balanced notation allows symmetry between addition and subtraction, positive and negative numbers (+3)
CON
 Thirds and fifths do not have finite representation (6)
 **Balanced notation halves thresholds for figure counts (1)
2/6 Standard, 4/6 Balanced
Base X0; 2^3*3*5
PRO
 Halves, thirds, fourths, fifths, sixths and eighths are all succinct (+8)
 **Balanced notation halves size of multiplication table (+7)
 **Balanced notation allows symmetry between addition and subtraction, positive and negative numbers (+3)
CON
 Subradix notation blunts divisibility transparency (3)
 **Balanced notation halves thresholds for figure counts (1)
3~4/6 Standard, 5/6 Balanced

wendy.krieger
And herein we see the value of doing things on the ground. You can't tell much difference between 16; and 18; from the numerical factors like in Stella's list. When you actually use them, they're as different as chalk and cheese.
For bases of the type \(a^2.b\), there are 13 significant integers which divide the cube of the base.
10; : 1, 1.4, 1.6, 2, 2.3, 2.8, 3, 4, 4.6, 5.4, 6, 8, 9,
16; : 1, 1.6, 1.9, 2, 2.4.9, 3, 4, 4.9, 6, 8, 9, 12, 13.9,
18; : 1, 1.5, 1.12, 2, 2.10, 3.4, 4, 5, 6.5, 8, 10, 12.10, 16
The numbers in 16; have inter ratios of 4/3 and 9/8. The next few powers of 18 provide larger numbers which serve to split the 4/3's into 9/8 and 32/27. 32/27 is then split by 9/8 into 9/8 and 256/243. In short, the gaps are bigger than in 18; but larger powers split these down to smaller ones.
The numbers in 18; vary between them, at the ratios of 5/4, 32/25, ie 1.25 vs 1.28. Subsequent powers of 18; add regulars very close to what's there. For example, the next pair of regulars are 6.8 and 3.2.10, very close to the 6.5 and 3.4 aleady there. The 32/25 is then split into 128/125 and 5/4. Eventually, it takes about ten bites at the cherry to split 5/4 by 125/128, 5/4 = 128/125 * 625/512, and 625/512 = 128/125 * 78125/65536. You have to get quite large powers before anything interesting happens.
16; is the smallest base where not all regulars are of the form 2^a 1.0^b. That is, there are numbers which are not already powers of 2, which never become powers of 2 on repeated doubling.
One of the things i found, especially when using 16; is that the significant integer of a number can never be the multiple of the base. So you can't have for example, a multiple of 12 out the front in base 12. If you want to have 2,3,5,7 in the number, you are pretty much restricted to getting junk 5,7 in the significant integer, and the 2's and 3's disappear in the 0's. This is why, for example, it's a good idea to treat the circle as 1 in bases like 12 and 120.
On the other hand, you can have 2^3.3.5.7 in base 18, as 2.10.12 in that base.
16;s divisors are spread more evenly over the logrithmic wheel than 18; or 10; which makes using coins of order 1, 2, 3, 6, 9, as the extended set, and 1, 3, 6 as a minimal set, more attractive. With 10; and 12; one is either vying for 1,2,5,10 or 1,3,6 or 1,2,4 (twopences and groats in a shilling), with little symmetry.
Twelfty.
Some of Stella's comments deserve mention.
A subradix no more blunts divisibility in this base than in any other. For example, the test for multiples of 8 are the last two digits in both 12 and 120. It's just things like 3 also rely on the last two digits in twelfty.
On the main, the last four digits of 12 permit divisibility tests for some 44 numbers. Although there are fewer fourdigit combinations in 120, it permits some 62 numbers. The last six digits give for 12, some 90 numbers. In twelfty, sixdigit tests give divisibilities for some 159 numbers.
Twelfty's 14 divisors are more uniformly spread around the logrithmic wheel, with no big gap between 6 and 10 as does 60. There are some 32 new divisors of 10000 not of 100, these do not interfere with the 15 already there. Likewise, the 50 orderthree divisors do double up with themselves, (eg 105 and 108), but do not interfere with the 47 that were previously there. They occupy all together, 77 of the 83 semitones. The 97 numbers are pretty much allow for cubes of the firstorder numbers, and assorted numbers.
Stella seems facinated with balanced notation. You can quite well do with basecomplement forms, and a leading 'M' digit for most uses. In any case, she assumes that someone is going to learn the full 10000 table, or even the 30.00 symmetric table. In practice, there are different rules of arithmetic, and alternating arithmetic works quite well with bases where the two subbases differ by 1 or 2.
For bases of the type \(a^2.b\), there are 13 significant integers which divide the cube of the base.
10; : 1, 1.4, 1.6, 2, 2.3, 2.8, 3, 4, 4.6, 5.4, 6, 8, 9,
16; : 1, 1.6, 1.9, 2, 2.4.9, 3, 4, 4.9, 6, 8, 9, 12, 13.9,
18; : 1, 1.5, 1.12, 2, 2.10, 3.4, 4, 5, 6.5, 8, 10, 12.10, 16
The numbers in 16; have inter ratios of 4/3 and 9/8. The next few powers of 18 provide larger numbers which serve to split the 4/3's into 9/8 and 32/27. 32/27 is then split by 9/8 into 9/8 and 256/243. In short, the gaps are bigger than in 18; but larger powers split these down to smaller ones.
The numbers in 18; vary between them, at the ratios of 5/4, 32/25, ie 1.25 vs 1.28. Subsequent powers of 18; add regulars very close to what's there. For example, the next pair of regulars are 6.8 and 3.2.10, very close to the 6.5 and 3.4 aleady there. The 32/25 is then split into 128/125 and 5/4. Eventually, it takes about ten bites at the cherry to split 5/4 by 125/128, 5/4 = 128/125 * 625/512, and 625/512 = 128/125 * 78125/65536. You have to get quite large powers before anything interesting happens.
16; is the smallest base where not all regulars are of the form 2^a 1.0^b. That is, there are numbers which are not already powers of 2, which never become powers of 2 on repeated doubling.
One of the things i found, especially when using 16; is that the significant integer of a number can never be the multiple of the base. So you can't have for example, a multiple of 12 out the front in base 12. If you want to have 2,3,5,7 in the number, you are pretty much restricted to getting junk 5,7 in the significant integer, and the 2's and 3's disappear in the 0's. This is why, for example, it's a good idea to treat the circle as 1 in bases like 12 and 120.
On the other hand, you can have 2^3.3.5.7 in base 18, as 2.10.12 in that base.
16;s divisors are spread more evenly over the logrithmic wheel than 18; or 10; which makes using coins of order 1, 2, 3, 6, 9, as the extended set, and 1, 3, 6 as a minimal set, more attractive. With 10; and 12; one is either vying for 1,2,5,10 or 1,3,6 or 1,2,4 (twopences and groats in a shilling), with little symmetry.
Twelfty.
Some of Stella's comments deserve mention.
A subradix no more blunts divisibility in this base than in any other. For example, the test for multiples of 8 are the last two digits in both 12 and 120. It's just things like 3 also rely on the last two digits in twelfty.
On the main, the last four digits of 12 permit divisibility tests for some 44 numbers. Although there are fewer fourdigit combinations in 120, it permits some 62 numbers. The last six digits give for 12, some 90 numbers. In twelfty, sixdigit tests give divisibilities for some 159 numbers.
Twelfty's 14 divisors are more uniformly spread around the logrithmic wheel, with no big gap between 6 and 10 as does 60. There are some 32 new divisors of 10000 not of 100, these do not interfere with the 15 already there. Likewise, the 50 orderthree divisors do double up with themselves, (eg 105 and 108), but do not interfere with the 47 that were previously there. They occupy all together, 77 of the 83 semitones. The 97 numbers are pretty much allow for cubes of the firstorder numbers, and assorted numbers.
Stella seems facinated with balanced notation. You can quite well do with basecomplement forms, and a leading 'M' digit for most uses. In any case, she assumes that someone is going to learn the full 10000 table, or even the 30.00 symmetric table. In practice, there are different rules of arithmetic, and alternating arithmetic works quite well with bases where the two subbases differ by 1 or 2.

anirudhkCasual Member
 Joined: Aug 8 2013, 11:58 AM
Base 888 (620 in dozenal)
2^3 * 3 * 37
Yeah, this is a pretty random base. I'm pretty sure no one in all of history has ever considered it, and nobody ever will. It is divisible by 24 though, making it a bit better than other bases around it.
Pros
Has many factors compared with it's close neighbors.
Has a better totient ratio compared to 3smooth bases ( not even by 1% though)
It's alpha neighbor is divisible by 7.
Cons
Has a large useless prime factor
It's number of factors are nothing compared to numbers such as 720 and 840.
It's useful factors are shared with these numbers, while it lacks some useful factors which divide these numbers.
Isn't divisible by 5, which is especially important for a number of this size.
Has a useless omega neighbor.
Can't be coded, due to large prime factor.
I would rate this base
2.2 out of 6 stars
2^3 * 3 * 37
Yeah, this is a pretty random base. I'm pretty sure no one in all of history has ever considered it, and nobody ever will. It is divisible by 24 though, making it a bit better than other bases around it.
Pros
Has many factors compared with it's close neighbors.
Has a better totient ratio compared to 3smooth bases ( not even by 1% though)
It's alpha neighbor is divisible by 7.
Cons
Has a large useless prime factor
It's number of factors are nothing compared to numbers such as 720 and 840.
It's useful factors are shared with these numbers, while it lacks some useful factors which divide these numbers.
Isn't divisible by 5, which is especially important for a number of this size.
Has a useless omega neighbor.
Can't be coded, due to large prime factor.
I would rate this base
2.2 out of 6 stars

icarusDozens Demigod
 Joined: Apr 11 2006, 12:29 PM
I am pleased I had the opportunity to use an umlaut today! The number 82 is intriguing to me only because it appears in some studies I did in base 360. It's also cool because it is close to a power of 3.
Note: I've put this and any other "randomly considered" base I submit on the [url=http://z13.invisionfree.com/DozensOnline/index.php?showtopic=621&view=findpost&p=22006428]Tour des Bases "Itinerary"[/url]. I have two others written up. These are much more compact. I'll let you all rate them!

dgoodmaniiiDozens Demigod
 Joined: May 21 2009, 01:45 PM
That's funny! Wonder why I never noticed that before?icarus @ Sep 30 2013, 02:48 PM wrote: Base 82  Duoöctagesimal</strong> (Dozenal: Base 6X;  hexdecimal [lulz!])
All numbers in my posts are dozenal unless stated otherwise.
For ten, I use or X; for elv, I use or E. For the digital/fractional/radix point, I use the Humphrey point, ";".
TGM for the win!
Dozenal Adventures
For ten, I use or X; for elv, I use or E. For the digital/fractional/radix point, I use the Humphrey point, ";".
TGM for the win!
Dozenal Adventures

icarusDozens Demigod
 Joined: Apr 11 2006, 12:29 PM
Don, who would think about SDNhexdecimal (sic!), being a fairly unremarkable, thus "random" base! I wouldn't feel bad about it, because I only noticed it after I began writing about it this morning! (hence the lulz).

icarusDozens Demigod
 Joined: Apr 11 2006, 12:29 PM
The 120s are a favorite neighborhood of mine. We have a highly divisible 120, the square of 11, the cube of 5, the seventh power of 2 all in a decade. The number 126 shows pretty well among these neighbors, but not nearly as deliciously as 120.

anirudhkCasual Member
 Joined: Aug 8 2013, 11:58 AM
792 (560 in dozenal)
2^3 * 3^2 * 11
This is sort of the 11version of 360, having the same layout as it, just replacing 11 with 5. It has 24 factors, which isn't too bad.
Pros
It has many factors (24).
It has a low totient ratio (30%)
It isn't a prime flank (791 is 7*113)
It's useful neighbor is omega.
Cons
It isn't divisible by 5, which is expected for a number of this size.
It doesn't even have a useful divisibility test for 5.
It's bigger than twice of 360, so that means if someone was going to consider it, they would see that 720 is much more efficient.
11 is not really useful
I would rate this base
3.6/6 stars
2^3 * 3^2 * 11
This is sort of the 11version of 360, having the same layout as it, just replacing 11 with 5. It has 24 factors, which isn't too bad.
Pros
It has many factors (24).
It has a low totient ratio (30%)
It isn't a prime flank (791 is 7*113)
It's useful neighbor is omega.
Cons
It isn't divisible by 5, which is expected for a number of this size.
It doesn't even have a useful divisibility test for 5.
It's bigger than twice of 360, so that means if someone was going to consider it, they would see that 720 is much more efficient.
11 is not really useful
I would rate this base
3.6/6 stars

anirudhkCasual Member
 Joined: Aug 8 2013, 11:58 AM
46189 ( 22891 in dozenal)
11 * 13 * 17 * 19
This base sure is a small prime hater. It gives values to bigger primes but doesn't have terminating expansions of the first 4 primes.
Pros
It's divisible by many primes.
Cons
Isn't even.
Isn't divisible by 3.
Isn't divisible by 5.
Isn't even divisible by 7.
Has high totient ratio (75%).
Few regular numbers.
I would rate this base
0.8/6 stars
11 * 13 * 17 * 19
This base sure is a small prime hater. It gives values to bigger primes but doesn't have terminating expansions of the first 4 primes.
Pros
It's divisible by many primes.
Cons
Isn't even.
Isn't divisible by 3.
Isn't divisible by 5.
Isn't even divisible by 7.
Has high totient ratio (75%).
Few regular numbers.
I would rate this base
0.8/6 stars