On Number Bases

__Prelude.__

I remember sitting in 7^{th} grade math class, and being introduced to the idea of other number bases besides Base 10. This got me to thinking: Why are we using Base 10 anyway? Is there something special about it? Might there not be a better/more efficient Number Base to use, and we simply use Base 10 because that is what is taught by tradition?

Thus began my inquiry into Number Base theory, and although carried out sporadically over the years since, due to other interests and obligations, it is something I have wanted to expound upon for some time. There are sources available on-line and in print which I have used for reference, primarily in the historical section, but the huge majority of the math and analysis is my own doing.

As this is an essay as opposed to a scholarly dissertation, references will be kept to a minimum. Familiarity with basic math, roots, powers, and logarithms is presumed, as well as graphic representation of formulae. As mentioned, there are many sources regarding this subject. Also there exist organizations in various countries, such as the Dozenal Society, which promote the use of other Number Bases.

__Defined.__

A Number Base can be defined as the amount of (number of) whole numbers one counts until starting over with one’s digits, and incrementing the next register (column). Each register up or down can only and always have the same numbers in order. In Base 10, these are 0,1,2,3,4,5,6,7,8,9, and then repeated as the next register is added (to), such as 10,11,12,13, etc.. In Base 2, it is simply 0,1, and then repeated 10,11, 100,101, 110, 111, 1000,1001, etc. If this system is maintained regularly for all positive and negative registers, then it is a true number base. This also means that power sequences are geometrically consistent and constant – that is 10[b] = 1 * 10[b], 100[b] = 10[b] * 10[b], 0.1[b] = 1 / 10[b], etc.

A consistent number base is necessary for precise calculations beyond basic arithmetic.

This can be expressed simply as: b^n = a, or log[b] (a) = n, where b = the number base, n = the exponent, and a = any number (variable).

In Base 10, b = 10, so 10^1=10, 10^0=1, 10^2=100, 10^-1=0.1, etc.

Logarithmically, log[10] 10 = 1, log[10] 1 = 0, log[10] 100 = 2, log[10] 0.1 = -1, etc.

The question then is: what is ‘n’ in 10^n = 2. This is not so obvious, hence the use of logarithms, where log[10] 2 = n, which in this case is 0.30103…. (most log values of whole numbers in any base are infinite non-repeating decimal expansions.)

In another number base, say Base 6, 6^n =2, or log[6] 2 = n, this value will be different than log

[10] 2, or log[8] 2, etc.

__History.__

Use of numbers probably started in prehistory with the need to somehow codify or represent the amount of objects or volume of materials being traded or stored. This was likely a system of tally marks made on some medium, such as charcoal on a piece of smooth bark, or indentations impressed in clay tablets. This might have had the appearance of ///// or //////// or ////////////, (essentially, Base 1). As you can see if you tried to count the digits, especially the 3^{rd} set, the chance for error is great. (Studies have shown most humans do not easily retain more than 7 to10 digits of any number. Hence the length of SSN’s and Phone #’s)

Once agriculture got started, the need for accurate calendars, to count the days in a month or year, would have quickly overwhelmed this system. With agriculture also came the possibility for the existence of cities, and the need for record keeping of very large amounts of things.

The simplest method is to group the tally marks, such as groups of 5. (This system is still in use.) The numbers above would now be ////\, ////\ ///, and ////\ ////\ ///. You can see it is easier to tell what the amounts are. However, it still requires the same number of strokes, and quickly becomes unwieldy. Basic arithmetic operations are also quite tedious in this system.

The Romans improved on the tally system by adopting symbols to represent fixed amounts. V represents 5, or one hand, and X represents 10, or 2 hands (2 V’s joined at the apexes.) They also used C for 100 (Latin Centum), M for 1000 (Latin Mille), and L for 50, D for 500. Even greater amounts could be indicated by placing a line over the symbol. The numbers above are now V, VIII, and XIII respectively. This is certainly easier to write, and to read. Math operations beyond addition are still quite difficult, and it it not a true number Base.

In the same general millenia, other cultures, such as the ancient Semites and Greeks, assigned numerical value to letters of their alphabet as a means of recording numbers. This improved the situation somewhat for small numbers, but was not a true register system; for example in Greek, 3 = gamma, but 30 = lambda, and 300 = tau. The numbers above would be epsilon, eta, and iota + gamma.

Meanwhile, and in some cases even a millennium earlier, the need for accurate time-keeping for civic and agricultural purposes created the need in the developing astrological/astronomical disciplines for a number system which could be easily multiplied, and divided, and capable of representing geometric diagrams in calculations. This arose in ancient Sumeria and Egypt, and possibly other cultures. What was critical here was divisibility, so a number system based on 60 was devised (Evenly divisible by 2,3,4,5,6,10,12,15, 20, and 30.). This allowed for more complex math, and took less writing or clay tablet space than previous methods, but was still not a complete number base. (We still use a derivative of this system for measuring time, angles, and the positions of astronomical objects in 2 and 3 dimensions.)

Other cultures are known to have used others number bases. The Maya used Base 20, and while it wasn’t derived from this source, we have the concept of the __Score__. Some NW Europeans, such as the Danes, used Base 12. (This is the source of our concept for the __Dozen__ and __Gross__.)

The final set of changes which led to the adaptation of a true number Base, in this case, Base 10, occurred over several centuries, starting with the Hindu astronomers/astrologers, who apparently came up with the concept of __zero__, which allowed for a true register system of numbers, zero acting a a placeholder in any register not having another value. When added to the basic semitic (Arabic) numbers, this became the Base 10 that expanded through trade to the Muslim world, and then to Renaissance Europe. This allowed far more complex math operations with few symbols, including modern algebra and trigonometry. Our original numbers above are now 5,8, and 13 – easily recognizable.

The French revolution in the later 1700’s attempted to impose Base 10 numbering on all phases of life; not just in computation, but in measurement and timekeeping – the Metric System. This was only partially successful. Due to its divisibility, Base 12 has continued to be used for timekeeping and circular measurement.

With the beginning of the age of electronic computing, and data storage being essentially limited to the on/off state (1 or 0), binary based number systems have come into use. Octal (Base 8) was used earlier on for designating registers, but Hexadecimal (Base 16) has become favored as processing power has increased. Of course, to use a larger Base beyond 10, numbers have to be added/invented. Encoders have opted to take the letters A – F and give them numerical values 11 – 15 respectively. (As if the confusion of 0 and O, 1 and I, 2 and Z, 5 and S weren’t bad enough already.

__Base 10.__

Before trying to evaluate Number Bases to see if any one would be better, let us determine what are the characteristics, positive and negative, of Base 10.

__Divisibility__. Even divisibility (no remainder) is always possible by 1 and the Number Base itself, in this case 10, and division by 0 is undefined. Among the remaining numbers 2-9, only 2 and 5, or 2 out of 8 (25 %) evenly divide 10. [Note: Percentages in parentheses are expressed in Base 10, for comparison purposes.]

__Primes__. Except for 2 and 5, any number ending in 1,3,7 or 9 could be a prime number (40 % possible). (Not all of them are of course).

__Digit addability__. In any number Base, the value immediately before 10 has digit addability. That is, adding the digits of any whole number will add up to that number or one of its multiples if it is evenly divisible by that number, For example, in Base 10 this would be 9. The number 576 is evenly divisible by 9, because 5+7+6=18, which is divisible by 9 (1+8=9). 579 is not divisible by 9, because 5+7+9=21, (2+1=3). Also, because 3 evenly divides 9, one can also tell by the same method if a number is evenly divisible by 3, if the digits add up to 3 or a multiple thereof.

__Divisor recognizability__. In Base 10, a number is divisible by 10 if it ends in 0, by 5 if it ends in 0 or 5, by 2 if it is even, by 9 if its digits add to a 9 multiple, by 3 if its digits add to a 3 multiple, by 6 if its digits add to a 3 multiple and is even, and by 4 if it ends in one of the 25 numbers between 1 and 100 that are evenly divisible by 4. This last one takes either a bit of memory, recognizing the pattern of an even number followed by a 0,4,or 8, or an odd number followed by a 2 or 6. (…20,24,28,32,36…..) Only 7 and 8 cannot be easily recognized as even dividers. (80 % recognizable).

__Decimal expansions of fractions__. ½ = 0.5, 1/3 = 0.*3333*, ¼ = 0.25, 1/5 = 0.2, 1/6 = 0.*6666*, 1/7 = 0.*142857*, 1/8 = 0.125, 1/9 = 0.*1111*, 1/10 = 0.1, 1/11 = 0.*0909*, 1/12 = 0.08*333*, 1/13 = 0.*076923*, 1/14 = 0.0*714258*, 1/15 = 0.0*6666*, 1/16 = 0.0625. (6 out of 15 non-repeating (40 %))

(Note: *Italicized Numbers* are infinitely repeating number sequences.)

Power Sequences (cyclic). 2: 0 1 4 9 6 5 6 9 4 1 0. 3: 0 1 8 7 4 5 6 3 2 9 0. 4: 0 1 6 1 6 5 6 1 6 1 0.

Idempotent: 0,1,5,6. Nilpotent: 0.

Groups. 1,3,7,9 form C4 group under multiplication, with 1 as identity element. This contains 1,9 as a C2 subgroup, with 1 as identity element. 6,2,4,8 form C4 group under multiplication, with 6 as identity element. This contains 6,4 as a C2 subgroup, with 6 as identity element. (32 % of multiplication operations are part of a group.)

If another number base is to be judged better than Base 10, at least some of the characteristics listed previously must be a improvement. More divisibility, digit addability, divisor recognizability, and non repeating decimal expansions would be advantageous, as well as less possible primes.

Since there are an infinite number of possible Bases, we should restrict our inquiry to those around Base 10. A smaller number base has less symbols, and would allow clearing of some typographical confusion. As an example: Base 8, with nos. 0,1,2,3,4,5,6,7 could be changed to 8,7,2,3,4,5,6,9, with 8 replacing 0, 7 replacing 1, and 9 replacing 7. This would eliminate the 0 – O and 1 – I confusion. The disadvantage of a smaller base is that numbers become longer (more digits), especially as one gets into millions or higher. Therefore we will start with Base 6 as the smallest.

A larger Base allows us to have shorter numbers, as there are more symbols to use in each register. The problem here is that new symbols have to be created, hopefully different enough from written language to avoid confusion. Also names for these numbers will have to be created for every day use. If one has too many symbols, they may become hard to distinguish from each other as well, unless they are very complicated symbols. But then, writing them becomes a bit of a chore. As a practical matter, We will put the upper limit at Base 16.

In terms of divisibility, any prime number Base would be less desirable, so that would eliminate Base 7, 11 and 13. And since division by 2 is the most common, we would want ½ to be a non-repeating decimal, so as interesting as they may be, this eliminates Base 9 and 15.

The primary divisors of Base 10 are 2 and 5, while for Base 6 it is 2 and 3, for Base 8 it is 2 and 4, and for Base 14 it is 2 and 7. Division by 3 or 4 is more common then 5, but not 7, so there would be no advantage to Base 14 over Base 10.

This leaves Base 6, 8, 12, and 16 for consideration. The binary related Bases 8 or 16 would be useful in interfacing with electronic computation, which will likely continue its expansion into all areas of life. The Bases with the greatest percentage of divisible factors, that is 6 or 12, would be more useful in working with numbers every day, particularly with circular and spherical angular measurement, as well as time keeping, where they are already being used to some extent.

(Evaluations of the other number bases are in the appendix for those who are curious.)

__Base 6__

Symbols: 0,1,2,3,4,5, (Or 8,7,2,3,4,5 to remove ambiguity with some letters).

__Divisibility__. Among numbers 2-5, 2 and 3 evenly divide 10. (This is 2/4, or 50 %.)

__Primes__. Except for 2 and 3, all potential prime numbers end in 1 or 5. (33.33 % possible)

__Digit Addability__. If the digits add up to 5, or a multiple of 5, then the number is evenly divisible by 5.

__Divisor recognizability__. Any number ending in 0 is divisible by 10, and is divisible by 3 if it ends in 0 or 3. If the number is even it is divisible by 2, It is divisible by 5 if the digits add up to 5. It is evenly divisible by 4 if it is one of the 9 numbers between 1 and 100. The pattern is an even number followed by an ending number of 0 or 4, or an odd number followed by an ending number of 2. (…20,24,32,…)

There are no numbers that cannot easily be recognized as even divisors. (100 % recognizable)

__Heximal expression of fractions__. ½ = 0.3, 1/3 = 0.2, ¼ = 0.13, 1/5 = 0.*1111*, 1/10 = 0.1, 1/11 = 0.*0505*, 1/12 = 0.043, 1/13 = 0.04, 1/14 = 0.0*333*, 1/15 = 0.*0313452421*, 1/20 = 0.03, 1/21 = 0.*02435*, 1/22 = 0.0*2323*, 1/23 = 0.0*2222*, 1/24 = 0.0213. (8 out of 15 non-repeating (53.33 %))

Power Sequences (cyclic). 2: 0 1 4 3 4 1 0. 3: 0 1 2 3 4 5 0. 4. 0 1 4 3 4 1 0.

Idempotent: 0,1,3,4. Nilpotent: 0.

Groups: 1,5 form a C2 group under multiplication, with 1 as the identity element. 4,2 form a C2 group under multiplication, with 4 as the identity element. (22.22% of multiplication operations are part of a group.)

__Advantages:__

Superior to Base 10 in all 4 measurable characteristics. (Divisibility, Divisor recognition, Fractional expression greater, Prime registers less.)

Could allow for conversion to all non-ambiguous symbols with respect to written language symbols.

__Disadvantages:__

Fewer symbols makes numbers longer, especially when many registers involved.

__Base 8__

Symbols: 0,1,2,3,4,5,6,7 (Or 8,7,2,3,4,5,6,9 to remove ambiguity with some letters).

__Divisibility__. Among numbers 2-7, 2 and 4 evenly divide 10. (This is 2/6, or 33.33 %.)

__Primes__. Except for 2, all potential prime numbers end in 1,3,5, or 7. (50 % possible)

__Digit Addability__. If the digits add up to 7, or a multiple of 7, then the number is evenly divisible by 7.

__Divisor recognizability__. Any number ending in 0 is divisible by 10, and is divisible by 4 if it ends in 0 or 4. If the number is even it is divisible by 2, It is divisible by 7 if the digits add up to 7.

Numbers 3,5,and 6 cannot easily be recognized as even divisors. (62.5 % recognizable)

__Octimal expression of fractions__. ½ = 0.4, 1/3 = 0.*2525*, ¼ = 0.2, 1/5 = 0.*1463*, 1/6 = 0.1*2525*, 1/7 = 0.*1111*, 1/10 = 0.1, 1/11 = 0.*0707*, 1/12 = 0.0*6314*, 1/13 = 0.*0564272135*, 1/14 = 0.0*5252*, 1/15 = 0.*0473*, 1/16 = 0.0*444*, 1/17 = 0.*0421*, 1/20 = 0.04. (4 out of 15 non-repeating (26.67 %))

Power Sequences (cyclic). 2: 0 1 4 1 0 1 4 1 0. 3: 0 1 0 3 0 5 0 7 0. 4. 0 1 0 1 0 1 0 1 0.

Idempotent: 0,1. Nilpotent: 0,4.

Groups: 1,3,5,7 form a V4 group under multiplication, with 1 as the identity element. This contains 3 C2 groups under multiplication, with 1 as the identity element 1,3; 1,5; and 1,7. (25 % of multiplication operations are part of a group.)

__Advantages:__

Superior to Base 10 in 1 measurable characteristic. (Divisibility)

Easily converts from and to Binary for electronic encoding.

Could allow for conversion to more non-ambiguous symbols with respect to written language symbols.

__Disadvantages:__

Inferior to Base 10 in 3 measurable characteristics. (Divisor recognizability, Non-repeating fractional expression, Possible prime registers greater.)

Fewer symbols makes numbers longer, especially when many registers involved.

__Base 12__

Symbols: 0,1,2,3,4,5,6,7,8,9,X,E (Some use an inverted 2 and reversed 3 for X and E, others use A and B respectively).

__Divisibility__. Among numbers 2-E, 2,3,4 and 6 evenly divide 10. (This is 4/10, or 40 %. )

__Primes__. Except for 2 and 3, all potential prime numbers end in 1,5,7 or E. (33.33 % possible)

__Digit Addability__. If the digits add up to E, or a multiple of E, then the number is evenly divisible by E.

__Divisor recognizability__. Any number ending in 0 is divisible by 10, and is divisible by 6 if it ends in in 0 or 6. If the number is even it is divisible by 2, it is divisible by 3 if it ends in 0,3,6,9, and it is divisible by 4 if it ends in 0,4,8. It is divisible by E if the digits add up to E. It is evenly divisible by 8 if it is one of 18 numbers between 1 and 100, where the pattern is an even number followed by an ending number of 0 or 8, or an odd number followed by an ending number of 4. (…20,28,34,…). It is evenly divisible by 9 if it is one of 16 numbers between 1 and 100, where the pattern is 3n followed by 0 or 9, 3n+1 followed by 6, or 3n+2 followed by 3. (….30,39,46,53,60,69,76,83,…)

Numbers 5,7, and X cannot easily be recognized as even divisors. (75 % recognizable)

__Duodecimal expression of fractions__. ½ = 0.6, 1/3 = 0.4, ¼ = 0.3, 1/5 = 0.*2497*, 1/6 = 0.2, 1/7 = 0.*186X35*, 1/8 = 0.16, 1/9 = 0.14, 1/X = 0.*12497*, 1/E = 0.*1111*, 1/10 = 0.1, 1/11 = 0.*0E0E*, 1/12 = 0.0*X35186*, 1/13 = 0.0*9724*, 1/14 = 0.09. (8 out of 15 non-repeating (53.33 %))

Power Sequences (cyclic). 2: 0 1 4 9 4 1 0 1 4 9 4 1 0. 3: 0 1 8 3 4 5 0 7 8 9 4 E 0.

4. 0 1 4 9 4 1 0 1 4 9 4 1 0.

Idempotent: 0,1,4,9. Nilpotent: 0,6.

Groups: 1,5,7,E form a V4 group under multiplication, with 1 as the identity element. This contains the 3 C2 groups under multiplication 1,5; 1,7; and 1,E also with 1 as identity element. 4,8 form a C2 group under multiplication, with 4 as the identity element. 9,3 form a C2 group under multiplication with 9 as the identity element.. (16.67 % of multiplication operations are part of a group.)

__Advantages:__

Superior to Base 10 in 3 measurable characteristics. (Divisibility, Fractional expression greater, Prime registers less.)

More symbols per register means shorter number lengths, especially as registers increase..

__Disadvantages:__

Inferior to base 10 in 1 measurable characteristic. (Divisor recognition.)

Requires creation of 2 additional symbols..

__Base 16__

Symbols: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

__Divisibility__. Among numbers 2-F, 2,4 and 8 evenly divide 10. (This is 3/14, or 21.42 %. )

__Primes__. Except for 2, all potential prime numbers end in 1,3,5,7,9,B,D, or F. (50 % possible)

__Digit Addability__. If the digits add up to F, or a multiple of F, then the number is evenly divisible by F. Since F is 3 * 5, it is also true that a number is evenly divisible by 3 if the digits add up to 3 or a multiple of 3, and if this number is even, it is also evenly divisible by 6. And if the digits add up to 5 or a multiple of 5, it is evenly divisible by 5, and if this number is also even, it is evenly divisible by A.

__Divisor recognizability__. Any number ending in 0 is divisible by 10, and is divisible by 8 if it ends in in 0 or 8. If the number is even it is divisible by 2, and is evenly divisible by 4 if it ends in 0,4,8, or C. Any number is divisible by 3,5,6,A, or F if the digits add up as described above. It is evenly divisible by C if it follows the pattern of 3n followed by an end register number of 0 or C, 3n+1 followed by 8, or 3n+2 followed by 4. (…30,3C,48,54,60,6C,78,84,…)

The numbers 7,9,B,D, and E cannot easily be recognized as even divisors. (68.75 % recognizable)

__Hexadecimal expression of fractions__. ½ = 0.8, 1/3 = 0.*5555*, ¼ = 0.4, 1/5 = 0.*3333* 1/6 = 0.2*AAAA*, 1/7 = 0.*249*, 1/8 = 0.2, 1/9 = 0.*1C7*, 1/A = 0.1*9999*, 1/B = 0.*1745D*, 1/C = 0.1*5555*, 1/D = 0.*13B*, 1/E = 0.*1249*, 1/F = 0.*1111*, 1/10 = 0.1. (4 out of 15 non-repeating (26.33 %))

Power Sequences (cyclic). 2: 0 1 4 9 0 9 4 1 0 1 4 9 0 9 4 1 0. 3: 0 1 8 B 0 D 8 7 0 9 8 3 0 5 8 F 0.

4. 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0.

Idempotent: 0,1. Nilpotent: 0,4,8,C.

Groups: 1,3,5,7,9,B,D,F form a C4XC2 group under multiplication, with 1 as the identity element. Within this 1,7,9,F form a V4 group under multiplication, with 1 as the identity element., and containing the 3 C2 groups 1,7; 1,9; and 1,F, all with 1 as the identity element. Also contained within are 2 C4 groups, both with 1 as identity element: 1,7,B,D, and 1,7,3,5. (25 % of multiplication operations are part of a group.)

__Advantages:__

Easily converts from and to Binary for electronic encoding.

Having 6 additional symbols greatly reduces number length, especially with many registers.

__Disadvantages:__

Inferior to Base 10 in all 4 measurable characteristics. (Divisibility, Divisor recognition, Fractional expression less, Prime registers greater.)

6 new symbols have to be added or created. Greater possible confusion with written language symbols.

__Conclusion__.

Adaptation of another number Base over Base 10 is dependent upon which of two characteristics is more desirable.

__Binary Convertibility__

Base 8 or 16 provide easy conversion to and from Binary and each other. The choice of which would then hinge on whether several more symbols and shorter registers is preferred, (Base 16), or the easier route of eliminating a couple symbols and possible symbolic confusion at the expense of somewhat longer registers. (Base 8) Neither has much advantage over Base 10 in ordinary daily use (other than multiplication and division by 2).

__Ease of Common Divisibility__

Base 6 or 12 provide more factors for even division, and overall shorter ‘decimalization’, and are easier to use (once learned) for most everyday computations, in relation to Base 10. The choice would then hinge on eliminating a few symbols and possibly symbolic confusion at the expense of significantly longer registers (Base 6), versus somewhat shorter numbers at the expense of adding 2 more symbols, which would seem preferable. (Comparing percentages of multiplication operations being parts of a group, Base 12 has the lowest of the bases considered, at 16.67 %. This seems to be comparable to its ease of divisibility.)

This is not nor probably ever will be an easy decision to make. The decision depends upon who is making it and their motives behind it, both of which are equally valid. If I were making it, I would choose Base 12, as we are already used to converting to and from the Binary related bases with respect to Base 10, and Base 12 conversion to the Binaries is actually a bit easier. This would however make common everyday math easier for just about everyone. The only problems: How would it be accomplished, and how much would be the initial expense?

__Appendix__

__Base 7__

Symbols: 0,1,2,3,4,5,6. (Or 8,7,2,3,4,5,6)

__Divisibility__. Among numbers 2-6, Nothing evenly divides 10. (This is 0/5, or 0 %. )

__Primes__. Except for 0, all numbers 1-6 can be prime. (85.71 % possible)

__Digit Addability__. If the digits add up to 6, or a multiple of 6, then the number is evenly divisible by 6. Since 6 is 2 * 3, it is also true that a number is evenly divisible by 2 if the digits add up to 2 or a multiple of 2, and if the digits add up to 3 or a multiple of 3, it is evenly divisible by 3.

__Divisor recognizability__. Any number ending in 0 is divisible by 10. Any number is divisible by 2,3, or 6 if the digits add up as described above.

The numbers 4 and 5 cannot easily be recognized as even divisors. (71.43 % recognizable)

__Heptimal expression of fractions__. ½ = 0.*3333*, 1/3 = 0.*2222*, ¼ = 0.*1515*, 1/5 = 0.*1254* 1/6 = 0.*1111*, 1/10 = 0.1, 1/11 = 0.*0606*, 1/12 = 0.*053*, 1/13 = 0.*0462*, 1/14 = 0.*0431162355*, 1/15 = 0.*0404*, 1/16 = 0.*03525631421*, 1/20 = 0.0*3333*, 1/21 = 0.*0316*, 1/22 = 0.*0303*. (14 out of 15 non-repeating (6.67 %))

Power Sequences (cyclic). 2: 0 1 4 2 2 4 1 0. 3: 0 1 1 6 1 6 6 0. 4. 0 1 2 4 4 2 1 0.

Idempotent: 0,1. Nilpotent: 0.

Groups: 1,2,3,4,5,6 form a C6 group under multiplication, with 1 as the identity element. Within this 1,2,4 form a C3 group under multiplication, with 1 as the identity element., and 1,6 form a C2 group with 1 as the identity element. (73.47 % of multiplication operations are part of a group.)

__Base 9__

Symbols: 0,1,2,3,4,5,6,7,8

__Divisibility__. Among numbers 2-8, 3 evenly divides 10. (This is 1/7, or 14.29 %. )

__Primes__. Except for 3, all potential prime numbers end in 1,2,4,5,7, or 8. (66.67 % possible)

__Digit Addability__. If the digits add up to 8, or a multiple of 8, then the number is evenly divisible by 8. Since 8 is 2 * 4, it is also true that a number is evenly divisible by 2 if the digits add up to 2 or a multiple of 2, and if the digits add up to 4 or a multiple of 4, it is evenly divisible by 4.

__Divisor recognizability__. Any number ending in 0 is divisible by 10. A number is evenly divisible by 3 if it ends in 0,3, or 6. Any number is divisible by 2,4, or 8 if the digits add up as described above. It is evenly divisible by 6 if it follows the pattern of 2n followed by an end register number of 0 or 6, or 2n+1 followed by 3. (…20,26,33,40,46,53,…)

The numbers 5 and 7 cannot be easily recognized as even divisors. (77.78 % recognizable)

__Enneimal expression of fractions__. ½ = 0.*4444*, 1/3 = 0.3, ¼ = 0.*2222*, 1/5 = 0.*1717* 1/6 = 0.1*4444*, 1/7 = 0.*125*, 1/8 = 0.*1111*, 1/10 = 0.1, 1/11 = 0.*0808*, 1/12 = 0.*07324*, 1/13 = 0.0*6666*, 1/14 = 0.*062*, 1/15 = 0.*057*, 1/16 = 0.*053*, 1/17 = 0.*0505*. (2 out of 15 non-repeating (13.33 %))

Power Sequences (cyclic). 2: 0 1 4 0 7 7 0 4 1 0. 3: 0 1 8 0 1 8 0 1 8 0 4. 0 1 7 0 4 4 0 7 1 0.

Idempotent: 0,1. Nilpotent: 0,3,6.

Groups: 1,2,4,5,7,8 form a C6 group under multiplication, with 1 as the identity element. Within this 1,4,7 form a C3 group under multiplication, with 1 as the identity element., and 1,8 form a C2 group with 1 as the identity element. (44.44 % of multiplication operations are part of a group.)

__Base 11__

Symbols: 0,1,2,3,4,5,6,7,8,9,X

__Divisibility__. Among numbers 2-X, Nothing evenly divides 10. (This is 0/9, or 0 %.)

__Primes__. Except for 0, all numbers ending 1 through X are possible primes. 90.91 % possible)

__Digit Addability__. If the digits add up to X, or a multiple of X, then the number is evenly divisible by X. Since X is 2 * 5, it is also true that a number is evenly divisible by 2 if the digits add up to 2 or a multiple of 2, and if the digits add up to 5 or a multiple of 5, it is evenly divisible by 5.

__Divisor recognizability__. Any number ending in 0 is divisible by 10. Any number is divisible by 2,5, or X if the digits add up as described above.

The numbers 3,4,6,7,8, and9 cannot be recognized as even divisors. (45.45 % recognizable)

__Hendecimal expression of fractions__. ½ = 0.*5555*, 1/3 = 0.*3737*, ¼ = 0.*2828*, 1/5 = 0.*2222* 1/6 = 0.*1919*, 1/7 = 0.*163*, 1/8 = 0.*1414*, 1/9 = 0.1*2626*, 1/X = 0.*1111*, 1/10 = 0.1, 1/11 = 0.*0X0X*, 1/12 = 0.*093425X17685*, 1/13 = 0.*087*, 1/14 = 0.*0808*, 1/15 = 0.*0762*. (1 out of 15 non-repeating (6.67 %))

Power Sequences (cyclic). 2: 0 1 4 9 5 3 3 5 9 4 1 0. 3: 0 1 8 5 9 4 7 2 6 3 X 0.

4. 0 1 5 4 3 9 9 3 4 5 1 0.

Idempotent: 0,1. Nilpotent: 0.

Groups: 1 through X form a C10 group under multiplication, with 1 as the identity element. Within this 1,X form a C2 group under multiplication, with 1 as the identity element, and 1,3,9,5,4 form a C5 group under multiplication, with 1 as identity element.. (82.64 % of multiplication operations are part of a group.)

__Base 13__

Symbols: 0,1,2,3,4,5,6,7,8,9,A,B,C

__Divisibility__. Among numbers 2-C, Nothing evenly divides 10. (This is 0/11, or 0 %. )

__Primes__. Except for 0, all numbers from 1 to C are potential primes. (92.31 % possible)

__Digit Addability__. If the digits add up to C, or a multiple of C, then the number is evenly divisible by C. Since C is 2 * 2 * 3, it is also true that a number is evenly divisible by 2 if the digits add up to 2 or a multiple of 2, by 3 if the digits add up to 3 or a multiple of 3, by 4 if the digits add up to 4 or a multiple of 4, and by 6, if the digits add up to 6 or a multiple of 6.

__Divisor recognizability__. Any number ending in 0 is divisible by 10. Any number is divisible by 2,3,4,6, or C if the digits add up as described above.

The numbers 5,7,8,9,A, and B cannot easily be recognized as even divisors. (53.85 % recognizable)

__Tridecimal expression of fractions__. ½ = 0.*6666*, 1/3 = 0.*4444*, ¼ = 0.*3333*, 1/5 = 0.*2525* 1/6 = 0.*2222*, 1/7 = 0.*1B1B*, 1/8 = 0.*1818*, 1/9 = 0.*15A*, 1/A = 0.*13B9*, 1/B = 0.*12495B837*, 1/C = 0.*1111*, 1/10 = 0.1, 1/11= 0.*0C0C*, 1/12 = 0.*0B36*, 1/10 = 0.*0A74*. (1 out of 15 non-repeating (6.67 %))

Power Sequences (cyclic). 2: 0 1 4 9 3 C A A C 3 9 4 1 0. 3: 0 1 8 1 C 8 8 5 5 1 C 5 C 0.

4. 0 1 3 3 9 1 9 9 1 9 3 3 1 0.

Idempotent: 0,1. Nilpotent: 0.

Groups: 1 through C form a C12 group under multiplication, with 1 as the identity element. Within this 1,C form a C2 group under multiplication, with 1 as the identity element, 1,3,9 form a C3 group, with 1 as identity element, and 1,3,4,9,A,C form a C6 group, with 1 as the identity element. (85.21 % of multiplication operations are part of a group.)

__Base 14__

Symbols: 0,1,2,3,4,5,6,7,8,9,A,B,C,D

__Divisibility__. Among numbers 2-D, 2 and 7 evenly divide 10. (This is 2/12, or 16.67 %. )

__Primes__. Except for 2 and 7, all potential prime numbers end in 1,3,5,9,B, or D. (42.86 % possible)

__Digit Addability__. If the digits add up to D, or a multiple of D, then the number is evenly divisible by D.

__Divisor recognizability__. Any number ending in 0 is divisible by 10, and is divisible by 7 if it ends in in 0 or 7. If the number is even it is divisible by 2. Any number is divisible by D if the digits add up as described above. It is evenly divisible by 4 if it follows the pattern of 2n followed by an end register number of 0,4,8 or C, or 2n+1 followed by 2,6, or A. (…20,24, 28,2C,32,36,3A,…)

The numbers 3,5,6,8,9,A,B, and C cannot easily be recognized as even divisors. (42.86 % recognizable)

__Tetradecimal expression of fractions__. ½ = 0.7, 1/3 = 0.*4949*, ¼ = 0.37, 1/5 = 0.*2B2B* 1/6 = 0.2*4949*, 1/7 = 0.2, 1/8 = 0.1A7, 1/9 = 0.*17AC63*, 1/A = 0.1*5858*, 1/B = 0.*13B65*, 1/C = 0.12*4949*, 1/D = 0.*1111*, 1/10 = 0.1, 1/11 = 0.*0D0D*, 1/12 = 0.0C37. (6 out of 15 non-repeating (40 %))

Power Sequences (cyclic). 2: 0 1 4 9 2 B 8 7 8 B 2 9 4 1 0. 3: 0 1 8 D 8 D 6 7 8 1 6 1 6 D 0.

4. 0 1 2 B 4 9 8 7 8 9 4 B 2 1 0.

Idempotent: 0,1,7,8. Nilpotent: 0.

Groups: 1,3,5,9,B,D form a C6 group under multiplication, with 1 as the identity element. Within this 1,9,B form a C3 group under multiplication, with 1 as the identity element., and a C2 group 1,D with 1 as the identity element. Also 8,2,4,6,A,C form a C6 group under multiplication with 8 as identity element, containing a C3 group of 8, 2 and 4, and a C2 group of 8,6, both with 8 as identity element.

(36.73 % of multiplication operations are part of a group.)

__Base 15__

Symbols: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E

__Divisibility__. Among numbers 2-E, 3 and 5 evenly divide 10. (This is 2/13, or 15.38 %. )

__Primes__. Except for 3 and 5, all potential prime numbers end in 1,2,4,7,8,B,D, or E. (53.33 % possible)

__Digit Addability__. If the digits add up to E, or a multiple of E, then the number is evenly divisible by E. Since E is 2 * 7, it is also true that a number is evenly divisible by 2 if the digits add up to 2 or a multiple of 2, and is evenly divisible by 7 if the digits add up to 7 or a multiple of 7.

__Divisor recognizability__. Any number ending in 0 is divisible by 10. A number is evenly divisible by 3 if it ends in 0,3,6,9 or C. A number is divisible by 5 if it ends in 0,5 or A. Any number is divisible by 2,7, or E if the digits add up as described above. It is evenly divisible by 6 if it follows the pattern of 2n followed by an end register number of 0,6 or C, or 2n+1 followed by 3 or 9 (…20,26,2C,33,39,…). It is evenly divisible by 9 if it follows the pattern of 3n followed by 0 or 9, 3n+1 followed by 3 or C, or 3n+2 followed by 6. (…30,39,43,4C,56,60,69,73,7C,86,…). A number is divisible by A if it follows the pattern of 2n followed by 0 or A, or 2n+1 followed by 5 (…20,2A,35,40,4A,55,…).

The numbers 4,8,B,C, and D cannot be easily recognized as even divisors. (66.67 % recognizable)

__Pentadecimal expression of fractions__. ½ = 0.*7777*, 1/3 = 0.5, ¼ = 0.*3B3B*, 1/5 = 0.3 1/6 = 0.2*7777*, 1/7 = 0.*2222*, 1/8 = 0.*1D1D*, 1/9 = 0.1A, 1/A = 0.1*7777*, 1/B = 0.*156C4*, 1/C = 0.1*3B3B*, 1/D = 0.*124936DAC5B8*, 1/E = 0.*1111*, 1/10 = 0.1, 1/11 = 0.*0E0E*. (4 out of 15 non-repeating (26.67 %))

Power Sequences (cyclic). 2: 0 1 4 9 1 A 6 4 4 6 A 1 9 4 1 0. 3: 0 1 8 C 4 5 6 D 2 9 A B 3 7 E 0.

4. 0 1 1 6 1 5 6 1 1 6 A 1 6 1 1 0.

Idempotent: 0,1,6,A. Nilpotent: 0.

Groups: 1,2,4,7,8,B,D,E form a C4XC2 group under multiplication, with 1 as the identity element. Within this 1,4,B,E form a V4 group under multiplication, with 1 as the identity element., and containing the 3 C2 groups 1,4; 1,B; and 1,E, all with 1 as the identity element. Also within are 2 C4 groups with 1 as the identity element, 1,4,2,8 and 1,4,7,D. There are also a C4 group of 6,9,3,C, with 6 as identity element, containing C2 group 6,9, with 6 as identity element, and another C2 group of A,5, with A as identity element. (37.33 % of multiplication operations are part of a group.)