NUMBER BASES

On Number Bases

Prelude.

I remember sitting in 7th 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 3rd 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 = 0.08333, 1/13 = 0.076923, 1/14 = 0.0714258, 1/15 = 0.06666, 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 = 0.0333, 1/15 = 0.0313452421, 1/20 = 0.03, 1/21 = 0.02435, 1/22 = 0.02323, 1/23 = 0.02222, 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 = 0.12525, 1/7 = 0.1111, 1/10 = 0.1, 1/11 = 0.0707, 1/12 = 0.06314, 1/13 = 0.0564272135, 1/14 = 0.05252, 1/15 = 0.0473, 1/16 = 0.0444, 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 = 0.0X35186, 1/13 = 0.09724, 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 = 0.2AAAA, 1/7 = 0.249, 1/8 = 0.2, 1/9 = 0.1C7, 1/A = 0.19999, 1/B = 0.1745D, 1/C = 0.15555, 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 = 0.03333, 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 = 0.14444, 1/7 = 0.125, 1/8 = 0.1111, 1/10 = 0.1, 1/11 = 0.0808, 1/12 = 0.07324, 1/13 = 0.06666, 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 = 0.12626, 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 = 0.24949, 1/7 = 0.2, 1/8 = 0.1A7, 1/9 = 0.17AC63, 1/A = 0.15858, 1/B = 0.13B65, 1/C = 0.124949, 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 = 0.27777, 1/7 = 0.2222, 1/8 = 0.1D1D, 1/9 = 0.1A, 1/A = 0.17777, 1/B = 0.156C4, 1/C = 0.13B3B, 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.)

2. GRAVITY

 

Gravity (Latin for Heaviness) is the central controlling factor of concern when discussing occupying the rest of the solar system (or pretty much anywhere else in space). I have found over the years that most people do not understand it fully, (which is partly the fault of the media,) and even the experts cannot come to a complete agreement as to exactly what it is. (I am also not claiming I understand it fully.)

It can however be described and illustrated in various ways. One misconception, after seeing astronauts floating around on space walks and in space habitats, is that there is no gravity in space. That is not true. What is happening is that the speed at which the astronauts are traveling is fast enough to to keep them from spiraling back into the earth’s surface, instead they go around the earth, and the centripetal force of the earth’s gravity is balanced by the “centrifugal force” of orbital motion. This same balance is what keeps the various bodies of the solar system orbiting each other in their respective locations.

In fact, gravity can be viewed as a field, following the same type of inverse square law that electrical fields can exhibit; that is, if one is twice as far from the center of gravitational attraction, the force will be one quarter as great, or, if one is twice as close, the force will be four times greater. One result of this is that the force on a planet close to the sun is much greater than on a planet far from it, and consequently a near planet must travel much faster than its distant neighbor to stay in orbit.

[For the math minded: this is Kepler’s 3rd law, simplified, where S{m} • p^2 = a^3, where S{m} is the sum of the masses, p is the period, and a is the average distance between the bodies. Left out are conversion constants needed to make the units in various measurement systems ‘line up’. If the mass of one of the objects is negligible compared to the other, such as a space vehicle compared to the earth, its mass can be ignored]

In theory, all gravitational fields extend to infinity. To illustrate, imagine a universe of whatever size you wish (larger than atomic size), containing only one hydrogen atom. This atom constitutes the sole gravitational source in this universe, and its field extends to the edge of the universe in all directions. Obviously this is a very weak gravitational field. Now imagine introducing a second hydrogen atom somewhere far away in the same universe, but at rest with respect to the first atom. (Absolute rest and motion are impossible to determine, as Einstein postulated.) This alters the universal gravitational field system, and sets up an attraction that will eventually result in the two atoms coming together, barring interference from some other source. (At least as close as possible until other forces, such as electromagnetism, take over.) With such weak attraction, depending on the size of your universe, it could take a very very long time for this to occur.

What happens in the real universe, and therefore in the solar system, is there are very many gravitational sources, so there exist points, really curved surfaces, where one gravitational field loses influence compared to another, so it is considered that the first field then ends at that point. For example: the earth has a gravitational sphere (“sphere” is used in the general sense of an area of influence, not necessarily having the shape of a perfect sphere.) in which all objects orbiting the earth exist, such as the moon. However, if one gets close enough to the moon, its gravity dominates, and it has its own sphere. Go far enough away from the earth, and you enter the sun’s gravitational sphere. The size of the various spheres is determined by the mass of the objects producing them and how far they are from other objects. Example: Even though Neptune is about 1/19 the mass of Jupiter, its sphere of gravitational influence is larger than Jupiter’s because of its much greater distance from the sun and other massive planets.

Based upon Einstein’s (and others) work, the 3 dimensions of space and 1 dimension of time are combined as 4 dimensions of space-time, called a Continuum. As gravity increases in the continuum, it has an effect on it, which can be observed in both space and time. Space becomes more and more ‘curved’ (this is curvature in 3 dimensions, not easily visualized), and time becomes slower and slower with respect to the rest of the universe. Or, it has also been said that the presence of great mass causes space to ‘curve’ and time to slow, which in turn cause gravity.

Effects

In considering inhabiting the solar system beyond earth, there is more to take into account than orbital mechanics, relative forces, and continuum effects. Specifically, the health of the human organism. It has been found that prolonged exposure to zero net gravity (weightlessness) causes muscles and bones to weaken, as well as the immune system to shut down, among other things. Humans (and other animals) are adapted to living in earth’s surface gravity, and the complete absence of it has many negative effects. What is not yet known is how much gravity is necessary to prevent these effects. For example, is the gravity of mars, at a little over 1/3 that of earth sufficient? How about the moon, at about 1/6 earth gravity? It may be true that Humans born and raised in differing gravitational fields will be physically different from each other, particularly in dimension and mass.

This brings up the question: how does one define a human? Is a human to be defined based only on physical appearance and biological characteristics? Or should mentality and consciousness be considered? How about moral and ethical standards? Or societal organization and interaction? This becomes an important consideration when one considers that the best and easier way to explore and colonize alternate environments is using robotics designed for these specific environments. If the human brain and spinal column can be excised intact and living, and then introduced into some type of cybernetic body, it would moot much of the health and environmental concerns related to space travel, and the bodies could be made interchangeable depending on conditions and locale. This might also greatly extend the span of human life. And there are likely those who would jump at this chance if it were available, particularly those with debilitating or terminal diseases.

But are they still human enough? It partially depends on the results and how humanity is defined, and largely on what humanity as a whole is willing to accept for survival.

Since, right now, humanity is limited to their current physical bodies, and there is not yet an effective way to control or alter gravity, Gravity, or lack of it, is still the most important and least controlled factor in considering colonization of space and other worlds. As it turns out, in this Solar System the surface gravity of the various planetary bodies and moons can be arranged in groups of similar intensity. For planets with huge atmospheres and no near solid surface, the surface is considered to be at the top of the visible atmosphere. (see the following chart)

[At this point it is convenient to introduce a term coined by others, namely ‘Planemo’, which stands for Planetary Mass Object. This avoids having to distinguish between planets, dwarf planets, moons, and asteroids when discussing ‘body’ characteristics. (as opposed to orbital characteristics) A Planemo is an astronomical body, small enough to not shine by its own light, like stars, which is spherical in overall form due to its gravity being sufficient to pull it into this shape, overriding the atomic structural forces of the materials of which it is made. The upper and lower size limits for planemos depend upon what substances they contain. Liquid and gas bodies, like Jupiter, can be much larger than solid bodies, like Earth (a silicate rock body surrounding an iron core). At the lower end, a planemo made of mostly ices, like water ice, like Enceladus, can be spherical at smaller sizes than silicate rock bodies.]

1. Jupiter.

With a surface gravity of around 2.38 that of Earth, this is the only planemo where most humans would have difficulty due to increased weight. It is an unlikely locale for colonization for other reasons.

2. Neptune, Saturn, Earth, Uranus, Venus.

These range from 1.12 to .87 of Earth’s gravity, essentially the planemos with roughly the same gravity as ours. Aside from Earth, they all are not as easily colonizable, 3 have extensive gas atmospheres, and Venus is way too hot, at least at the surface.

3. Mars, Mercury.

These 2 planemos have almost the same surface gravity, at about .38 Earths. Both are relatively near, in the inner Solar System. Mars is currently the favored target for colonization.

4. Io, Luna (an official name for Earth’s moon – The Moon seems a bit Earth Centric), Ganymede, Titan, Europa, Callisto.

These 6 planemos, all moons of larger planemos, range around 1/6 to 1/8 Earth gravity (.18 to .13). Io and Luna have rocky surfaces, and the rest are covered in deep layers of ice. Titan is a moon of Saturn, and the rest are moons of Jupiter.

These are the primary targets of colonization for gravitational reasons; the rest of the planemos are are less than 10% of Earth’s gravity. A human raised on Earth might not be able to easily distinguish their weight difference among them.

5. Eris, Triton.

Around .08 Earth gravity, Eris is a distant Kuiper belt object, Triton a moon of Neptune.

6. Pluto, Makemake, Haumea, Titania, Oberon.

Around .06 to .04 Earth gravity. The first 3 are distant planemos and the last 2 moons of Uranus.

7. Charon, Ceres, Ariel, Rhea, Dione, Umbriel, Iapetus.

Around .03 to .02 Earth gravity. Charon is a moon of Pluto, Ariel and Umbriel of Uranus, and Rhea, Dione, and Iapetus of Saturn. Ceres is the only planemo in the asteroid belt, and is of interest for other reasons.

8. Tethys, Enceladus, Miranda, Mimas.

Around .015 to .0065 Earth gravity. Miranda is a moon of Uranus, the rest are of Saturn. Mimas is currently the smallest rounded planemo known. Neptune’s moon Proteus is larger, but is not rounded (due to gravity).

 

 

 

 

Planemo Surface Gravity – Solar System Objects

(descending order)

 

Planemo                                    Surface Gravity                                      Weight Compared to Earth

 

Jupiter                                          2.357                                                                       235.7             353.6         471.4

 

Neptune                                         1.121                                                                    112.1             168.2          224.2

Saturn                                             1.064                                                                   106.4             159.6          212.8

Earth                                                1.000                                                                  100.0            150.0           200.0

Uranus                                              0.886                                                                  88.6             132.9           177.2

Venus                                                 0.874                                                                 87.4             131.1           174.8

 

Mars                                                    0.378                                                                37.8             56.7                75.6

Mercury                                              0.377                                                               37.7              56.6               75.4

 

Io                                                            0.183                                                              18.3              27.5              36.6

Luna                                                       0.165                                                             16.5               24.8             33.0

Ganymede                                              0.147                                                             14.7              21.9            29.2

Titan                                                         0.138                                                            13.8               20.6            27.5

Europa                                                      0.134                                                           13.4                20.1          26.8

Callisto                                                      0.126                                                         12.6                18.9            25.2

 

Eris                                                            0.0815                                                          8.15               12.3          16.4

Triton                                                        0.0795                                                          7.95                11.9          15.9

 

Pluto                                                            0.061                                                             6.1                  9.15       12.2

Makeake                                                      0.051                                                             5.1                 7.65        10.2

Haumea                                                       0.0449                                                           4.49               6.75          9.0

Titania                                                          0.0398                                                           3.98               5.97        7.96

Oberon                                                         0.0357                                                            3.57               5.36       7.14

 

Charon                                                           0.0285                                                           2.85                4.28      5.70

Ceres                                                              0.0275                                                            2.75                4.13     5.50

Ariel                                                                0.0275                                                           2.75                4.13     5.50

Rhea                                                                0.0269                                                           2.69                4.04     5.38

Dione                                                               0.0235                                                           2.35                3.53    4.70

Umbriel                                                          0.0234                                                            2.34                3.51    4.68

Iapetus                                                            0.0224                                                            2.24                3.36   4.48

 

Tethys                                                             0.0148                                                            1.48               2.22     2.96

Enceladus                                                      0.0113                                                             1.13              1.70     2.26

Miranda                                                          0.0082                                                             .82                1.23     1.64

Mimas                                                             0.0065                                                              .65                 .98       1.3

1. Y?

Y leave the protection of the earth to live in harsher more demanding environments in the rest of the solar system?  One could also ask:  Y not?  But that would probably not satisfy very many.

This is a question that has been asked over the past several decades, particularly with the added comment that not all the problems have yet been solved here on earth.  Many possible responses to this have been offered,  and they all have some degree of validity.  However, at the most basic level there is a simple equation that sums up the entire problem for the entire earth, which is:  Q = R/P, where Q = the Quality of life, R = the Resources available, and P = total Population.  While it is certainly possible to discover new resources on earth, use them more efficiently, conserve, and to some degree recycle them, there is a limit to what the earth can supply.  And more extensive development of resources here will come at an increased degradation and destruction of the natural environment that has developed over eons, on land, at sea, or within the earth.

On the other hand, the world population keeps increasing, so far at an ever increasing rate.  There does not seem to be much recognition of this as a problem by most individuals I have met, and no one seems to want to do anything about it.  To return to a population level that would allow everyone on the planet a decent quality of life by modern standards, requires substantial population reduction from present levels.  How is that to be done?  A number of wars could probably be easily started, particularly with atomic weapons,  but that would certainly mess up the environment.  Or medical procedures and treatments could be withheld, or even basic food – but that would be very unpopular, and probably lead to social instability and riots.  And any talk of birth control, if not actively resisted, results in the view that it is OK for someone else, and would meet with intense reaction if enforced by the authorities.

So there is only really one option left: expansion. The frontiers left on the earth are constantly shrinking, and would eventually lead to colonization of the ocean surface, (not necessarily a good idea, considering the ocean temperature problems we are having even now.) and possibly in artificial subterranean environments.

What is left is moving outward, into the solar system, where the resources are vastly greater. (And yes, a time could come when even they were used up. But comparing the volume of the earth with the volume of the usable solar system, puts any such possibility millions or more years into the future.) It has its dangers, but the rewards are vastly greater from many points of view.  For one thing, it makes the extinction of humanity from a single natural or human cause much more difficult.  Such a move would also allow for greater human freedom by allowing societies of different standards to exist on different worlds or in space. No one need remain in the tyranny of a particular social/ political/ economic/ religious system.