### Post by on Aug 15, 2015 17:36:31 GMT -8

Introduction

What are the basics of choosing a real number base system? There are bases for complex numbers... but ugh they are lengthier and much slower to calculate in than typical integer bases; it's nicer to just write a + b * i for complex numbers, where a and b are real coefficients. Integer bases, of course, can still deal with any fractions as well, whether they recur digits endlessly or not. They can also approximate irrational numbers, so they are still versatile and useful enough.

Arguments

Time and time again, researching what makes a base convenient to calculate with turns up its prime factors. The smallest base with three prime factors is trigesimal (base thirty), followed by duoquadragesimal (base forty-two). The current system (decimal) is a lot smaller than these....

How many symbols for quantities is too many? In other words, you need 'em for 11, 13, 17, 21, and so on for big bases; at what point is this just a mental mess? It can also mean spending days creating the multiplication table, let alone attempting to study and memorize its patterns. Let's assume thirty is too high; its multiplication table has 900 entries, while the one for decimal has only 100. Making sure the base has the first two prime factors (2 and 3) is thus the objective (decimal has the primes 2 and 5 instead). This limits the choices to 6, 12, 18, and 24. We notice that octodecimal does fall in this list (base eighteen), so can it cut the mustard? The number eighteen itself has six factors (1, 2, 3, 6, 9, and 18 in decimal); eggs have been sold in packages of 6*3 (eighteen), and 2*9 is also possible; the smallest number more divisible than 12 and 18 is 24 (8 factors). Even base 24's multiplication table has 576 numbers... I have made it, but not without errors. 24 is doable, although teaching it to young lings may just cause confusion among some symbols, such as 6, b, and G.

Octodecimal representations

Custom symbols: 0, 1, ...., 8, 9 as usual, then t, r, U, H, C, A, J, and N for ten through seventeen.

Fractions

1/2 = 0.9

1/3 = 0.6

2/3 = 0.U

1/4 = 0.49

3/4 = 0.H9

1/6 = 0.3

5/6 = 0.A // Because of the primes 2 and 3 in eighteen, all the common fractions terminate, unlike in decimal, binary, octal, vigesimal, or hexadecimal.

Let's try translating 0.7777777.... in decimal into octodecimal. It's only a floating number (no integer component), so we multiply by eighteen and subtract off the integer part (which becomes the first Od. digit after its floating point). 0.7777... * 18 = 12.6 + 1.26 + 0.126 + 0.0126 + .... = 13.86 + 0.1386 + ... = 13.9986 + 0.00139986 + .... = 14.0000, and the 14 changes to C. The recurring stuff and floating part are already gone, so the octodecimal representation is 0.C, which means 14/18 in decimal. This reduces to 7/9.

Ok, what was I talking about with "cold facts"? Even with a good base, some problems just get ugly. Most of the integers of size 150 or so have primes other than 2 or 3, so their fractions (reciprocals) still recur (let alone larger ones). Among the bases 6, 12, 18, and 24, I have decided octodecimal has the most interesting recurring properties:

1/5^2, or 1/17, is 0.0UN50UN50UN5.... (I admit this is 1/25 in decimal, which is 0.04)

1/7 = 0.2t52t52t52t52t52t52t52...

1/7^2, or 1/2H, is 0.06r06r06r06r06r06r06r...

1/7^3 = 1/111 = 0.00N00N00N00N00N00...

1/H = one thirteenth = 0.16Jr16Jr16Jr16Jr16Jr...

Integers

Unlike in decimal, where 10 = 2*5, 2*5 is t in octodecimal and 10 = 2*3*3 = 6*3 = 2*9, that's kinda useful.

Note that adding a zero in octodecimal means multiplying by eighteen, not ten. 100 in Od. is NOT 180 in decimal, 180(decimal) is 10 * t = t0 in octodecimal. Counting above 95 in octodecimal looks like this: 96, 97, 98, 99, 9t, 9r, 9U, 9H, 9C, 9A, 9J, 9N, t0, t1, t2 (nope, not near 100 yet), ...., JH, JC, JA, JJ, JN, N0, N1, N2, N3, .... NJ, NN, 100.

6*8*9 = 6 * 40 because 9 is half of eighteen; 9 * x = 10 * x / 2

6*30 = 100 so 6*40 = 100 + 60 = 160(Od.); this number is 432 in decimal. 9 occupies the role that 5 has for decimal (and that 6 has for dozenal, base twelve).

The number of the BEAST, 666 in decimal, is 210(Od.), easy one to remember.

Here's an interesting bit: 7^3, which is 343 in decimal, is 111 in octodecimal.

Thus, in Od.: 7^4 = 777, 7^6 = 12321, and 7^9 = 1367631. Octodecimal can make numbers appear smaller than they are for people who are used to decimal. That 7^9, 1367631, is 40353607 in decimal.

1JH00000(Od.) = 1180980000(Dec.). The effect is still more subtle than with a bigger base such as 30, however. For those huge numbers, does scientific notation work in Od.? Oh yeah. 6.CU1rUA * 10^47 is roughly a googol when written in Od. Power towers of eighteens are pretty strong as well. A googolplexian (googolduplex) is about 10^(10^(10^47.t9r))(Od.). On the other hand, the octodecimal equivalent of a googolduplex, 10^(10^(10^100))(Od.) translates to 18^(18^(18^324)) in decimal, which is roughly 10^(10^(10^406.807)). Octodecimal should be more useful for expressing huge or tiny orders of magnitude.

Irrational Numbers

Although there are bases in which *some* irrational numbers have finite representations (such as Pi being 10 in base Pi), almost all of them still run on endlessly with no patterns in these bases. Even just counting in a non-integer base is also pretty annoying, let alone multiplying or dividing, because "10" and "15" and "20" and so on don't represent whole numbers. Octodecimal yields the following initial digits:

Golden ratio = 1.r246H486rH3Ut5NCUA6Utt....

e = 2.UJH0668016t8AC9044rHrHNtU164...

e^(1/e) = 1.801582J1H66tt85U4...

Pi = 3.29AHCN0J771811t...

Sqrt(3) = 1.H335HC0A0U5t108...

What are the basics of choosing a real number base system? There are bases for complex numbers... but ugh they are lengthier and much slower to calculate in than typical integer bases; it's nicer to just write a + b * i for complex numbers, where a and b are real coefficients. Integer bases, of course, can still deal with any fractions as well, whether they recur digits endlessly or not. They can also approximate irrational numbers, so they are still versatile and useful enough.

Arguments

Time and time again, researching what makes a base convenient to calculate with turns up its prime factors. The smallest base with three prime factors is trigesimal (base thirty), followed by duoquadragesimal (base forty-two). The current system (decimal) is a lot smaller than these....

How many symbols for quantities is too many? In other words, you need 'em for 11, 13, 17, 21, and so on for big bases; at what point is this just a mental mess? It can also mean spending days creating the multiplication table, let alone attempting to study and memorize its patterns. Let's assume thirty is too high; its multiplication table has 900 entries, while the one for decimal has only 100. Making sure the base has the first two prime factors (2 and 3) is thus the objective (decimal has the primes 2 and 5 instead). This limits the choices to 6, 12, 18, and 24. We notice that octodecimal does fall in this list (base eighteen), so can it cut the mustard? The number eighteen itself has six factors (1, 2, 3, 6, 9, and 18 in decimal); eggs have been sold in packages of 6*3 (eighteen), and 2*9 is also possible; the smallest number more divisible than 12 and 18 is 24 (8 factors). Even base 24's multiplication table has 576 numbers... I have made it, but not without errors. 24 is doable, although teaching it to young lings may just cause confusion among some symbols, such as 6, b, and G.

Octodecimal representations

Custom symbols: 0, 1, ...., 8, 9 as usual, then t, r, U, H, C, A, J, and N for ten through seventeen.

Fractions

1/2 = 0.9

1/3 = 0.6

2/3 = 0.U

1/4 = 0.49

3/4 = 0.H9

1/6 = 0.3

5/6 = 0.A // Because of the primes 2 and 3 in eighteen, all the common fractions terminate, unlike in decimal, binary, octal, vigesimal, or hexadecimal.

Let's try translating 0.7777777.... in decimal into octodecimal. It's only a floating number (no integer component), so we multiply by eighteen and subtract off the integer part (which becomes the first Od. digit after its floating point). 0.7777... * 18 = 12.6 + 1.26 + 0.126 + 0.0126 + .... = 13.86 + 0.1386 + ... = 13.9986 + 0.00139986 + .... = 14.0000, and the 14 changes to C. The recurring stuff and floating part are already gone, so the octodecimal representation is 0.C, which means 14/18 in decimal. This reduces to 7/9.

Ok, what was I talking about with "cold facts"? Even with a good base, some problems just get ugly. Most of the integers of size 150 or so have primes other than 2 or 3, so their fractions (reciprocals) still recur (let alone larger ones). Among the bases 6, 12, 18, and 24, I have decided octodecimal has the most interesting recurring properties:

1/5^2, or 1/17, is 0.0UN50UN50UN5.... (I admit this is 1/25 in decimal, which is 0.04)

1/7 = 0.2t52t52t52t52t52t52t52...

1/7^2, or 1/2H, is 0.06r06r06r06r06r06r06r...

1/7^3 = 1/111 = 0.00N00N00N00N00N00...

1/H = one thirteenth = 0.16Jr16Jr16Jr16Jr16Jr...

Integers

Unlike in decimal, where 10 = 2*5, 2*5 is t in octodecimal and 10 = 2*3*3 = 6*3 = 2*9, that's kinda useful.

Note that adding a zero in octodecimal means multiplying by eighteen, not ten. 100 in Od. is NOT 180 in decimal, 180(decimal) is 10 * t = t0 in octodecimal. Counting above 95 in octodecimal looks like this: 96, 97, 98, 99, 9t, 9r, 9U, 9H, 9C, 9A, 9J, 9N, t0, t1, t2 (nope, not near 100 yet), ...., JH, JC, JA, JJ, JN, N0, N1, N2, N3, .... NJ, NN, 100.

6*8*9 = 6 * 40 because 9 is half of eighteen; 9 * x = 10 * x / 2

6*30 = 100 so 6*40 = 100 + 60 = 160(Od.); this number is 432 in decimal. 9 occupies the role that 5 has for decimal (and that 6 has for dozenal, base twelve).

The number of the BEAST, 666 in decimal, is 210(Od.), easy one to remember.

Here's an interesting bit: 7^3, which is 343 in decimal, is 111 in octodecimal.

Thus, in Od.: 7^4 = 777, 7^6 = 12321, and 7^9 = 1367631. Octodecimal can make numbers appear smaller than they are for people who are used to decimal. That 7^9, 1367631, is 40353607 in decimal.

1JH00000(Od.) = 1180980000(Dec.). The effect is still more subtle than with a bigger base such as 30, however. For those huge numbers, does scientific notation work in Od.? Oh yeah. 6.CU1rUA * 10^47 is roughly a googol when written in Od. Power towers of eighteens are pretty strong as well. A googolplexian (googolduplex) is about 10^(10^(10^47.t9r))(Od.). On the other hand, the octodecimal equivalent of a googolduplex, 10^(10^(10^100))(Od.) translates to 18^(18^(18^324)) in decimal, which is roughly 10^(10^(10^406.807)). Octodecimal should be more useful for expressing huge or tiny orders of magnitude.

Irrational Numbers

Although there are bases in which *some* irrational numbers have finite representations (such as Pi being 10 in base Pi), almost all of them still run on endlessly with no patterns in these bases. Even just counting in a non-integer base is also pretty annoying, let alone multiplying or dividing, because "10" and "15" and "20" and so on don't represent whole numbers. Octodecimal yields the following initial digits:

Golden ratio = 1.r246H486rH3Ut5NCUA6Utt....

e = 2.UJH0668016t8AC9044rHrHNtU164...

e^(1/e) = 1.801582J1H66tt85U4...

Pi = 3.29AHCN0J771811t...

Sqrt(3) = 1.H335HC0A0U5t108...