In the existing number system (base 10 for starters), there are additive and multiplicative digital roots of positive integers. The additive digital root is obtained by adding the digits of the number, and then adding the digits of the result, and continuing this process until a single digit number is obtained. This number is the additive digital root of the original number. For example, from the number 1278, we add the digits to get 18, and then add these digits to get 9. So 9 is the additive digital root of 1278. Similarly, a multiplicative digital root of 1278 is 1x2x7x8=112 which reduces to 2. So 2 is the multiplicative digital root of 1278.
In ENS, the additive roots always reduce to a number >1. However, due to the existence of a zero digit in ENS, multiplicative inverses often reduce to zero. In ANS, on the otherhand, there is no zero digit. Therefore all numbers must also reduce to multiplicative roots > 1. Also the additive roots can differ between ANS and ENS. For example, the number 100 in ENS is 9A in ANS. (Remember that A is the digit ten in ANS). The additive root of 100 in ENS is 1 whereas in ANS the digit sum of 9A is 19 which reduces to A as the ANS additive root of 9A.
As mentioned, multiplicative roots in ENS reduce to zero if the number has a zero digit. Therefore a number such as 407 has an ENS multiplicative root of 0. In ANS, 407 is 3A7, whose digits multiply to give 210 which is 1AA in ANS which reduces to 9A, to 8A, to 7A and so on to 1A which has the ANS multiplicative root of A.
The number of steps required to reduce a number to its additive or multiplicative roots also can vary between ENS and ANS as seen.
In base A=(ANS base ten), for example, 11 reduces to 2, and this is the first number that requires 2 steps. The first number requiring 3 steps is 1A, since it reduces first to 1+A=11 which reduces to 2. Similarly for other numbers and bases.
The following table is the number where a certain #steps first occurs for
additive digital roots using ANS in bases 2 to 10. To begin with, the values
are in the ENS system in base 10.
#steps---> 1 2 3 4 5 6 7
Base ---------------------------------
2 1 3 4 6 14 254 ...
3 1 4 6 12 120 ... ...
4 1 5 8 20 1364 ... ...
5 1 6 10 30 19530 ... ...
6 1 7 12 42 ... ... ...
7 1 8 14 56 ... ... ...
8 1 9 16 72 ... ... ...
9 1 10 18 90 ... ... ...
10 1 11 20 110 ... ... ...
If b is the base, then:
2 steps occurs first at b+1 = ANS value of 11 (base b)
3 steps occurs first at b+b = ANS value of 1b (base b)
4 steps occurs first at b(b+1) = b^2+b = ANS value of bb (base b)
Beyond this, it seems difficult to determine the pattern if you look at these
numbers only in ENS base 10. However if you convert these numbers to the
selected bases using ANS, the pattern becomes clear:
#steps-> 1 2 3 4 5 6 7
Base --------------------------------------
2 1 11 12 22 222 2222222 ...
3 1 11 13 33 3333 ... ...
4 1 11 14 44 44444 ... ...
5 1 11 15 55 555555 ... ...
6 1 11 16 66 ... ... ...
7 1 11 17 77 ... ... ...
8 1 11 18 88 ... ... ...
9 1 11 19 99 ... ... ...
A 1 11 1A AA ... ... ...
The values marked ? are quite large. From the above values, notice that for a selected base, the value depends on the value to its left. The sum of the digits equals the number to the left. For example, in base 2 under 6 steps, the number is 2222222. The sum of the digits is 14 in base 10, and this is 222 in base 2. Similarly for all other numbers. So the missing values are easy to determine. For example, the value for base 2 and 7 steps, must have a digit sum equal to 2222222(base 2). This number turns out to be the large value of (2127-1)*2 since this is the number 222...222 in base 2 and it has 127 digits equal to 2 and therefore the sum in base 10 is 254. Similarly for all other values in the table. Therefore the additive table above is completely determined for all bases in ANS.
The following table similar to the previous ones, and is the number where
a certain #steps first occurs for multiplicative digital roots using ANS in
bases 2 to 10. Once again we first examine the values in base 10 of the ENS
system.
#steps-> 1 2 3 4 5 6 7 8 9 10
Base --------------------------------------------
2 1 3 +6 x x x x x x x
3 1 4 8 +12 27 39 768 ? ? ?
4 1 5 11 +16 +20 64 84 ? ? ?
5 1 6 13 +20 +25 +30 70 155 ? ?
6 1 7 16 +24 +30 +36 +42 174 1043 ?
7 1 8 18 27 +35 +42 +49 +56 133 343
8 1 9 21 31 +40 +48 +56 +64 +72 182
9 1 10 23 35 +45 +54 +63 +72 +81 +90
10 1 11 26 39 +50 +60 +70 +80 +90 +100
There are some interesting patterns. There is an area in the middle of this
table within which the number is equal to #steps * base. However, this is
isolated to a certain part of the table as seen above (such numbers are
flagged with +).
The values flagged with x, cannot occur. For base 2, all digits must be
either 1 or 2. Therefore the product must be a power of 2 which is
11....112 in ANS (base 2). Therefore the resultant product (and hence the
multiplicative root) will always be 2.
Therefore, the maximum number of steps possible is 3 for base 2.
To examine further, convert the decimal ENS values above to the equivalent ANS values in the selected base, and we get the following table (extended to base 16) :
#steps-> 1 2 3 4 5 6 7 8 9 10
Base ------------------------------------------------------
2 1 11 +22 x x x x x x x
3 1 11 22 +33 223 333 223333 ? ? ?
4 1 11 23 +34 +44 334 444 ? ? ?
5 1 11 23 +35 +45 +55 235 555 ? ?
6 1 11 24 +36 +46 +56 +66 446 4455 ?
7 1 11 24 36 +47 +57 +67 +77 247 667
8 1 11 25 37 +48 +58 +68 +78 +88 266
9 1 11 25 38 +49 +59 +69 +79 +89 +99
A 1 11 26 39 +4A +5A +6A +7A +8A +9A
11 1 11 26 3A +4.11.+5.11 +6.11 +7.11 +8.11 +9.11
12 1 11 27 3.11 +4.12 +5.12 +6.12 +7.12 +8.12 +9.12
13 1 11 27 3.11 +4.13 +5.13 +6.13 +7.13 +8.13 +9.13
14 1 11 28 3.12 +4.14 +5.14 +6.14 +7.14 +8.14 +9.14
15 1 11 28 3.13 +4.15 +5.15 +6.15 +7.15 +8.15 +9.15
16 1 11 29 3.14 +4.16 +5.16 +6.16 +7.16 +8.16 +9.16
There is some order to the above table. The values flagged with + are once again, equal to the product of the #steps times the base. There are also several numbers (but not all) whose digit product equals the number to its immediate left (similar to the additive table). Notice also that the numbers 36 (bases 6 & 7) and 3.11 (bases 12 & 13) are repeated in column 4. This suggests a periodic pattern with length of 6 in column 4. But why is this column slightly irregular compared to column 3 which has consistent pairs of numbers? And why the irregular numbers near the top of the columns? Other than these patterns, i have not been able to determine a general pattern to all elements of the table, as was found for the additive table. Also there are elements of the table that i have not found yet (marked ?). Can anyone help with this?
Lots of new territory to explore.
Enjoy.