Lets say we represent .4389
| 3n | value | Cumulative Total | Balance | Percentage Error | ||
| -1 | 1/3 | .33333333 | + | .333333 | -.105567 | |
| -2 | 1/9 | .11111111 | + | .444444 | +.005544 | |
| -3 | 1/27 | .03703703 | 0 | .444444 | ||
| -4 | 1/81 | .01234567 | 0 | .444444 | ||
| -5 | 1/243 | .00411226 | - | .440329 | +.001429 | |
| -6 | 1/729 | .00137174 | - | .438957 | +.000057 | |
| -7 | 1/2187 | .00045724 | 0 | .438957 | ||
| -8 | 1/6561 | .00015241 | 0 | .438957 | ||
| -9 | 1/19683 | .00005080 | - | .4389062 | +.0000062 |
.4389 = 0.++00--00-
.25483 =
| 3n | value | 1/6 value | Cumulative Total | Balance | Percentage Error | ||
| -1 | 1/3 | .33333333 | .0555555 | + | .333333 | -.078503 | 31% |
| -2 | 1/9 | .11111111 | .0185185 | - | .222222 | +.032608 | 12.79% |
| -3 | 1/27 | .03703703 | .0061728 | + | .259259 | -.004429 | 1.73% |
| -4 | 1/81 | .01234567 | .00205761 | 0 | |||
| -5 | 1/243 | .00411226 | .00068587 | - | .255144 | -.000314 | .12% |
| -6 | 1/729 | .00137174 | .00022862 | 0 | |||
| -7 | 1/2187 | .00045724 | .000076207 | - | .2546868 | +.0001432 | .056% |
| -8 | 1/6561 | .00015241 | .000025402 | + | .25483921 | -.00000921 | .0036% |
| -9 | 1/19683 | .00005080 | .000008467 |
.25483 = 0.+-+0-0-+
Lets say the number 0.1415936
| 3n | value | Cumulative Total | Balance | Percentage Error | ||
| -1 | 1/3 | .33333333 | 0 | 0 | -.1415936 | 100% |
| -2 | 1/9 | .11111111 | + | .11111111 | -.0304825 | 21.52% |
| -3 | 1/27 | .03703703 | + | .14814814 | +.0065545 | 4.6% |
| -4 | 1/81 | .01234567 | - | .13580246 | -.0057911 | 4.08% |
| -5 | 1/243 | .00411226 | + | .13991769 | -.0016759 | 1.18% |
| -6 | 1/729 | .00137174 | + | .14128943 | -.0003041 | 0.21% |
| -7 | 1/2187 | .00045724 | + | .14174668 | +.00015308 | 0.10% |
| -8 | 1/6561 | .00015241 | - | .14159426 | -.00000066 | <.0001% |
| -9 | 1/19683 | .00005080 | 0 | .14159426 | -.00000066 |
Therefore the value of .1415936 = 0.0++-+++-0
Lets say the number is 0.7182818
This number is larger than .5, so we shall find
the 1s complement of it. (subtract from 1) which is equal to 1 - 0.7182818 =
0.2817182
.2817182 =
An interesting quirk was noticed while resolving this number manually and under the algorithm. There was a difference. The difference showed up in the 15th and 16th place. The same number seemed to resolve to two different digits. The difference was startling at first but then has been understood. There is no overlap.
Manual resolution.
| 3n | value | 1/6 value | Cumulative Total | Balance | Remarks | ||
| -1 | 1/3 | .33333333 | .0555555 | + | .33333333 | +.0516151 | |
| -2 | 1/9 | .11111111 | .0185185 | 0 | |||
| -3 | 1/27 | .03703703 | .0061728 | - | .29629629 | +.0145780 | |
| -4 | 1/81 | .01234567 | .00205761 | - | .28395061 | +.00223238 | |
| -5 | 1/243 | .00411226 | .00068587 | - | .27983539 | -.00801719 | |
| -6 | 1/729 | .00137174 | .00022862 | + | .28120713 | -.00051107 | |
| -7 | 1/2187 | .00045724 | .000076207 | + | .28166438 | -.00005382 | |
| -8 | 1/6561 | .00015241 | .000025402 | 0 | |||
| -9 | 1/19683 | .00005080 | .000008467 | + | .28171518 | -.00000302 | |
| -10 | 1/59049 | .00001693508 | .0000028225 | 0 | |||
| -11 | 1/177147 | .000005645029 | .00000094083 | + | .28172083 | +.00000263 | |
| -12 | 1/531441 | .000001881676 | .00000031361 | - | .281718949 | +.000000749 | |
| -13 | 1/1594323 | .0000006272254 | .00000010453 | - | .2817183217 | +.000000121 | violation of rule here |
| -14 | 1/4782969 | .0000002090751 | .000000034845 | 0 | |||
| -15 | 1/14348907 | .0000000696917 | .000000011615 | - | .2817182520 |
We had resolved that number completely by the
15th place.
.2817182 = 0.+0---++0+0+--0-
However, when we use the algorithm, the number
resolved upto the 16th place like this.
.2817182 =
| 3n | value | 1/6 value* | Cumulative Total | Balance | Percentage Error | ||
| -1 | 1/3 | .33333333 | .0555555 | + | .33333333 | +.0516151 | 18% |
| -2 | 1/9 | .11111111 | .0185185 | 0 | |||
| -3 | 1/27 | .03703703 | .0061728 | - | .29629629 | +.0145780 | 5% |
| -4 | 1/81 | .01234567 | .00205761 | - | .28395061 | +.00223238 | .79% |
| -5 | 1/243 | .00411522 | .00068587 | - | .27983539 | -.00188281 | .66% |
| -6 | 1/729 | .00137174 | .00022862 | + | .28120713 | -.00051107 | .18% |
| -7 | 1/2187 | .00045724 | .000076207 | + | .28166438 | -.00005382 | .019% |
| -8 | 1/6561 | .00015241 | .000025402 | 0 | |||
| -9 | 1/19683 | .00005080 | .000008467 | + | .28171518 | -.00000302 | .001% |
| -10 | 1/59049 | .00001693508 | .0000028225 | 0 | |||
| -11 | 1/177147 | .000005645029 | .00000094083 | + | .28172083 | +.00000263 | .0009% |
| -12 | 1/531441 | .000001881676 | .00000031361 | - | .281718949 | +.000000749 | .0002% |
| -13 | 1/1594323 | .0000006272254 | .00000010453 | - | .2817183218 | +.000000121 | .00004% |
| -14 | 1/4782969 | .0000002090751 | .000000034845 | - | .28171811274 | -.0000000872 | .000030% |
| -15 | 1/14348907 | .0000000696917 | .000000011615 | + | .28171818243 | -.0000000175 | .0000062% |
| -16 | 1/43046721 | .0000000232305 | .0000000038717 | + | .28171820566 | +.0000000056 | .0000019% |
.2817182 = .+0---++0+0+---++
There is a difference between the two.
.2817182 = 0.+0---++0+0+--0-
(Manual)
.2817182 = 0.+0---++0+0+---++
(Algorithm)
It appears that the manual value resolved
completely within 15th place whereas the algorithm went on to 16th.
Actually
The Algorithmic resolution is correct.
Even at the 15th place its value is closer to the target value than the manual
result.
This quirk appeared because we are working with decimal numbers.
And we are ignoring the fact that .2817182 is really also .28171820
There is however no overlap between the two resolutions - they are entirely different numbers.