© Michel Dulac 2006-2024
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Later, another characteristic has taken form, when I wondered how to stop this never-ending process of such a division. I realized while working with the base 10 that starting from a division like 1 ÷ 9, where I increased my dividend by one figure ( the same figure)each time I repeated the division, I found that those divisions generate all the figures of the base to finally have a correct answer I mean the obtaining of the desired zero, involving the end of the periodical.
Example : Creation and stop of periodicals in base 10
1 ÷ 9 = .11111 ..... 111111 ÷ 9 = 12345.66666 .....
11 ÷ 9 = 1.22222 ..... 1111111 ÷ 9 = 123456.77777 .....
111 ÷ 9 = 12.33333 ..... 11111111 ÷ 9 = 1234567.88888 .....
1111 ÷ 9 = 123.44444 ..... 111111111 ÷ 9 = 1234567_9
11111 ÷ 9 = 1234.55555 1111111111 ÷ 9 = 1234567_90
(The repetition of the same figure to the dividend, generating all the figures of the base 10)The base 10, as well as at the level of the other bases, all the figures of a base are generated in this type of division [ see example: creation and stop of periodicals, at point 3 above] but the complementary figure which in addition to that, used as a dividend, is equal to the divisor in value, but this, only for a sequence of the quotient where the quotient reaches zero or the rest is equal to zero. For the base 10, the repetition of a same number ten times is necessary to complete this kind of sequences portraits at the level of the quotient, 21 times for the base 21.
Example: - If the dividend is constituted of the number 1 [repeated 10 times, at the level of the base 10] then it will be the 8 which will be absent from the generated sequence portrait because 1 + 8 = 9 (9 being the divisor).
- If the dividend is constituted of the number 2 [repeated 10 times, at the level of the base 10] then it will be the 7 that will be absent from the generated sequence portrait because 2 + 7 = 9 (9 being the divisor).
And this is true for all figures in the base 10, except for the 3 and 6, and, of course, the divider itself [see list of sequences portraits at the base 10 at point 4.1]. For the base 21, not all figures will generate such sequences3 [See the list of sequences portraits at the base 21 at point 4.3].
3 Sequences that I call Sequence portrait because in those sequences all the figures of the base 21 are generated, except the complementary figure affiliated to the appropriate sequence portrait .
