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Topic: base 3 (Read 3896 times) |
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Christine
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I don't remember how to do this type of operation:
Repeated permutations of (0,1,2) (base 3)
5/26 = 0.012…,
7/26 = 0.021…,
11/26 = 0.102…,
15/26 = 0.120…,
19/26 = 0.201…,
21/26 = 0.210…
How do you find these rational numbers?
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Grimbal
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Re: base 3
« Reply #1 on: Mar 31st, 2013, 1:16pm » |
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26 is 27-1. 27 is 3^3 (3 because of the base, 3 because of 3 digits).
If you took 27, you would just have 3 digits. Taking 26 makes it periodic.
5 is 123, 7 is 213, 11 is 2013, etc.
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| « Last Edit: Mar 31st, 2013, 1:17pm by Grimbal » |
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Christine
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Re: base 3
« Reply #2 on: Mar 31st, 2013, 10:53pm » |
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on Mar 31st, 2013, 1:16pm, Grimbal wrote:26 is 27-1. 27 is 3^3 (3 because of the base, 3 because of 3 digits).
If you took 27, you would just have 3 digits. Taking 26 makes it periodic.
5 is 123, 7 is 213, 11 is 2013, etc. |
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It's still not clear to me.
I'll take another example: Let's find fractions 0. ...
so that we get permutations of {0,1,2,3}
75 -> 10234, 99 -> 12034, 108 -> 12304, 78 -> 10324,
135 -> 20134, 147 -> 21034, etc.
what are these rational numbers?
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| « Last Edit: Mar 31st, 2013, 10:54pm by Christine » |
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towr
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Re: base 3
« Reply #3 on: Mar 31st, 2013, 11:12pm » |
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1/(bn-1) =  i=1..inf (1/bn)i
1/(10-1) = 1/9 = i=1..inf 0.1i = 0.111...
1/(100-1) = 1/99 = i=1..inf 0.01i = 0.010101...
1/(1000-1) = 1/999 = i=1..inf 0.001i = 0.001001001...
1/(3-1) = 1/2 = i=1..inf 0.13i = 0.111...3
1/(9-1) = 1/8 = i=1..inf 0.013i = 0.010101...3
1/(27-1) = 1/26 = i=1..inf 0.013i = 0.001001001...3
1/(4-1) = 1/3 = i=1..inf 0.14i = 0.111...4
1/(16-1) = 1/15 = i=1..inf 0.014i = 0.010101...4
1/(64-1) = 1/63 = i=1..inf 0.0014i = 0.001001001...4
So to repeat k digits x..z in base b, take 1/(bk-1) * x..z
And of course you can shift it by dividing by bm and add some not repeating term if you want.
So, e.g.
0.1023...4
= 10234 * i=1..inf 0.00014i
= 10234 * 1/(44-1)
= 10234 * 1/(256-1)
= 75/255
= 15/51
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| « Last Edit: Apr 1st, 2013, 6:57am by towr » |
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Grimbal
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Re: base 3
« Reply #4 on: Apr 1st, 2013, 6:06am » |
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on Mar 31st, 2013, 10:53pm, Christine wrote:
75 -> 10234, 99 -> 12034, 108 -> 12304, 78 -> 10324,
135 -> 20134, 147 -> 21034, etc.
what are these rational numbers? |
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Short answer: the denominator is 44-1 = 255.
75/255 = 0.294117647058824 = 0.10231023...4
99/255 = 0.12031203...4
As towr mentioned, the fraction needs to be simplified.
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| « Last Edit: Apr 1st, 2013, 6:08am by Grimbal » |
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Christine
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Posts: 159
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Re: base 3
« Reply #5 on: Apr 1st, 2013, 9:59am » |
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Thanks Grimbal and towr. I understand better now
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whizen
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Re: base 3
« Reply #6 on: May 30th, 2013, 2:25pm » |
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Loved this question and the answers.
Christine...
Where do you get all these questions? ( I have seen some more interesting ones from you in the forum )
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