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0.2 in base 6
0.22222..... in base 7
0.252525.... in base 8
Here is the demonstration of how I arrived at both the base 6 and base 7 answers; base 8 was done the same way - I'm just too lazy to type it in as well :-).
The way that you can determine 1/3 = 0.33333.... in base 10 is to perform the long division:
0.33333..... --------- 3 | 1.00000 9 ---- 10 9 --- 10
So if we were to perform the same operation in base 6, knowing that the 3 times table (in base 6) is as follows
3 x 0 = 03 x 1 = 33 x 2 = 103 x 3 = 133 x 4 = 203 x 5 = 23the division would look like: 0.2 ------ 3 | 1.0 1 0 ---- 0
This can be confirmed by doing the multiplication:
3 x 0.2 in base 6 (ensuring that the result is 1)is the same as (3 x 6^0) x (2 x 6^(-1)) converted to base 10 (3 x 1) x (2 x 1/6) 3 x 1/3 1The multiplication confirmed the division.....----------------------------------------In base 7, the 3 times table is:3 x 0 = 03 x 1 = 33 x 2 = 63 x 3 = 123 x 4 = 153 x 5 = 213 x 6 = 24The long division is ..... 0.222..... ------- 3 | 1.000 6 ---- 10 6 --- 10To confirm this with multiplication 3 x 0.22222..... in base 7can be written as (in base 10) (3 x 7^0) x (2x7^(-1) + 2x7^(-2) + 2x7^(-3) + .... ) 3 x (2/7 + 2/49 + 2/343 + 2/2401 + .....) 3 x (0.3331945 + .....)which is equal to 1.
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