You can represent the number fifteen in base ten as 15_{10}, in base nine as 169, in base 8 as 17_{8}, in base 7 as 21_{7}. Why 21_{7}? In base 7 there is no numeral 8. In order to represent fifteen, you will need two sevens and one 1.
How do you generalize the process? To convert a base ten number to base 7, think about the columns: in base 7 they are ones, sevens, forty-nines, three-hundred forty-threes, and so on. Why these columns? They represent 7^{0}, 7^{1}, 7^{2}, 7^{4}, and so forth. Create a table for yourself:
4 | 3 | 2 | 1 |
7^{3} | 7^{2} | 7^{1} | 7^{0} |
343 | 49 | 7 | 1 |
The first row represents the column number. The second row represents the power of 7. The third row represents the decimal value of each number in that row.
To convert from a decimal value to base 7, here ...
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