When you ask if there's an "equation," there is (as in the article mentioned in the first reply). In terms of actually converting the base 10 number into a base 9 in practice, though, the equation doesn't provide any mystical secret in figuring out how to do this other than what you probably already know. By hand, it's still just a cyclic process of dividing a number (the original number or the remainder of a previous digit's division) by decreasing exponents of the base (9, in this example).Each digit in base 9, going from RIGHT TO LEFT, represents 9^0 (or 1), 9^1 (or 9), 9^2 (81), etc. Make enough columns so that the result 9^x is below the number to be converted, but 9^(x+1) is greater than the number to be converted (you can use as many digits as you want, but if the resulting exponential number is greater than the number to be converted, you'll just end up with leading zeros).
If we use 1234 as an example base 10 number to convert, then:
9^0 = 1
9^1 = 9
9^2 = 81
9^3 = 729
9^4 = 6561. Stop! 6561 is larger than 1234, so we don't need a 9^4 digit. So you will have 4 digits, representing:
9^3 9^2 9^1 9^0
Divide the original base 10 number by the leftmost digit's value, put in the whole number result, and repeat, for the next digit, with the remainder of the previous division, and the next column's digit value.
So for 1234:
1234 / 9^3 = 1234 / 729 = 1, remainder 505
505 / 9^2 = 505 / 81 = 6, remainder 19
19 / 9^1 = 19 / 9 = 2, remainder 1
1 / 9^0 = 1 / 1 = 1
so 1234 converted to base 9 is 1621
check: (1 x 729) + (6 x 81) + (2 x 9) + (1 x 1) =
729 + 486 + 18 + 1 = 1234
As far as I know, there's nothing more elegant or simple than this. Even computers and calculators will use this basic algorithym to make the transformation. They just do it trillions of times faster.
Hope this helps...
Kevin
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