We have two responses for you

Johanna,

If you go to our quandaries and queries section and type in the word base in the search box you will find a number of previously answered questions on bases that should help you. You may have to go from base 5 to base 10 and then to base 9 however.

Penny

Well Johanna, there are quick ways for some conversions, but not for others.

Because I (like you I presume) work and think in base ten most of the time, I would first convert to base 10, then to the target base. For example, convert 40242 in base 5 to base 9.

When we use different bases, we usually write a subscript after the number to indicate it isn't a base 10 number. e.g. 40242 in base 5 is written 40242₅.

(a) convert to base 10 by repeatedly multiplying digits and sums by fives:
40142₅ = (((4 * 5 + 0) * 5 + 2) * 5 + 4) * 5 + 2
= ((20*5 + 2) * 5 + 4) * 5 + 2
= (102 * 5 + 4) * 5 + 2
= 514 * 5 + 2
= 2572₁₀.

(b) convert to base 9 by repeatedly dividing by 9 and recording the remainders:
2572 / 9 = 285 remainder 7
285 / 9 = 31 remainder 6
31 / 9 = 3 remainder 4
3 is less than 9 so stop.
(read the last quotient (3) and all remainders backwards): 3467₉.

So 40242₅ = 2572₁₀ = 3467₉.

There are faster ways for particular combinations of bases. For example, 4E20 in hexadecimal (base 16) is easily converted to base 2: 0100 1110 0010 0000. Because 16 is just 2⁴, so each base 16 number turns into a four digit (with leading zeros) base 2 number. But this kind of trick doesn't work for just any two bases.

Stephen La Rocque.