# Base 9 to Base 3 Conversion w/o using base 10 or Calculator

I'd like too thank anyone who takes the time to answer this question. I recently came out of an exam and there was one question that i could not answer that i would really like to know how to do. the question was

"Convert the base 9 number 46287852013 to base 3" (Dont convert to base 10)

of course trying to convert this to base 10 would take a long time so i was wondering is there any other way of approaching this.

Again thanks a lot!

• Hint: every digit in base 9 corresponds to exactly 2 digits in base 3. So you can make a short table that will let you convert any number. – Nate Eldredge Oct 11 '17 at 20:23

Consider the number $205$ in base $9$. This is the same as $(2 \times 9^2) + (0 \times 9^1) + (5 \times 9^0)$ in base 10.
Now $9=3^2$ so we can rewrite this as $(2 \times 3^4) + (0 \times 3^2) + (5 \times 3^0)$ but to keep all of our coefficients below $3$, we should split the terms and bring in odd powers of $3$ to get $(2 \times 3^4) + (0 \times 9^1) + (1 \times 3^1) + (2 \times 3^0)$.
In effect, all we need to do is convert every number $0$ to $8$ to base $3$ with two digits (as $9=3^2$) and can replace all digits individually; e.g. $8 \mapsto 22$, $2 \mapsto 02$.
Hence $46287852013$ becomes $1120022221221202000110$ in base $3$.