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I have number 220111021.0021101is in Base-3, I need to convert it to base-9. How I can convert the fractional value into Base-9 using shortcut method.

Kaur Navpreet
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    Do you know how to convert a two digit base 3 number into base 9? What about four digit numbers? It is not really possible for us to write an answer that would help you understand this unless you tell us more about what you already understand or where this is going wrong. What is your understanding on "shortcut method" for instance? Could you do this if not for the decimal point? – Milo Brandt Jan 16 '20 at 04:07
  • Note that $\sum_{n}a_n3^n=\sum_n[a_{2n-1}3^{2n-1}+a_{2n}3^{2n}]=\sum_n[3a_{2n-1}3^{2n-2}+a_{2n}3^{2n}]=\sum_n[3a_{2n-1}9^{n-1}+a_{2n}9^{n}]=\sum_n[a_{2n}+3a_{2n+1}]9^n$. Note also that since $a_{2n},a_{2n+1}\leq2$, then $a_{2n}+3a_{2n+1}\leq 2+3\cdot 2=8$. So, $\sum_n[a_{2n}+3a_{2n+1}]9^n$ is the base $9$ representation. – MoonLightSyzygy Jan 16 '20 at 04:27
  • In other words, you only need to scan the digits, from left to right, with a window of size $2$-digits $[x,y]$. You start such that $x$ fall in a position corresponding to an odd power of $3$. You can append a zero on the left of the number if necessary. Then each time, replace the two digits in the window by the single base-$9$ digit $3x+y$. Continue by moving the $2$-digit window to the next pair of digits and repeat. – MoonLightSyzygy Jan 16 '20 at 04:33

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