# Converting base 10 fractions into other bases

How do you convert

$14\frac{8}{13}$

into base 3?

I was able to convert $\frac{3}{7}$ into base 3 by constantly multiplying by 3 and dividing the numerator by denominator until I finally got a repetition, but this method doesn't seem to work for $14\frac{8}{13}$.

The correct answer is supposed to be $112.\overline{121}_{3}$

• Um, fractions work the same in all bases. There's an integer above the fraction bar and an integer below. You can write those integers in any basis you like; doing that is independent of the fact that they're in a fraction. – Henning Makholm Nov 5 '11 at 15:33
• I'm not sure if I understand, are you saying I can just convert the top number and the bottom number seperately? – Arvin Nov 5 '11 at 15:35
• Sorry, I didn't see the supposed correct answer. I was confused because you described the input as a "base 10 fraction", and the equivalent of that in base 3 would be $\left[112\frac{22}{111}\right]_3$. You can do long division $22\div 111$ in base 3 to get the positional notation. – Henning Makholm Nov 5 '11 at 15:52
• Thanks for that, so I can convert the 14 seperately to the $\frac{8}{13}$ since they are just two numbers multiplied – Arvin Nov 5 '11 at 16:05
• Oops, yes, I meant added. – Arvin Nov 5 '11 at 17:57

Just like you convert a fraction to decimal. $8\cdot 3=24=13(1)+11,$ so the first digit (ternit?) is $1. \ \ 11\cdot 3=33=13(2)+7,$ so the second digit is $2. \ \ 3\cdot 7=21=13(1)+8$ and so on. I get $112.\overline{121}_{3}$

Alternately you can notice that $\frac{8}{13}=\frac{16}{3^3-1}$, so the repeat is $3$ digits long and is $16_{10}=121_3$

• Is your answer only for $\frac{8}{3}$? – Arvin Nov 5 '11 at 15:50
• Thanks, no worries, I am able to convert the 14 into base 3, I didn't know you were supposed to do the 14 and the fraction separately. – Arvin Nov 5 '11 at 16:12
• @Arvin: I was only describing how to do the fraction, $\frac{8}{13}$, not $\frac{8}{3}$. I then appended the result for the whole number (now corrected to $112_3$) – Ross Millikan Nov 5 '11 at 16:16
• Typo, I meant to write $\frac{8}{13}$. Thanks! – Arvin Nov 5 '11 at 17:58

You convert 8 and 13 into base 3, and do a long division in that base,

          0.202           8 = 101
------------           -----
111 ) 101              13 = 111
22.2
-----
1.100
0.222
101  (repeats on three places)


So the answer is 112.202 202 202