TITLE: Converting decimal fractions to base N
QUESTION [6 upvotes]: I find converting from base N to base 10 very easy. I have no problem converting integers in base 10 to base N but when I have to find, for example, 0.5 base 3. I am not sure how I am supposed to do it. I can't do division with remainders any more.
How is this done?

REPLY [9 votes]: Instead of using division with remainder as you do for integers, you can use multiplication for the decimal part:
$0.5 * 3 = 1.5$
Use the integer part as your first digit after the comma, and repeat with the decimal part:
$0.5 * 3 = 1.5$
You see that this will continue forever, so we have $0.5_{10} = 0.\overline{1}_3$.
Another example: 
$$0.4 * 3 = 1.2\\
0.2*3 = 0.6\\
0.6*3=1.8\\
0.8*3=2.4\\
0.4*3 = 1.2$$
And now we're repeating already, so $0.4_{10} = 0.\overline{1012}_3$