Converting decimal fractions to base N 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?
 A: 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 $0.4_{10} = 0.\overline{1012}_3$
A: The principle is the same as for integers. To convert $0.5$ in base $10$ to base $3$, we'll be running through (negative) powers of $3$ and successively subtracting them from the given number.
How many times can $\frac13$ fit into $0.5$? Once, so the first digit after the decimal point is $1$.

$0.1..._3$

Subtract $\frac13$ from $0.5$ to get $0.1666$. How many times can $\frac19$ fit into $0.1666$? Once, so the next digit is again $1$.

$0.11..._3$

Subtract $\frac19$ from $0.1666$ to get $0.0555$. How many times can $\frac1{27}$ fit into $0.0555$? Once, so the next digit is again $1$.

$0.111..._3$

Continue along this vein and you will see that $0.5$ in base $3$ is $0.\bar1_3$. This corresponds to the geometric series $\frac13 + \frac19 + \frac1{27} + \ldots = \frac12$.
A: To convert 23 (base 10) to base 3, what do you do? 
You write down powers of 3: 27, 9, 3, 1, and see what's the largest one that's no bigger than 11. That's 9, so you divide, to get 2 with a remainder of 5, and write
2XX
Now the next power that's less than 4 is 3, so you divide (quotient 1, remainder 2) and you have
21X
Finally, there's 1: you divide, get 2 with remainder 0, and your answer is 
212  (base 3)
What about converting a fraction? 
You write down powers of 3: 1/3, 1/9, 1/27, etc. 
Since 1/3 fits into 1/2, you get
.1XXXXX
with a remainder of 1/6. 1/9 is less than 1/6, so you get
.11XXX
with a remainder of 1/6 - 1/9 = 1/18
Now 1/27 is less then 1/18, so you get a 1 in the 1/27ths place, and you have
.111XX
and you continue in this manner (until you feel the approximation is good enough). 
A: You can do several things.  $\frac 12$ will not terminate in base $3$, but you can multiply it by some large power of $3$ and convert the whole part.  So $\frac 12=\frac {40.5}{81}, 40_{10}=1111_3, \frac 12 \approx 0.1111_3$  You can note the repeat and say $\frac 12=0.\overline 1_3$  You can just do long division in base $3$ for fractions.
