Converting between Bases

The subtraction method works as follows: keep subtracting powers of the destination base from the number, until you get 0. Take note of which powers were used. Finally, write the multiples of all these subtracted powers in a string, which is your result!

Since we will frequently convert numbers to base $2$, it is useful to keep a handy list of some common powers of $2$:

$2^0$	$2^1$	$2^2$	$2^3$	$2^4$	$2^5$	$2^6$	$2^7$	$2^8$	$2^9$	$2^{10}$	$2^{11}$	$2^{12}$	$2^{13}$	$2^{14}$	$2^{15}$	$2^{16}$
$1$	$2$	$4$	$8$	$16$	$32$	$64$	$128$	$256$	$512$	$1,024$	$2,048$	$4,096$	$8,192$	$16,384$	$32,768$	$65,536$

Example: To convert $2619_{10}$ to base 2, we do the following:

Note that the biggest power of $2$ that goes into $2619_{10}$ is $2^{11} = 2048$, so we subtract to get $2619 - 2048 = 571$. The biggest power that goes into $571$ is $2^9 = 512$, so we get $571 - 512 = 59$. Keeping this way, we see that $59 - 32 = 27$, $27 - 16 = 11$, $11 - 8 = 3$, $3 - 2 = 1$, $1 - 1 = 0$. Next on, we will build up the resulting number.