Converting between Bases

Let's also practice converting to base 16. First, a handy list of common powers of $16$:

$16^0$	$16^1$	$16^2$	$16^3$	$16^4$	$16^5$	$16^6$	$16^7$	$16^8$
$1$	$16$	$256$	$4,096$	$65,536$	$1,048,576$	$16,777,216$	$268,435,456$	$4,294,967,296$

Example: To convert $100,000_{10}$ to base $16$, we do:

$100000_{10} = 1\cdot65536 + 8\cdot4096 + 6\cdot256 + 10\cdot16 = 1\cdot16^{4} + 8\cdot16^{3} + 6\cdot16^{2} + 10\cdot16^1 + 0\cdot16^0 = 186\mathrm{A}0_{16}.$

Note: Base $16$ has $6$ more symbols besides the digits $0-9$: $\mathrm A$ stands for $10$, $\mathrm B$ stands for $11$, $\mathrm C$ stands for $12$, $\mathrm D$ stands for $13$, $\mathrm E$ stands for $14$, and $\mathrm F$ stands for $15$. Examples demonstrating the use of these hexadecimal symbols:

• $13_{10} = 13\cdot1 = 13\cdot16^0 = \mathrm D_{16}$.
• $242_{10} = 15\cdot16 + 2\cdot1 = 15\cdot16^1 + 2\cdot16^0 = \mathrm F2_{16}$.