2.2 Real numbers in other number bases

Let $x$ be a non-negative real number. In the previous section, we saw that we can use powers of $10$ to write

where each of $a_{0},a_{1},a_{2}\ldots,a_{n}$ and $b_{1},b_{2},b_{3}\ldots$ is one of the ten symbols $0,1,2,\ldots,9$ (called digits).

Rather than using powers of $10$ and the digits $0,1,2,\ldots,9$, we can use powers of $2$, for example, to write

where each of $a_{0},a_{1},a_{2}\ldots,a_{n}$ and $b_{1},b_{2},b_{3}\ldots$ is one of the bits $0$ or $1$. This is called a binary representation (or base $2$ representation) of $x$, and is important in computing.

Then, for example, because $13=\bm{1}\times 2^{3}+\bm{1}\times 2^{2}+\bm{0}\times 2^{1}+\bm{1}\times 2^{0}$. The binary representation of $13$ is $1101_{2}$, where the subscript $2$ indicates that we are working with base $2$.

We will stick to using the familiar decimal numeral system in this course. You can read more about binary if you are interested. Other bases are possible, too, for example the hexadecimal numeral system uses base $16$.