Converting between Bases

Suppose you see the binary number $01001110$. What is its equivalent decimal (base $10$) number? Conversely, how do you represent the decimal number $2619$ in the binary system?

We will learn two methods: the subtraction method and the division remainder method. The division remainder method is the easiest and fastest of the two.

Before we see how these methods work, note that a subscript might be used sometimes to indicate in what base a number is written. For example, the number $101_{10}$ is the number $101$ in the decimal system, while the number $101_{2}$ is the binary sequence $101$ written in base 2. [BTW, $101_{2}$ is equal to the number $5_{10}$ in our decimal system.]

• Other examples of subscripts you will encounter are base $8$ (= 'octal'), like $75_{8}$, and base $16$ (= 'hexa-decimal'), like $2\mathrm{B}_{16}$.
• The symbols of the octal system are: $0$, $1$, $2$, $3$, $4$, $5$, $6$, and $7$ (a total of 8 symbols,) and the symbols of the hexadecimal system are: $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$, $\mathrm{D}$, $\mathrm{E}$, and $\mathrm{F}$ (a total of 16 symbols.)