Numerical Analysis

Binary Numbers

Internally, computers store information using electronic switches. These switches can be on (represented by the number ${1}$) or off (represented by the number ${1}$). If we can convert a decimal number into a binary number, then we can store that number in a computer.

Definition. An ${n}$-digit binary number is a sequence

$$(b_{n-1}, \ldots, b_1, b_0)_2$$

where ${b_i \in \set{0,1}}$ for ${0 \leq i \lt n.}$

To distinguish between number systems, we will use the notation ${(x_{n-1}, \ldots, x_1, x_0)_b,}$ where ${x}$ is a digit in the number system and ${b}$ is the system's base. For example, the notation ${1011_2}$ represents the binary number ${1011,}$ while the notation ${1011_{10}}$ represents the decimal number ${1011.}$

Decimal to Binary

Converting Integers

We convert decimal integers to binary by dividing by ${2}$ successively and recording the remainders until we reach a quotient of ${0.}$ For example, the number ${53:}$

Reading the recorded remainders from top to bottom, we have the binary number ${101011_2.}$

Converting Rationals

Converting rationals to binary is similar to converting integers, except the steps are reversed. We multiply by two successively and record the integer parts, moving away from the decimal point towards the right. For example, the number ${0.7:}$

Notice that the process repeats after four steps and continues indefinitely. Thus,

where the overbar notation denotes infinitely repeated digits. From these last two sections, we see that

Floating Point Representation