3a. Binary Integer Examples

Examples of Binary Integers

Example 1: Overflowing Complement Systems

Incorrect example with overflow:

$$\begin{array}{r}20\\ +14\\ 34\end{array}$$

equals

$$\begin{array}{r}010100\\ +001110\\ 100010\end{array}$$

in a 6-bit 2’s complement system (leftmost bit as sign), which has $2^6=64$ possible values ($-32$ to $32$). This is incorrect because the leading bit in the result signals negation, but $30+24$ is not negative. There was an overflow error.

Example 2:

$-3$ in 2’s complement with flip and increment steps:

$$3:011⟹100⟹101$$

Adding a 5-bit 2’s complement number to a 3-bit 2’s comp number $9+(-3):$

$$\begin{array}{r}01001\\ +101\\ \text{results in 14 if you just add assuming leading zeroes}\end{array}$$

How to actually add numbers represented by different bit complements:

• move leading bit (sign) to match with larger
• fill spaces with ones (negative) and zeroes (for positive)
• (sign extension)

$$\begin{array}{c}01001\\ +101\\ \\ \text{becomes}\\ \begin{array}{r}\\ 01001\\ +11101\\ 00110\end{array}\end{array}$$