Question

34. Find the one’s complement representations, using bit strings of length six, of the following integers.

(a) 22 b) 31 c) −7 d) −19

Short answer

The one’s complement representations, using bit strings of length six

$$\begin{gathered} \left(a\right)010110 \\ \left(b\right)011111 \\ \left(c\right)111000 \\ \left(d\right)101100 \end{gathered}$$

Step-by-step solution

Step 1:

(a) You obtain the binary expansion of an integer by consecutively dividing the integer by 2 until you obtain 0

$\begin{array}{r}22=2.11+0\\ 11=2.5+1\\ 5=2.2+1\\ 2=2.2+0\\ 1=2.0+1\end{array}$

The successive remainders of each division represent the binary expansion from right to left.

$10110$

Since 22 is positive, we add a 0 to the left of the binary expansions (to signify a positive number)

$010110$

Step 2:

(b) You obtain the binary expansion of an integer by consecutively dividing the integer by 2 until you obtain 0

$\begin{array}{c}31=2.15+1\\ 15=2.7+1\\ 7=2.3+1\\ 3=2.1+1\\ 1=2.0+1\end{array}$

The successive remainders of each division represents the binary expansion from right to left.

$11111$

Since 31 is positive, we add a 0 to the left of the binary expansions (to signify a positive number)

$011111$

Step 3:

(c) You obtain the binary expansion of an integer by consecutively dividing the integer by 2 until you obtain 0

$\begin{array}{r}7=2.3+1\\ 3=2.1+1\\ 1=2.0+1\end{array}$

The successive remainders of each division represents the binary expansion from right to left.

$111$

Since one’s complement needs to contain 6 bits, the expansion will need to contain 5 bits, thus add 0’s in front of the binary expansion until we obtain 5 bits.

$00111$

Since -7 is negative, replace every 1 by 0 and every 0 by 1:

$11000$

Since -7 is negative, we add a 1 to the left of the previous result (to signify a negative number).

$111000$.

Step 4:

(d) You obtain the binary expansion of an integer by consecutively dividing the integer by 2 until you obtain 0

$\begin{array}{r}19=2.9+1\\ 9=2.4+1\\ 4=2.2+1\\ 2=2.1+0\\ 1=2.0+1\end{array}$

The successive remainders of each division represents the binary expansion from right to left.

$10011$

Since -19 is negative, replace every 1 by 0 and every 0 by 1:

$01100$

Since -19 is negative, we add a 1 to the left of the previous result (to signify a negative number).

$101100$