Question

Find the two’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

$$\begin{gathered} a\left)010110 \\ b\right)011111 \\ c\left)111001 \\ d\right)101101 \end{gathered}$$

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 represents 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

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

c) 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:

d) 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 two’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, subtract 00111 from 100000

$(100000{)}_2-(00111{)}_2=(11001{)}_2$

Note: you could determine the difference by using $(00111{)}_2+(11001{)}_2=(100000{)}_2$

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

111001

Step 4:

e) 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, subtract 10011 from 100000

$(100000{)}_2-(10011{)}_2=(01101{)}_2$

Note: you could determine the difference by using $(10011{)}_2+(01101{)}_2=(100000{)}_2$

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

101101