Question

Multiply (1110)2 and (1010)2 by working through each step of the algorithm for multiplication given in the text.

Short answer

$(10001100{)}_2$

Step-by-step solution

Step 1:

$\begin{array}{r}a=(1110{)}_2\ast (1010{)}_2\\ \\ b_0=0\text{so}{c}_0=0,p=0\\ \\ b_1=1_{\text{so}}{c}_1=\left(11100{)}_2\text{shifted 1place so it becomes}\right(11100{)}_{2}p=(11100{)}_2\\ \\ b_2=0\text{so}{c}_2=0p=(11100{)}_2\\ \\ b_3=1\text{so}{c}_3=\left(11100{)}_2\text{shifted}3\text{places so it becomes}\right(1110000{)}_2\\ \\ p=(11100{)}_2+(1110000{)}_2=(10001100{)}_2\\ \\ \text{Hence}(1110{)}_2\ast (1010{)}_2=p=(10001100{)}_2\end{array}$

Step 2:

$\begin{array}{r}a{b}_n\ast 2^0=(1110{)}_2\ast 0\ast 2^0=(000{)}_2\\ \\ a{b}_1\ast 2^1=(1110{)}_2\ast 1^{\ast }{2}^1=(11100{)}_2\\ \\ a{b}_2\ast 2^2=(1110{)}_2\ast 0\ast 2^2=(000000{)}_2\\ \\ a{b}_3\ast 2^3=(1110{)}_2\ast 1^{\ast }{2}^3=(1110000{)}_2\end{array}$

Now add those and numbers from each round up carry when need

$\text{Hence}(1110{)}_2\ast (1010{)}_2=(10001100{)}_2$