Chapter 16 - Cumulative Review - Page 568: 26

The multiplication of the binary numbers $110001$ and $10111$ is $10001100111$.

Multiply the binary numbers as, $\begin{align} & \underline{\begin{align} & \underline{\begin{matrix} {} & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{1 1 0 0 0 1} \\ \text{x} & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{1 0 1 1 1} \\ \end{matrix}} \\ & \begin{matrix} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{1 1 0 0 0 1} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{1 1 0 0 0 1} \\ \text{ }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{1 1 0 0 0 1} \\ \,\,\,\,\,\,\,\,\text{0 0 0 0 0 0} \\ \text{1 1 0 0 0 1} \\ \end{matrix} \\ & \,\,\,\,\,\,\,\,\,\,\text{1 1 0 0 0 1} \\ \end{align}} \\ & \,\,\,\,\,\,\,\,\,\text{1 0 0 0 1}\,\text{ 1 1 0 0 1 }\,\text{1 }\,\,\text{1} \\ \end{align}$ Now, check the result by decimal multiplication: The equivalent decimal notation for the above binary number will be obtained by writing down the powers of two from right to left and adding them as, $\begin{align} & 110001=1\times {{2}^{5}}+1\times {{2}^{4}}+0\times {{2}^{3}}+0\times {{2}^{2}}+0\times {{2}^{1}}+1\times {{2}^{0}} \\ & =32+16+0+1 \\ & =49 \end{align}$ Also, $\begin{align} & 10111=1\times {{2}^{4}}+0\times {{2}^{3}}+1\times {{2}^{2}}+1\times {{2}^{1}}+1\times {{2}^{0}} \\ & =16+4+2+1 \\ & =23 \end{align}$ And, $\begin{align} & 10001100111=1\times {{2}^{10}}+0\times {{2}^{9}}+0\times {{2}^{8}}+0\times {{2}^{7}}+1\times {{2}^{6}}+1\times {{2}^{5}}+0\times {{2}^{4}}+0\times {{2}^{3}}+1\times {{2}^{2}}+1\times {{2}^{1}}+1\times {{2}^{0}} \\ & =1024+64+32+4+2+1 \\ & =1127 \end{align}$ Since, $49\times 27=1127$ Thus, the obtained result is correct.