## Elementary Technical Mathematics

The sum of the binary numbers $1110101$ and $10011$ is $10001000$.
Add the binary numbers as, $\begin{matrix} Carry, & 1& 1& & 1& 1& 1& \\ & 1& 1& 1& 0& 1& 0& 1&\\ +& & & 1& 0& 0& 1& 1& \\ & \text{ }\!\!\_\!\! & \text{ }\!\!\_\!\! &\text{ }\!\!\_\!\! &\text{ }\!\!\_\!\! &\text{ }\!\!\_\!\! &\text{ }\!\!\_\!\! &\text{ }\!\!\_\!\! \\ & 10& 0& 0& 1& 0& 0& 0& \\ \end{matrix}$ Now, check the result by decimal addition. Now, 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} & 1110101=1\times {{2}^{6}}+1\times {{2}^{5}}+1\times {{2}^{4}}+0\times {{2}^{3}}+1\times {{2}^{2}}+0\times {{2}^{1}}+1\times {{2}^{0}} \\ & =64+32+16+4+0+1 \\ & =117 \end{align} Also, \begin{align} & 10011=1\times {{2}^{4}}+0\times {{2}^{3}}+0\times {{2}^{2}}+1\times {{2}^{1}}+1\times {{2}^{0}} \\ & =16+2+1 \\ & =19 \end{align} And, \begin{align} & 10001000=1\times {{2}^{7}}+0\times {{2}^{6}}+0\times {{2}^{5}}+0\times {{2}^{4}}+1\times {{2}^{3}}+0\times {{2}^{2}}+0\times {{2}^{1}}+0\times {{2}^{0}} \\ & =128+8 \\ & =136 \end{align} Since, $117+19=136$ Thus, the obtained result is correct. Hence the sum of the binary numbers is $10001000$.