## Thinking Mathematically (6th Edition)

The multiplication of two given numbers in base two is ${{1001110}_{\text{two}}}$.
Since the computation involves base two, the only digits which are allowed are 0 and 1. The procedure to multiply two numbers in base two is same as in base ten. ${{1101}_{\text{two}}}$ $\underline{\times {{110}_{\text{two}}}}$ Hence, first multiply ${{0}_{\text{two}}}$ in first column from right, with ${{1}_{\text{two}}}$ which is above it in the same column: ${{0}_{\text{two}}}\times {{1}_{\text{two}}}={{0}_{\text{two}}}$. Now, write ${{0}_{\text{two}}}$ in the first column from right, below the horizontal line: ${{1101}_{\text{two}}}$ $\underline{\times {{110}_{\text{two}}}}$ ${{0}_{\text{two}}}$ Similarly, multiply ${{0}_{\text{two}}}$ with the remaining digits of ${{1101}_{\text{two}}}$: ${{1101}_{\text{two}}}$ $\underline{\times {{110}_{\text{two}}}}$ ${{0000}_{\text{two}}}$ Repeat whole procedure, but with ${{1}_{\text{two}}}$. First, place the symbol $\times$ below ${{0}_{\text{two}}}$in ${{0000}_{\text{two}}}$. ${{1101}_{\text{two}}}$ $\underline{\times {{110}_{\text{two}}}}$ ${{0000}_{\text{two}}}$ $\times$ Now, multiply ${{1}_{\text{two}}}$ in second column from right, with ${{1}_{\text{two}}}$ in the first column from right: ${{1}_{\text{two}}}\times {{1}_{\text{two}}}={{1}_{\text{two}}}$ Write ${{1}_{\text{two}}}$ below${{0}_{\text{two}}}$in ${{0000}_{\text{two}}}$: ${{1101}_{\text{two}}}$ $\underline{\times {{110}_{\text{two}}}}$ ${{0000}_{\text{two}}}$ $1\times$ Similarly, multiply ${{1}_{\text{two}}}$ with the remaining digits of ${{1101}_{\text{two}}}$: ${{1101}_{\text{two}}}$ $\underline{\times {{110}_{\text{two}}}}$ ${{0000}_{\text{two}}}$ $1101\times$ Once again multiply ${{1}_{\text{two}}}$ with ${{1101}_{\text{two}}}$ in the same manner, after placing two symbols of $\times$ as shown: ${{1101}_{\text{two}}}$ $\underline{\times {{110}_{\text{two}}}}$ ${{0000}_{\text{two}}}$ $1101\times$ $+\underline{1101\times \times }$ Now, add together ${{0000}_{\text{two}}}$, ${{1101}_{\text{two}}}$ and ${{1101}_{\text{two}}}$ in the manner shown above: ${{1101}_{\text{two}}}$ $\underline{\times {{110}_{\text{two}}}}$ ${{0000}_{\text{two}}}$ $1101\times$ $+\underline{1101\times \times }$ ${{1001110}_{\text{two}}}$ Now, to check whether the above obtained solution is correct, perform the multiplication by converting each number to base ten: ${{1101}_{\text{two}}}=13$, ${{110}_{\text{two}}}=6$and ${{1001110}_{\text{two}}}=78$. Since, $13\times 6$ indeed equals 78, the solution obtained is correct. Hence, the multiplication of two given numbers in base two is ${{1001110}_{\text{two}}}$.