Answer
The multiplication of two given numbers in base two is \[{{1001110}_{\text{two}}}\].
Work Step by Step
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 \[{{1001110}_{\text{two}}}\].
