How to Multiply Binary Numbers

Find \(110 \times 101\)

Write it out like a normal long multiplication: $$\begin{align} 110 & \\ \times\,\, 101 & \\ \end{align}$$ Now just like you're used to we'll start with the digit on the bottom right and multiply it by each of the top digits starting from the right side. First we get \(1\times 0 = 0\) so we write that down. $$\begin{align} 110 & \\ \times\,\, 101 & \\ \overline{=\quad 0} & \\ \end{align}$$ Then we get \(1\times 1 = 1\). $$\begin{align} 110 & \\ \times\,\, 101 & \\ \overline{=\quad 10} & \\ \end{align}$$ And then \(1\times 1 = 1\) again. $$\begin{align} 110 & \\ \times\,\, 101 & \\ \overline{=\quad 110} & \\ \end{align}$$ Now we move onto the middle bottom digit, we add a 0 just like in normal long multiplication before we begin: $$\begin{align} 110 & \\ \times\,\, 101 & \\ \overline{=\quad 110} & \\ 0 & \\ \end{align}$$ Then we get three 0s since \(0\times\) anything is 0. $$\begin{align} 110 & \\ \times\,\, 101 & \\ \overline{=\quad 110} & \\ 0000 & \\ \end{align}$$ Now we move onto our bottom left digit. We write two 0s to take care of the place, just like always. $$\begin{align} 110 & \\ \times\,\, 101 & \\ \overline{=\quad 110} & \\ 0000 & \\ 00 & \\ \end{align}$$ Then we get \(1\times 0 = 0,\quad 1\times 1 = 1,\quad 1\times 1 = 1\) To give us: $$\begin{align} 110 & \\ \times\,\, 101 & \\ \overline{=\quad 110} & \\ 0000 & \\ 11000 & \\ \end{align}$$ Now we add up our three rows of answers to get our final answer: $$\begin{align} 110 & \\ \times\,\, 101 & \\ \overline{=\quad 110} & \\ 0000 & \\ +\quad 11000 & \\ \overline{11110} & \\ \end{align}$$ So our answer is \(11110\) We can check this by doing it in decimal. We know that \(110_2 = 6_{10}\) and \(101_2 = 5_{10}\) and \(5\times 6 = 30\) Now we check and see that indeed \(11110_2 = 30_{10}\) and we must have done everything right!