Question 4.4.1: Adding in Base 2 Add 11012 + 1112/...
Adding in Base 2
Add \begin{array}{r} \ {1101_2 } \\ + 111_2 \\ \hline \end{array}
Begin by adding the digits in the right-hand, or units, column. From previous discussion, and as can be seen in Table 4.7, 1_2 + 1_2 = 10_2 . Place the 0 under the units column and carry the 1 to the 2’s column, the second column from the right.
\begin{array}{llll}& 2³ & 2² & 2 & 1 \\ &↓ & ↓ & ↓ & ↓ \\& 1 & 1 &^{1}0 &1 \\ + & &1 & 1 & 1 \\ \hline &&&&0_2 \end{array}\text{Place value of columns}Now add the three digits in the 2’s column, 1_2 + 0_2 + 1_2. Treat it as (1_2 + 0_2) + 1_2. Therefore, add 1_2 + 0_2 to get 1_2, then add 1_2 + 1_2 to get 10_2.
Place the 0 under the 2’s column and carry the 1 to the 2² column (the third column from the right).
\begin{array}{llll} &1 &^{1}1& ^{1}0 &1\\ + & &1& 1& 1 \\\hline &&&0 &0_2\end{array}Now add the three 1’s in the 2² column to get (1_2 + 1_2) + 1_2 = 10_2 + 1_2 = 11_2. Place the 1 under the 2² column and carry the 1 to the 2³ column (the fourth column from the right).
\begin{array}{llll} ^{1}1 &^{1}1& ^{1}0& 1\\ + & 1 &1 &1 \\\hline &1 &0& 0_2\end{array}Now add the two 1’s in the 2³ column, 1_2 + 1_2 = 10_2. Place the 10 as follows.
\begin{array}{llll} &^{1}1 &^{1}1 &^{1}0& 1\\ + &&1 &1 &1 \\\hline 1 &0 &1 &0& 0_2\end{array}Therefore, the sum is 10100_2.
| Table 4.7 Base 2 Addition Table | ||
| + | 0 | 1 |
| 0 | 0 | 1 |
| 1 | 1 | 10 |