BINARY SYSTEM & ARITHMATIC

The binary number system  is used in the computer systems. The digits 0 and 1 are combined to get different binary numbers like 1001, 11000110 etc. In a binary number, a digit 0 or 1 is called a bit. For example, 1001 is a 4-bit binary number, and, 11000110 is an 8-bit binary number. All kinds of data, be it alphabets, numbers, symbols, sound data or video data, are represented as combination of bits i.e. 0’s and 1’s. Each character is a unique combination of bits. We shall now discuss how to perform basic arithmetic operations in the binary number system. 

 

2.1. Binary Addition

Binary addition involves adding of two or more binary numbers. The binary addition rules are used while performing the binary addition. Table 1 shows the binary addition rules.

Table 1. Binary Addition Rules for two inputs

Binary addition of three inputs, when all the inputs are 1, follows the rule shown in Table 2.

Table 2. Binary Addition Rules for three inputs

Addition of the binary numbers involves the following steps-

1. Start addition by adding the bits in unit column (the rightmost column). Use the rules of binary addition.
2. The result of adding bits of a column is a sum with or without a carry.
3. Write the sum in the result of that column.
4. If carry is present, the carry is carried-over to the addition of the next left column.
5. Repeat steps 2-4 for each column, i.e., the tens column, hundreds column and so on.

Let us now understand binary addition with the help of some examples.

Example 1. Add 10 and 01. Verify the answer with the help of decimal addition.

When we add 0 and 1 in the unit column, sum is 1 and there is no carry. The sum 1 is written in the unit column of the result. In the tens column, we add 1 and 0 to get the sum 1. There is no carry. The sum 1 is written in the tens column of the result.

Example 2. Add 01 and 11. Verify the answer with the help of decimal addition.

When we add 1 and 1 in the unit column, sum is 0 and carry is 1. The sum 0 is written in the unit column of the result. The carry is carried-over to the next column, i.e. the tens column. In the tens column, we add 0, 1 and the carried-over 1, to get sum 0 and carry 1. The sum 0 is written in the tens column of the result. The carry 1 is carried-over to the hundreds column. In the hundreds column, the result is 1.

Example 3. Add 11 and 11. Verify the answer with the help of decimal addition.

When we add 1 and 1 in the unit column, sum is 0 and carry is 1. The sum 0 is written in the unit column of the result. The carry is carried-over to the tens column. In the tens column, we add 1, 1 and the carried-over 1. We use the rule for binary addition of three inputs shown in Table 2. We get sum 1 and carry 1. The sum 1 is written in the tens column of the result. The carry 1 is carried-over to the hundreds column. In the hundreds column, the result is 1.

Example 4. Add 1101 and 1111. Verify the answer with the help of decimal addition.

Example 5. Add 10111, 11100 and 11. Verify the answer with the help of decimal addition.