Binary Arithmetic

          BINARY ARITHMETIC

The arithmetic operations—addition, subtraction, multiplication and division, performed on the binary numbers is called binary arithmetic. In computer, the basic arithmetic operations performed on the binary numbers is—

·         Binary addition, and

·         Binary subtraction.

In the following subsections, we discuss the binary addition and the binary subtraction operations.

                  Binary Addition

Binary addition involves addition of two or more binary numbers. The binary addition rules are used while performing the binary addition. Table1 shows the binary addition rules.

Binary Addition Rules 2-Input

Binary addition of three inputs follows the rule shown in Table 2.

Binary Addition Rules 3-Input

Addition of the binary numbers involves the following steps—

1. Start addition by adding the bits in unit column (the right-most 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 a 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.

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.

                  Binary Subtraction

Binary subtraction involves subtracting of two binary numbers. The binary subtraction rules are used while performing the binary subtraction. The binary subtraction rules are shown in Table 3, where “Input 2” is subtracted from “Input 1.”

Binary Subtraction Rules

The steps for performing subtraction of the binary numbers are as follows—

1.        Start subtraction by subtracting the bit in the lower row from the upper row, in the unit column.

2.        Use the binary subtraction rules. If the bit in the upper row is less than lower row, borrow 1 from the upper row of the next column (on the left side). The result of subtracting two bits is the difference.

3.        Write the difference in the result of that column.

4.        Repeat steps 2 and 3 for each column, i.e., the tens column, hundreds column and so on.

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

Example 1: Subtract 01 from 11. Verify the answer with the help of decimal subtraction.

When we subtract 1 from 1 in the unit column, the difference is 0. Write the difference in the unit column of the result. In the tens column, subtract 0 from 1 to get the difference 1. Write the difference in the tens column of the result.

Example 2: Subtract 01 from 10. Verify the answer with the help of decimal subtraction.

When we subtract 1 from 0 in the unit column, we have to borrow 1 from the left column since 0 is less than 1. After borrowing from the left column, 0 in the unit column becomes 10, and, 1 in the left column becomes 0. We perform 10-1 to get the difference 1. We write the difference in the unit column of the result. In the tens column, subtract 0 from 0 to get the difference 0. We write the difference 0 in the tens column of the result.

Example 3: Subtract 0111 from 1110. Verify the answer with the help of decimal subtraction.

When we do 0–1 in the unit column, we have to borrow 1 from the left column since 0 is less than 1. After borrowing from the left column, 0 in the unit column becomes 10, and, 1 in the left column becomes 0. We perform 10-1 to get the difference 1. We write the difference in the unit column of the result. In the tens column, when we do 0-1, we again borrow 1 from the left column. We perform 10-1 to get the difference 1. We write the difference in the tens column of the result. In the hundreds column, when we do 0-1, we again borrow 1 from the left column. We perform 10-1 to get the difference 1. We write the difference in the hundreds column of the result. In the thousands column, 0–0 is 0. We write the difference 0 in the thousands column of the result.

Example 4: Subtract 100110 from 110001. Verify the answer with the help of decimal subtraction.