Appendix E / E.3

E.3 ARITHMETIC OPERATIONS ON BINARY NUMBERS

Binary numbers are used by digital computers and communication systems to perform all arithmetic calculations. Because each digit can have only two possible values, the arithmetic operations are straightforward and correspond directly to logic operations.

The two most fundamental operations—addition and multiplication—are described below.

Addition

Binary addition follows the same principle as decimal addition: we begin with the least significant bit (right-most digit) and proceed leftward, carrying any overflow into the next higher bit position.

The basic rules of binary addition are:

An example of a binary addition is shown in Equation (E.3).

(E.3)

The basic procedure is the same as decimal addition, beginning with the right-most digit of each of the numbers. In this example, 0+0=0. We then proceed to the next digit to the right. 1+1=2₁₀=10₂. The zero is the value of the addition at this location; the one is carried to the left. The next digit of the sum is formed from 0+1+1=10₂. Once again, a zero is recorded in the current location and the one is carried to the next location to the left. In the final digit, we have 0+0+1=1.

Notice that binary addition can also be described using logic operations: the sum corresponds to the XOR of the input bits, and the carry corresponds to their AND. A circuit that performs this operation is called a half-adder.

Multiplication

Binary multiplication is performed in essentially the same manner as decimal multiplication: each digit of one number is multiplied by every digit of the other, and the partial products are then summed. However, the process is greatly simplified because each binary digit is either 0 or 1.

The basic rule is:

$0\times 0=0,0\times 1=0,1\times 0=0,1\times 1=1$

An example is shown below. In one sense, binary multiplication is actually much simpler than decimal multiplication, because a one-bit binary multiplication is an AND operation.

In this example, the product of 2₁₀ and 6₁₀ is 12₁₀.

Conceptually, binary multiplication is equivalent to the logical AND operation when applied bit-by-bit. In hardware implementations, AND gates and adders perform these operations directly.