3
Digital Arithmetic
Having discussed different methods of numeric and alphanumeric data representation in the first two
chapters, the next obvious step is to study the rules of data manipulation. Two types of operation
that are performed on binary data include arithmetic and logic operations. Basic arithmetic operations
include addition, subtraction, multiplication and division. AND, OR and NOT are the basic logic
functions. While the rules of arithmetic operations are covered in the present chapter, those related to
logic operations will be discussed in the next chapter.
3.1 Basic Rules of Binary Addition and Subtraction
The basic principles of binary addition and subtraction are similar to what we all know so well in
the case of the decimal number system. In the case of addition, adding ‘0’ to a certain digit produces
the same digit as the sum, and, when we add ‘1’ to a certain digit or number in the decimal number
system, the result is the next higher digit or number, as the case may be. For example, 6 + 1 in decimal
equals ‘7’ because ‘7’ immediately follows ‘6’ in the decimal number system. Also, 7 + 1 in octal
equals ‘10’ as, in the octal number system, the next adjacent higher number after ‘7’ is ‘10’. Similarly,
9 + 1 in the hexadecimal number system is ‘A’. With this background, we can write the basic rules of
binary addition as follows:
1. 0 + 0 = 0.
2. 0 + 1 = 1.
3. 1 + 0 = 1.
4. 1 + 1 = 0 with a carry of ‘1’ to the next more significant bit.
5. 1 + 1 + 1 = 1 with a carry of ‘1’ to the next more significant bit.
Table 3.1 summarizes the sum and carry outputs of all possible three-bit combinations. We have
taken three-bit combinations as, in all practical situations involving the addition of two larger bit
Digital Electronics: Principles, Devices and Applications Anil K. Maini
© 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03214-5
Digital Arithmetic
Having discussed different methods of numeric and alphanumeric data representation in the first two
chapters, the next obvious step is to study the rules of data manipulation. Two types of operation
that are performed on binary data include arithmetic and logic operations. Basic arithmetic operations
include addition, subtraction, multiplication and division. AND, OR and NOT are the basic logic
functions. While the rules of arithmetic operations are covered in the present chapter, those related to
logic operations will be discussed in the next chapter.
3.1 Basic Rules of Binary Addition and Subtraction
The basic principles of binary addition and subtraction are similar to what we all know so well in
the case of the decimal number system. In the case of addition, adding ‘0’ to a certain digit produces
the same digit as the sum, and, when we add ‘1’ to a certain digit or number in the decimal number
system, the result is the next higher digit or number, as the case may be. For example, 6 + 1 in decimal
equals ‘7’ because ‘7’ immediately follows ‘6’ in the decimal number system. Also, 7 + 1 in octal
equals ‘10’ as, in the octal number system, the next adjacent higher number after ‘7’ is ‘10’. Similarly,
9 + 1 in the hexadecimal number system is ‘A’. With this background, we can write the basic rules of
binary addition as follows:
1. 0 + 0 = 0.
2. 0 + 1 = 1.
3. 1 + 0 = 1.
4. 1 + 1 = 0 with a carry of ‘1’ to the next more significant bit.
5. 1 + 1 + 1 = 1 with a carry of ‘1’ to the next more significant bit.
Table 3.1 summarizes the sum and carry outputs of all possible three-bit combinations. We have
taken three-bit combinations as, in all practical situations involving the addition of two larger bit
Digital Electronics: Principles, Devices and Applications Anil K. Maini
© 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03214-5