COMPUTER ARITHMETIC IS A BRANCH OF COMPUTER ENGINEERING.

Computer Arithmetic

Introduction

• Recap:

• Binary numbers are a number system with base 2.

• Information represented inside a computer takes binary values.

• Previous lecture dealt with the conversions between different number systems.

• This lecture deals with basic mathematical operations (such as addition, subtraction, multiplication and division) for binary numbers.


                       

Binary Addition

• Addition in the decimal number system.

• Add values rightmost position (least significant).

• If this addition is grater than 10, 1 is carried to the 2^nd position and added.

• This process is carried for all the positions.

• Binary addition follows the same set of rules.

• If the addition is greater that 2, 1 is carried to the 2^nd next position.


                            

Binary Subtraction

• Similar to subtraction in the decimal number system.

• Inverse of addition.

• If the values cannot be subtracted, borrow from the next position.

• Subtraction table,

• 0 − 0 = 0

• 1 − 0 = 1

• 1 − 1 = 0

• 0 − 1 = 1 with a borrow of 1.

Multiplication & Division

• Similar to multiplication and division in the decimal number system.

• Rules of binary multiplication,

• 0 × 0 = 0

• 0 × 1 = 0

• 1 × 0 = 0

• 1 × 1 = 1.

• Rules of binary division,

• 0 ÷ 1 = 0

• 1 ÷ 1 = 1.

Self studying

• How these values are represented in a computer.

• How much space each value takes when storing.

• What happens if a binary operation provides a result which exceeds the allocated space (Overflow)?

• What happens if an operation provides a result that is too small for the allocated space (Underflow)?

• How are negative numbers represented in a computer?

Complimentary Arithmetic

• Complements are used in digital computers for simplifying,

                • the subtraction operation

                • the logical manipulation.

• Two types of compliments for each base 𝑏 system.

                • 𝑟’𝑠 compliment

                • (𝑟 − 1)’𝑠 compliment

                • Example: For binary numbers, 2’s complement and 1’s complement.
• Given a number 𝑁 in base 𝑟 having 𝑛 digits, (𝑟 − 1)’s complement of 𝑁,
                                              𝑁′ = 𝑟^𝑛 − 1 − 𝑁

• Given a number 𝑁 in base 𝑟 having 𝑛 digits, 𝑟’s complement of 𝑁,
                                             𝑁′ = 𝑟^𝑛 − 𝑁 for 𝑁 ≠ 0; 0 otherwise

• Comparing (𝑟 − 1)’s compliment, 𝑟’s compliment can be obtained by adding 1 to the (𝑟 − 1)’s compliment.


                              

Summary

• Perform basic binary operations (addition, subtraction, multiplication and division).

• Explain overflow and underflow.

• Explain how signs works in binary representations in a computer.

• To perform complement operations r’s complement and (r-1)’s complement.

Next Lecture:- Differentiation