Unit 4: Computer Arithmetic
In this unit, you will build upon your knowledge of computer instructions and digital logic design to discuss the role of computer arithmetic in hardware design. We will also discuss the designs of adders, multipliers, and dividers. You will learn that there are two types of arithmetic operations performed by computers: integer and floating point. Finally, we will discuss floating point details for carrying out operations with real numbers.
Completing this unit should take you approximately 5 hours.
Upon successful completion of this unit, you will be able to:
- show the block diagram for a simple adder and show the outputs for a simple 4 bit addition;
- show how to use a 4 bit adder to perform multiplication, subtraction, and division; and
- design a full adder from a half adder.
4.1: Number Representation
Read this article on the representation of real numbers.
Read these sections on the representation of integers and real numbers. Earlier, you read about number systems and the representation of numbers used for computing. This will give you a chance to review that material. Computer architecture comprises components which perform the functions of storage of data, transfer of data from one component to another, computations, and interfacing to devices external to the computer. Data is stored in terms of units, called words. A word is made up of a number of bits, typically, depending on the computer, 32 bits or 64 bits. Words keep getting longer, with larger numbers of bits. Instructions are also stored in words. Before, you saw examples of how instructions are stored in a word or words. Now, you will see how numbers are stored in words.
4.2: Addition and Subtraction Hardware
Study this article to learn how addition is implemented and carried out at the gate level. Nowadays, computers are architected using larger components. For example, to perform addition and subtraction, computer architects utilize ALUs, arithmetic logic units. You can design a computer without knowing the details of an ALU or of an adder, similar to using a calculator to find the square root of a number without knowing how to manually compute the square root (or in computer science terminology, without knowing the algorithm that the calculator performs to find the square root). However, we want you to have the strongest foundation in your study of computer architecture. Knowing the underlying algorithm for larger components, you will be able to better use them in constructing larger components, for example, using half adders to construct a full adder. A half adder takes 2 bits as input and outputs a sum and a carry bit; a full adder takes 2 operand bits and a carry bit as input, and outputs a sum bit and a carry bit. To add two floating point numbers, one or both need to be put in a form such that they have the same exponent. Then, the mantissas are added. Lastly, the result is normalized. We will cover floating point addition later. Subtraction is not discussed explicitly, because it is done via addition by reversing the sign of the number to be subtracted (subtrahend) and adding the result to the number subtracted from (minuend). You have to be careful when subtracting floating point numbers, because of the possible large round off error in the result.
Study this article to learn how multiplication is performed using basic steps that are carried out by elemental logic components, such as an adder and shifter. Two numbers are loaded into registers (fast word storage) and multiplication is carried out by repeated addition and shifting (moving the bits in a register to the right or left). Multiplication by an integer is performed by shifting the bits in the word to the left. Each shift multiplys by 2. Note that the ALU consists of the adder and associated circuitry that can also do ands, ors, inversions and other operations, so multiplications can be done by the adder or even by programs steps which can do the same process. A hardware multiplier can also be used, which is faster than using the adder and shifter or using programmming. To multiply two floating point numbers, add the exponents, multiply the mantissas, normalize the mantissa of the product, and adjust the exponent accordingly. Floating point multiplication will be covered later.
Read this article which gives another overview of multipliers.
4.4: Floating Point Arithmetic
Read these sections, which explain error analysis when numbers are represented using a fixed number of digits. This issue mostly arises when using floating-point numbers. Real numbers are represented using a fixed number of bits. The number of bits is the precision of the representation. The accuracy of the representation is described in terms of the difference between the actual number and its representation using a fixed number of bits. This difference is the error of the representation. The accuracy becomes more significant because computations can cause the error to get so large that the result is meaningless and potentially even high risk depending on the application. These sections also explain how programming languages approach number representation.
Read this article to get the basics of division. Again one can divide by an integer by shifting right. Each shift is a divide by 2. Note also that using the add and shift hardware or using an algorithm to do the same thing is slower that using a hardware divider.
This article nicely sums up what the ALU does. It includes block diagrams of the ALU, not present in the other articles.
4.6: Case Study: Floating Point Arithmetic in an x86 Processor
Read this article to learn about minimizing roundoff and overflow in floating point arithmetic using extended precision.
Unit 4 Assessment
- Receive a grade
Take this assessment to see how well you understood this unit.
- This assessment does not count towards your grade. It is just for practice!
- You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
- You can take this assessment as many times as you want, whenever you want.