Computer Arithmetic

1.1 ARITHMETIC AND LOGIC UNIT

 The arithmetic/logic unit (ALU) contains circuitry that executes the arithmetic and logical operations. The unit can perform four arithmetic operations: addition, subtraction, multiplication, and division. Its logical operations usually involve making comparisons that test for three conditions: the equal-to condition, the less-than condition, and the greater-than condition. The computer can test for more than one condition at once, so it can discern three other conditions as well: less-than-or-equal-to, greater-than-or-equal-to, and less-than-or-greater-than (not-equal-to). All of the other elements of the computer system—control unit, registers, memory, I/O—are there mainly to bring data into the ALU for it to process and then to take the results back out. We have, in a sense, reached the core or essence of a computer when we consider the ALU. In Figure 1.1 showing the ALU is interconnected with the rest of the processor. Data are presented to the ALU in registers, and the results of an operation are stored in registers. These registers are temporary storage locations within the processor that are connected by signal paths to the ALU.

Figure1.1 ALU Input and Output

1.2 INTEGER REPRESENTATION

The fact that computers are finite has important design implications. It means that computers can never faithfully represent the sets of integers or real numbers, both of which are infinite. Since these are generally what we work with in physics and mathematics, it’s important to understand how we approximately represent integers and reals on a computer.

1.2.1 SIGN-MAGNITUDE REPRESENTATION

The sign-magnitude binary format is the simplest conceptual format. To represent a number in sign-magnitude, we simply use the leftmost bit to represent the sign, where 0 means positive, and the remaining bits to represent the magnitude (absolute value).

1.2.2 TWO’S COMPLIMENT

Two's complement number representation is used for signed numbers on most modern computers. This notation allows a computer to add and subtract numbers using the same operations (thus we do not need to implement adders and subtractors). We can characterize two's complement notation as:

A fixed number of bits are used to represent numbers

The most significant bit is called the sign bit

This same notation is used to represent both positive and negative numbers.

1.3 INTEGER ARITHMETIC

This section examines common arithmetic functions on integer representations.

1.3.1 ADDITION AND SUBTRACTION OF INTEGER

Rule of ADDING and SUBTRACTING INTEGERS.

1.3.2 MULTIPLICATION AND DIVISIONOF INTEGER

Rule of MULTIPLYING and DIVIDING INTEGERS.

1.4 FLOATING POINT

With a fixed-point notation (e.g., twos complement) it is possible to represent a range of positive and negative integers centered on 0. By assuming a fixed binary or radix point, this format allows the representation of numbers with a fractional component as well.

1.5 FLOATING POINT ARITHMETIC 

1.5.1 FLOATING POINT ADDITION AND SUBTRACTION 

For addition and subtraction, it is necessary to ensure that both operands have the same exponent value. There are four basic phases of the algorithm for addition and subtraction:

1. Check for zeros.

2. Align the significands.

3. Add or subtract the significands.

4. Normalize the result.

1.5.2 FLOATING POINT MULTIPLICATION AND DIVISION

Check for zeros

Add/subtract exponents

Multiply/divide significands (watch sign)

Normalize

Rounding

CONCLUSION 

Computer arithmetic is a very old field that is now blossoming with recent discoveries. These ideas have come from education and changes in the technology and number systems. Education plays a key part in this advancement; understanding through proofs is key to insuring correctness in implementations. It is also vital that these descriptions express the underlying meaning of the concepts. The field of computer arithmetic is forever expanding around these discoveries and elegant proofs of basic concepts.
BOOK

1.College Algebra 4th Edition William Hart 1943

2.Data Communications Principle Richard D. Giltin "Calculations of performance - Binary Signals"

3.Digital Electronics and Logic Design B. SOMANATHAN NAIR "Bolean Algebra and Logic Simplification"

EBOOK

1.Computer Organization and Architecture Willian Stallings 7th Edition.

WEBSITE

1.Addition and Subtraction of Integer http://mypages.iit.edu/~smart/dvorber/lesson4.htm

2.Multiplication and Division of Integers http://www.mathplanet.com/education/pre-algebra/explore-and-understand-integers/multiplying-and-dividing-with-integers

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