data representation and computer arithmetic

• Trending Now
• Foundational Courses
• Data Science
• Practice Problem
• Machine Learning
• System Design
• DevOps Tutorial
• Computer Organization and Architecture Tutorial

Basic Computer Instructions

• What is Computer
• Issues in Computer Design
• Difference between assembly language and high level language
• Addressing Modes
• Difference between Memory based and Register based Addressing Modes
• Computer Organization | Von Neumann architecture
• Harvard Architecture
• Interaction of a Program with Hardware
• Simplified Instructional Computer (SIC)
• Instruction Set used in simplified instructional Computer (SIC)
• Instruction Set used in SIC/XE
• RISC and CISC in Computer Organization
• Vector processor classification
• Essential Registers for Instruction Execution
• Introduction of Single Accumulator based CPU organization
• Introduction of Stack based CPU Organization
• Machine Control Instructions in Microprocessor
• Very Long Instruction Word (VLIW) Architecture

Input and Output Systems

• Computer Organization | Different Instruction Cycles
• Machine Instructions
• Computer Organization | Instruction Formats (Zero, One, Two and Three Address Instruction)
• Difference between 2-address instruction and 1-address instructions
• Difference between 3-address instruction and 0-address instruction
• Register content and Flag status after Instructions
• Debugging a machine level program
• Vector Instruction Format in Vector Processors
• Vector instruction types

Instruction Design and Format

• Introduction of ALU and Data Path
• Computer Arithmetic | Set - 1
• Computer Arithmetic | Set - 2
• Difference between 1's Complement representation and 2's Complement representation Technique
• Restoring Division Algorithm For Unsigned Integer
• Non-Restoring Division For Unsigned Integer
• Computer Organization | Booth's Algorithm
• How the negative numbers are stored in memory?

Microprogrammed Control

• Computer Organization | Micro-Operation
• Microarchitecture and Instruction Set Architecture
• Types of Program Control Instructions
• Difference between CALL and JUMP instructions
• Computer Organization | Hardwired v/s Micro-programmed Control Unit
• Implementation of Micro Instructions Sequencer
• Performance of Computer in Computer Organization
• Introduction of Control Unit and its Design
• Computer Organization | Amdahl's law and its proof
• Subroutine, Subroutine nesting and Stack memory
• Different Types of RAM (Random Access Memory )
• Random Access Memory (RAM) and Read Only Memory (ROM)
• 2D and 2.5D Memory organization

Input and Output Organization

• Priority Interrupts | (S/W Polling and Daisy Chaining)
• I/O Interface (Interrupt and DMA Mode)
• Direct memory access with DMA controller 8257/8237
• Computer Organization | Asynchronous input output synchronization
• Programmable peripheral interface 8255
• Synchronous Data Transfer in Computer Organization
• Introduction of Input-Output Processor
• MPU Communication in Computer Organization
• Memory mapped I/O and Isolated I/O

Memory Organization

• Introduction to memory and memory units
• Memory Hierarchy Design and its Characteristics
• Register Allocations in Code Generation
• Cache Memory
• Cache Organization | Set 1 (Introduction)
• Multilevel Cache Organisation
• Difference between RAM and ROM
• What's difference between CPU Cache and TLB?
• Introduction to Solid-State Drive (SSD)
• Read and Write operations in Memory
• Instruction Level Parallelism
• Computer Organization and Architecture | Pipelining | Set 1 (Execution, Stages and Throughput)
• Computer Organization and Architecture | Pipelining | Set 3 (Types and Stalling)
• Computer Organization and Architecture | Pipelining | Set 2 (Dependencies and Data Hazard)
• Last Minute Notes Computer Organization

COA GATE PYQ's AND COA Quiz

• Computer Organization and Architecture
• Digital Logic & Number representation
• Number Representation
• Microprocessor
• GATE CS Preparation

Computer Arithmetic | Set – 1

  Negative Number Representation

• Sign Magnitude

Sign magnitude is a very simple representation of negative numbers. In sign magnitude the first bit is dedicated to represent the sign and hence it is called sign bit.

Sign bit ‘1’ represents negative sign.

Sign bit ‘0’ represents positive sign.

In sign magnitude representation of a n – bit number, the first bit will represent sign and rest n-1 bits represent magnitude of number.

For example,

• +25 = 011001

Where 11001 = 25

And 0 for ‘+’

• -25 = 111001

And 1 for ‘-‘.

Range of number represented by sign magnitude method = -(2 n-1 -1) to +(2 n-1 -1) (for n bit number)

But there is one problem in sign magnitude and that is we have two representations of 0

+0 = 000000

– 0 = 100000

• 2’s complement method

To represent a negative number in this form, first we need to take the 1’s complement of the number represented in simple positive binary form and then add 1 to it.

For example:

(-8)10 = (1000)2

1’s complement of 1000 = 0111

Adding 1 to it, 0111 + 1 = 1000

So, (-8)10 = (1000)2

Please don’t get confused with (8)10 =1000 and (-8)10=1000 as with 4 bits, we can’t represent a positive number more than 7. So, 1000 is representing -8 only.

Range of number represented by 2’s complement = (-2 n-1 to 2 n-1 – 1)

  Floating point representation of numbers

• 32-bit representation floating point numbers IEEE standard

• Floating point numbers are usually normalized
• Exponent is adjusted so that leading bit (MSB) of mantissa is 1
• Since it is always 1 there is no need to store it
• Scientific notation where numbers are normalized to give a single digit before the decimal point like in decimal system e.g. 3.123 x 10 3

For example, we represent 3.625 in 32 bit format. Changing 3 in binary=11 Changing .625 in binary

Writing in binary exponent form

3.625=11.101 X 2 0

On normalizing

11.101 X 2 0 =1.1101 X 2 1

On biasing exponent = 127 + 1 = 128

(128)10=(10000000) 2

For getting significand Digits after decimal = 1101 Expanding to 23 bit = 11010000000000000000000 Setting sign bit As it is a positive number, sign bit = 0 Finally we arrange according to representation 

• 64-bit representation floating point numbers IEEE standard

  Writing in binary exponent form 3.625 = 11.101 X 2 0 On normalizing 11.101 X 2 0 = 1.1101 X 2 1 On biasing exponent 1023 + 1 = 1024 (1024)10 = (10000000000)2 So 11 bit exponent = 10000000000 52 bit significand = 110100000000 …………. making total 52 bits Setting sign bit = 1 (number is negative) So, final representation 1 10000000000 110100000000 …………. making total 52 bits by adding further 0’s    Converting floating point into decimal Let’s convert a FP number into decimal 1 01111100 11000000000000000000000 The decimal value of an IEEE number is given by the formula: (1 -2s) * (1 + f) * 2 ( e – bias ) where

• s, f and e fields are taken as decimal here.
• (1 -2s) is 1 or -1, depending upon sign bit 0 and 1
• add an implicit 1 to the significand (fraction field f), as in formula

Again, the bias is either 127 or 1023, for single or double precision respectively.   First convert each individual field to decimal. 

• The sign bit s is 1
• The e field contains 01111100 = (124)10
• The mantissa is 0.11000 … = (0.75)10

Putting these values in formula (1 – 2) * (1 + 0.75) * 2 124 – 127 = ( – 1.75 * 2 -3 ) = – 0.21875     This article has been contributed by Anuj Batham.  

Computer arithmetic is the branch of computer science that deals with the representation and manipulation of numerical quantities in a computer system. Here are some basic concepts and operations involved in computer arithmetic:

• Number systems: Computers use different number systems to represent numerical quantities, including binary (base 2), decimal (base 10), and hexadecimal (base 16) systems. In binary system, each digit can only be either 0 or 1, while in decimal system, each digit can be any of the 10 digits from 0 to 9.
• Arithmetic operations: The basic arithmetic operations used in computer arithmetic are addition, subtraction, multiplication, and division. These operations are usually performed using arithmetic circuits within the CPU.
• Overflow: In computer arithmetic, overflow occurs when the result of an arithmetic operation is too large to be represented in the available number of bits. This can result in incorrect or unexpected results.
• Floating-point arithmetic: Floating-point arithmetic is used to represent and perform operations on non-integer numbers. It involves representing a number as a combination of a mantissa (or significand) and an exponent.
• Round-off errors: Round-off errors occur in floating-point arithmetic due to the limited precision of the number representation. This can result in small inaccuracies in the computed results.
• Bitwise operations: Bitwise operations are used to manipulate individual bits in a number. The basic bitwise operations include AND, OR, XOR, and NOT.
• Two’s complement representation: Two’s complement representation is a method of representing negative numbers in binary. In this representation, the most significant bit is used as a sign bit, with 0 indicating a positive number and 1 indicating a negative number.

Overall, computer arithmetic is a fundamental aspect of computer science and is used in a wide range of applications, including scientific computing, financial analysis, and digital signal processing.

Please Login to comment...

Similar reads.

• Computer Organization & Architecture

Improve your Coding Skills with Practice

What kind of Experience do you want to share?

Data representation 1: Introduction

This course investigates how systems software works: what makes programs work fast or slow, and how properties of the machines we program impact the programs we write. We discuss both general ideas and specific tools, and take an experimental approach.

Textbook readings

• How do computers represent different kinds of information?
• How do data representation choices impact performance and correctness?
• What kind of language is understood by computer processors?
• How is code you write translated to code a processor runs?
• How do hardware and software defend against bugs and attacks?
• How are operating systems interfaces implemented?
• What kinds of computer data storage are available, and how do they perform?
• How can we improve the performance of a system that stores data?
• How can programs on the same computer cooperate and interact?
• What kinds of operating systems interfaces are useful?
• How can a single program safely use multiple processors?
• How can multiple computers safely interact over a network?
• Six problem sets
• Midterm and final
• Starting mid-next week
• Attendance checked for simultaneously-enrolled students
• Rough breakdown: >50% assignments, <35% tests, 15% participation
• Course grading: A means mastery

Collaboration

Discussion, collaboration, and the exchange of ideas are essential to doing academic work, and to engineering. You are encouraged to consult with your classmates as you work on problem sets. You are welcome to discuss general strategies for solutions as well as specific bugs and code structure questions, and to use Internet resources for general information.

However, the work you turn in must be your own—the result of your own efforts. You should understand your code well enough that you could replicate your solution from scratch, without collaboration.

In addition, you must cite any books, articles, online resources, and so forth that helped you with your work, using appropriate citation practices; and you must list the names of students with whom you have collaborated on problem sets and briefly describe how you collaborated. (You do not need to list course staff.)

On our programming language

We use the C++ programming language in this class.

C++ is a boring, old, and unsafe programming language, but boring languages are underrated . C++ offers several important advantages for this class, including ubiquitous availability, good tooling, the ability to demonstrate impactful kinds of errors that you should understand, and a good standard library of data structures.

Pset 0 links to several C++ tutorials and references, and to a textbook.

Each program runs in a private data storage space. This is called its memory . The memory “remembers” the data it stores.

Programs work by manipulating values . Different programming languages have different conceptions of value; in C++, the primitive values are integers, like 12 or -100; floating-point numbers, like 1.02; and pointers , which are references to other objects.

An object is a region of memory that contains a value. (The C++ standard specifically says “a region of data storage in the execution environment, the contents of which can represent values”.)

Objects, values, and variables

Which are the objects? Which are the values?

Variables generally correspond to objects, and here there are three objects, one for each variable i1 , i2 , and i3 . The compiler and operating system associate the names with their corresponding objects. There are three values, too, one used to initialize each object: 61 , 62 , and 63 . However, there are other values—for instance, each argument to the printf calls is a value.

What does the program print?

i1: 61 i2: 62 i3: 63

C and C++ pointer types allow programs to access objects indirectly. A pointer value is the address of another object. For instance, in this program, the variable i4 holds a pointer to the object named by i3 :

There are four objects, corresponding to variables i1 through i4 . Note that the i4 object holds a pointer value, not an integer. There are also four values: 61 , 62 , 63 , and the expression &i3 (the address of i3 ). Note that there are three integer values, but four values overall.

What does this program print?

i1: 61 i2: 62 i3: 63 value pointed to by i4: 63

Here, the expressions i3 and *i4 refer to exactly the same object. Any modification to i3 can be observed through *i4 and vice versa. We say that i3 and *i4 are aliases : different names for the same object.

We now use hexdump_object , a helper function declared in our hexdump.hh helper file , to examine both the contents and the addresses of these objects.

Exactly what is printed will vary between operating systems and compilers. In Docker in class, on my Apple-silicon Macbook, we saw:

But on an Intel-based Amazon EC2 native Linux machine:

The data bytes look similar—identical for i1 through i3 —but the addresses vary.

But on Intel Mac OS X: 103c63020 3d 00 00 00 |=...| 103c5ef60 3e 00 00 00 |>...| 7ffeebfa4abc 3f 00 00 00 |?...| 7ffeebfa4ab0 bc 4a fa eb fe 7f 00 00 |.J......| And on Docker on an Intel Mac: 56499f239010 3d 00 00 00 |=...| 56499f23701c 3e 00 00 00 |>...| 7fffebf8b19c 3f 00 00 00 |?...| 7fffebf8b1a0 9c b1 f8 eb ff 7f 00 00 |........|

A hexdump printout shows the following information on each line.

• An address , like 4000004010 . This is a hexadecimal (base-16) number indicating the value of the address of the object. A line contains one to sixteen bytes of memory starting at this address.
• The contents of memory starting at the given address, such as 3d 00 00 00 . Memory is printed as a sequence of bytes , which are 8-bit numbers between 0 and 255. All modern computers organize their memory in units of 8-bit bytes.
• A textual representation of the memory contents, such as |=...| . This is useful when examining memory that contains textual data, and random garbage otherwise.

Dynamic allocation

Must every data object be given a name? No! In C++, the new operator allocates a brand-new object with no variable name. (In C, the malloc function does the same thing.) The C++ expression new T returns a pointer to a brand-new, never-before-seen object of type T . For instance:

This prints something like

The new int{64} expression allocates a fresh object with no name of its own, though it can be located by following the i4 pointer.

What do you notice about the addresses of these different objects?

• i3 and i4 , which are objects corresponding to variables declared local to main , are located very close to one another. In fact they are just 4 bytes part: i3 directly abuts i4 . Their addresses are quite high. In native Linux, in fact, their addresses are close to 2 47 !
• i1 and i2 are at much lower addresses, and they do not abut. i2 ’s location is below i1 , and about 0x2000 bytes away.
• The anonymous storage allocated by new int is located between i1 / i2 and i3 / i4 .

Although the values may differ on other operating systems, you’ll see qualitatively similar results wherever you run ./objects .

What’s happening is that the operating system and compiler have located different kinds of object in different broad regions of memory. These regions are called segments , and they are important because objects’ different storage characteristics benefit from different treatment.

i2 , the const int global object, has the smallest address. It is in the code or text segment, which is also used for read-only global data. The operating system and hardware ensure that data in this segment is not changed during the lifetime of the program. Any attempt to modify data in the code segment will cause a crash.

i1 , the int global object, has the next highest address. It is in the data segment, which holds modifiable global data. This segment keeps the same size as the program runs.

After a jump, the anonymous new int object pointed to by i4 has the next highest address. This is the heap segment, which holds dynamically allocated data. This segment can grow as the program runs; it typically grows towards higher addresses.

After a larger jump, the i3 and i4 objects have the highest addresses. They are in the stack segment, which holds local variables. This segment can also grow as the program runs, especially as functions call other functions; in most processors it grows down , from higher addresses to lower addresses.

Experimenting with the stack

How can we tell that the stack grows down? Do all functions share a single stack? This program uses a recursive function to test. Try running it; what do you see?

Page Statistics

Table of contents.

• Introduction to Functional Computer
• Fundamentals of Architectural Design

Data Representation

• Instruction Set Architecture : Instructions and Formats
• Instruction Set Architecture : Design Models
• Instruction Set Architecture : Addressing Modes
• Performance Measurements and Issues
• Computer Architecture Assessment 1
• Fixed Point Arithmetic : Addition and Subtraction
• Fixed Point Arithmetic : Multiplication
• Fixed Point Arithmetic : Division
• Floating Point Arithmetic
• Arithmetic Logic Unit Design
• CPU's Data Path
• CPU's Control Unit
• Control Unit Design
• Concepts of Pipelining
• Computer Architecture Assessment 2
• Pipeline Hazards
• Memory Characteristics and Organization
• Cache Memory
• Virtual Memory
• I/O Communication and I/O Controller
• Input/Output Data Transfer
• Direct Memory Access controller and I/O Processor
• CPU Interrupts and Interrupt Handling
• Computer Architecture Assessment 3

Course Computer Architecture

Digital computers store and process information in binary form as digital logic has only two values "1" and "0" or in other words "True or False" or also said as "ON or OFF". This system is called radix 2. We human generally deal with radix 10 i.e. decimal. As a matter of convenience there are many other representations like Octal (Radix 8), Hexadecimal (Radix 16), Binary coded decimal (BCD), Decimal etc.

Every computer's CPU has a width measured in terms of bits such as 8 bit CPU, 16 bit CPU, 32 bit CPU etc. Similarly, each memory location can store a fixed number of bits and is called memory width. Given the size of the CPU and Memory, it is for the programmer to handle his data representation. Most of the readers may be knowing that 4 bits form a Nibble, 8 bits form a byte. The word length is defined by the Instruction Set Architecture of the CPU. The word length may be equal to the width of the CPU.

The memory simply stores information as a binary pattern of 1's and 0's. It is to be interpreted as what the content of a memory location means. If the CPU is in the Fetch cycle, it interprets the fetched memory content to be instruction and decodes based on Instruction format. In the Execute cycle, the information from memory is considered as data. As a common man using a computer, we think computers handle English or other alphabets, special characters or numbers. A programmer considers memory content to be data types of the programming language he uses. Now recall figure 1.2 and 1.3 of chapter 1 to reinforce your thought that conversion happens from computer user interface to internal representation and storage.

• Data Representation in Computers

Information handled by a computer is classified as instruction and data. A broad overview of the internal representation of the information is illustrated in figure 3.1. No matter whether it is data in a numeric or non-numeric form or integer, everything is internally represented in Binary. It is up to the programmer to handle the interpretation of the binary pattern and this interpretation is called Data Representation . These data representation schemes are all standardized by international organizations.

Choice of Data representation to be used in a computer is decided by

• The number types to be represented (integer, real, signed, unsigned, etc.)
• Range of values likely to be represented (maximum and minimum to be represented)
• The Precision of the numbers i.e. maximum accuracy of representation (floating point single precision, double precision etc)
• If non-numeric i.e. character, character representation standard to be chosen. ASCII, EBCDIC, UTF are examples of character representation standards.
• The hardware support in terms of word width, instruction.

Before we go into the details, let us take an example of interpretation. Say a byte in Memory has value "0011 0001". Although there exists a possibility of so many interpretations as in figure 3.2, the program has only one interpretation as decided by the programmer and declared in the program.

• Fixed point Number Representation

Fixed point numbers are also known as whole numbers or Integers. The number of bits used in representing the integer also implies the maximum number that can be represented in the system hardware. However for the efficiency of storage and operations, one may choose to represent the integer with one Byte, two Bytes, Four bytes or more. This space allocation is translated from the definition used by the programmer while defining a variable as integer short or long and the Instruction Set Architecture.

In addition to the bit length definition for integers, we also have a choice to represent them as below:

• Unsigned Integer : A positive number including zero can be represented in this format. All the allotted bits are utilised in defining the number. So if one is using 8 bits to represent the unsigned integer, the range of values that can be represented is 28 i.e. "0" to "255". If 16 bits are used for representing then the range is 216 i.e. "0 to 65535".
• Signed Integer : In this format negative numbers, zero, and positive numbers can be represented. A sign bit indicates the magnitude direction as positive or negative. There are three possible representations for signed integer and these are Sign Magnitude format, 1's Compliment format and 2's Complement format .

Signed Integer – Sign Magnitude format: Most Significant Bit (MSB) is reserved for indicating the direction of the magnitude (value). A "0" on MSB means a positive number and a "1" on MSB means a negative number. If n bits are used for representation, n-1 bits indicate the absolute value of the number. Examples for n=8:

Examples for n=8:

0010 1111 = + 47 Decimal (Positive number)

1010 1111 = - 47 Decimal (Negative Number)

0111 1110 = +126 (Positive number)

1111 1110 = -126 (Negative Number)

0000 0000 = + 0 (Postive Number)

1000 0000 = - 0 (Negative Number)

Although this method is easy to understand, Sign Magnitude representation has several shortcomings like

• Zero can be represented in two ways causing redundancy and confusion.
• The total range for magnitude representation is limited to 2n-1, although n bits were accounted.
• The separate sign bit makes the addition and subtraction more complicated. Also, comparing two numbers is not straightforward.

Signed Integer – 1’s Complement format: In this format too, MSB is reserved as the sign bit. But the difference is in representing the Magnitude part of the value for negative numbers (magnitude) is inversed and hence called 1’s Complement form. The positive numbers are represented as it is in binary. Let us see some examples to better our understanding.

1101 0000 = - 47 Decimal (Negative Number)

1000 0001 = -126 (Negative Number)

1111 1111 = - 0 (Negative Number)

• Converting a given binary number to its 2's complement form

Step 1 . -x = x' + 1 where x' is the one's complement of x.

Step 2 Extend the data width of the number, fill up with sign extension i.e. MSB bit is used to fill the bits.

Example: -47 decimal over 8bit representation

As you can see zero is not getting represented with redundancy. There is only one way of representing zero. The other problem of the complexity of the arithmetic operation is also eliminated in 2’s complement representation. Subtraction is done as Addition.

More exercises on number conversion are left to the self-interest of readers.

• Floating Point Number system

The maximum number at best represented as a whole number is 2 n . In the Scientific world, we do come across numbers like Mass of an Electron is 9.10939 x 10-31 Kg. Velocity of light is 2.99792458 x 108 m/s. Imagine to write the number in a piece of paper without exponent and converting into binary for computer representation. Sure you are tired!!. It makes no sense to write a number in non- readable form or non- processible form. Hence we write such large or small numbers using exponent and mantissa. This is said to be Floating Point representation or real number representation. he real number system could have infinite values between 0 and 1.

Representation in computer

Unlike the two's complement representation for integer numbers, Floating Point number uses Sign and Magnitude representation for both mantissa and exponent . In the number 9.10939 x 1031, in decimal form, +31 is Exponent, 9.10939 is known as Fraction . Mantissa, Significand and fraction are synonymously used terms. In the computer, the representation is binary and the binary point is not fixed. For example, a number, say, 23.345 can be written as 2.3345 x 101 or 0.23345 x 102 or 2334.5 x 10-2. The representation 2.3345 x 101 is said to be in normalised form.

Floating-point numbers usually use multiple words in memory as we need to allot a sign bit, few bits for exponent and many bits for mantissa. There are standards for such allocation which we will see sooner.

• IEEE 754 Floating Point Representation

We have two standards known as Single Precision and Double Precision from IEEE. These standards enable portability among different computers. Figure 3.3 picturizes Single precision while figure 3.4 picturizes double precision. Single Precision uses 32bit format while double precision is 64 bits word length. As the name suggests double precision can represent fractions with larger accuracy. In both the cases, MSB is sign bit for the mantissa part, followed by Exponent and Mantissa. The exponent part has its sign bit.

It is to be noted that in Single Precision, we can represent an exponent in the range -127 to +127. It is possible as a result of arithmetic operations the resulting exponent may not fit in. This situation is called overflow in the case of positive exponent and underflow in the case of negative exponent. The Double Precision format has 11 bits for exponent meaning a number as large as -1023 to 1023 can be represented. The programmer has to make a choice between Single Precision and Double Precision declaration using his knowledge about the data being handled.

The Floating Point operations on the regular CPU is very very slow. Generally, a special purpose CPU known as Co-processor is used. This Co-processor works in tandem with the main CPU. The programmer should be using the float declaration only if his data is in real number form. Float declaration is not to be used generously.

• Decimal Numbers Representation

Decimal numbers (radix 10) are represented and processed in the system with the support of additional hardware. We deal with numbers in decimal format in everyday life. Some machines implement decimal arithmetic too, like floating-point arithmetic hardware. In such a case, the CPU uses decimal numbers in BCD (binary coded decimal) form and does BCD arithmetic operation. BCD operates on radix 10. This hardware operates without conversion to pure binary. It uses a nibble to represent a number in packed BCD form. BCD operations require not only special hardware but also decimal instruction set.

• Exceptions and Error Detection

All of us know that when we do arithmetic operations, we get answers which have more digits than the operands (Ex: 8 x 2= 16). This happens in computer arithmetic operations too. When the result size exceeds the allotted size of the variable or the register, it becomes an error and exception. The exception conditions associated with numbers and number operations are Overflow, Underflow, Truncation, Rounding and Multiple Precision . These are detected by the associated hardware in arithmetic Unit. These exceptions apply to both Fixed Point and Floating Point operations. Each of these exceptional conditions has a flag bit assigned in the Processor Status Word (PSW). We may discuss more in detail in the later chapters.

• Character Representation

Another data type is non-numeric and is largely character sets. We use a human-understandable character set to communicate with computer i.e. for both input and output. Standard character sets like EBCDIC and ASCII are chosen to represent alphabets, numbers and special characters. Nowadays Unicode standard is also in use for non-English language like Chinese, Hindi, Spanish, etc. These codes are accessible and available on the internet. Interested readers may access and learn more.

1. Track your progress [Earn 200 points]

Mark as complete

2. Provide your ratings to this chapter [Earn 100 points]

Data Representation, Number Systems and Arithmetic Processes

Cite this chapter.

• M. G. Hartley 4 ,
• M. Healey 5 &
• P. G. Depledge 6  

Part of the book series: Macmillan Computer Science Series ((COMPSS))

31 Accesses

The basic concepts of digital computers have been introduced in the first two chapters in a largely descriptive way. In this chapter and the next, the fundamental techniques necessary to construct a working computer to implement the concepts discussed are presented. The importance of the binary technique has been stressed as a practical method of achieving the inherent speed and accuracy by removing tight tolerances from the actual voltage levels used by the electronics. The representation of non-numeric data, such as alphabetic characters, is also important. This chapter reviews the commonly used techniques.

Having established the concept of binary representation of information, then the manipulation of binary data can be treated as a logical two-state process — for example, on or off, open or shut, true or false. For simplicity the two logical states are represented as 0 or 1, drawing an immediate parallel with binary numbers. Indeed it is shown in some detail in the next chapter how logic circuits are used to implement binary arithmetic units, such as an adder.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Unable to display preview.  Download preview PDF.

Bibliography

Richard, R. K., Arithmetic Operations in Digital Computers , Van Nostrand, New York, 1955.

Google Scholar  

Download references

Author information

Authors and affiliations.

Information Systems Engineering Group, Department of Electrical Engineering and Electronics, University of Manchester Institute of Science and Technology, UK

M. G. Hartley

Department of Electrical and Electronic Engineering, University College, Cardiff, UK

Application Engineering Organisation, Hewlett-Packard Ltd, USA

P. G. Depledge

You can also search for this author in PubMed   Google Scholar

Copyright information

© 1988 M. G. Hartley, M. Healey and P. G. Depledge

About this chapter

Hartley, M.G., Healey, M., Depledge, P.G. (1988). Data Representation, Number Systems and Arithmetic Processes. In: Mini and Microcomputer Systems. Macmillan Computer Science Series. Palgrave, London. https://doi.org/10.1007/978-1-349-19315-8_3

Download citation

DOI : https://doi.org/10.1007/978-1-349-19315-8_3

Publisher Name : Palgrave, London

Print ISBN : 978-0-333-41759-1

Online ISBN : 978-1-349-19315-8

eBook Packages : Computer Science Computer Science (R0)

Share this chapter

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Publish with us

Policies and ethics

• Find a journal
• Track your research

Data Representation in Computer: Number Systems, Characters, Audio, Image and Video

• Post author: Anuj Kumar
• Post published: 16 July 2021
• Post category: Computer Science
• Post comments: 0 Comments

Table of Contents

• 1 What is Data Representation in Computer?
• 2.1 Binary Number System
• 2.2 Octal Number System
• 2.3 Decimal Number System
• 2.4 Hexadecimal Number System
• 3.4 Unicode
• 4 Data Representation of Audio, Image and Video
• 5.1 What is number system with example?

What is Data Representation in Computer?

A computer uses a fixed number of bits to represent a piece of data which could be a number, a character, image, sound, video, etc. Data representation is the method used internally to represent data in a computer. Let us see how various types of data can be represented in computer memory.

Before discussing data representation of numbers, let us see what a number system is.

Number Systems

Number systems are the technique to represent numbers in the computer system architecture, every value that you are saving or getting into/from computer memory has a defined number system.

A number is a mathematical object used to count, label, and measure. A number system is a systematic way to represent numbers. The number system we use in our day-to-day life is the decimal number system that uses 10 symbols or digits.

The number 289 is pronounced as two hundred and eighty-nine and it consists of the symbols 2, 8, and 9. Similarly, there are other number systems. Each has its own symbols and method for constructing a number.

A number system has a unique base, which depends upon the number of symbols. The number of symbols used in a number system is called the base or radix of a number system.

Let us discuss some of the number systems. Computer architecture supports the following number of systems:

Binary Number System

Octal number system, decimal number system, hexadecimal number system.

A Binary number system has only two digits that are 0 and 1. Every number (value) represents 0 and 1 in this number system. The base of the binary number system is 2 because it has only two digits.

The octal number system has only eight (8) digits from 0 to 7. Every number (value) represents with 0,1,2,3,4,5,6 and 7 in this number system. The base of the octal number system is 8, because it has only 8 digits.

The decimal number system has only ten (10) digits from 0 to 9. Every number (value) represents with 0,1,2,3,4,5,6, 7,8 and 9 in this number system. The base of decimal number system is 10, because it has only 10 digits.

A Hexadecimal number system has sixteen (16) alphanumeric values from 0 to 9 and A to F. Every number (value) represents with 0,1,2,3,4,5,6, 7,8,9,A,B,C,D,E and F in this number system. The base of the hexadecimal number system is 16, because it has 16 alphanumeric values.

Here A is 10, B is 11, C is 12, D is 13, E is 14 and F is 15 .

Data Representation of Characters

There are different methods to represent characters . Some of them are discussed below:

The code called ASCII (pronounced ‘􀀏’.S-key”), which stands for American Standard Code for Information Interchange, uses 7 bits to represent each character in computer memory. The ASCII representation has been adopted as a standard by the U.S. government and is widely accepted.

A unique integer number is assigned to each character. This number called ASCII code of that character is converted into binary for storing in memory. For example, the ASCII code of A is 65, its binary equivalent in 7-bit is 1000001.

Since there are exactly 128 unique combinations of 7 bits, this 7-bit code can represent only128 characters. Another version is ASCII-8, also called extended ASCII, which uses 8 bits for each character, can represent 256 different characters.

For example, the letter A is represented by 01000001, B by 01000010 and so on. ASCII code is enough to represent all of the standard keyboard characters.

It stands for Extended Binary Coded Decimal Interchange Code. This is similar to ASCII and is an 8-bit code used in computers manufactured by International Business Machines (IBM). It is capable of encoding 256 characters.

If ASCII-coded data is to be used in a computer that uses EBCDIC representation, it is necessary to transform ASCII code to EBCDIC code. Similarly, if EBCDIC coded data is to be used in an ASCII computer, EBCDIC code has to be transformed to ASCII.

ISCII stands for Indian Standard Code for Information Interchange or Indian Script Code for Information Interchange. It is an encoding scheme for representing various writing systems of India. ISCII uses 8-bits for data representation.

It was evolved by a standardization committee under the Department of Electronics during 1986-88 and adopted by the Bureau of Indian Standards (BIS). Nowadays ISCII has been replaced by Unicode.

Using 8-bit ASCII we can represent only 256 characters. This cannot represent all characters of written languages of the world and other symbols. Unicode is developed to resolve this problem. It aims to provide a standard character encoding scheme, which is universal and efficient.

It provides a unique number for every character, no matter what the language and platform be. Unicode originally used 16 bits which can represent up to 65,536 characters. It is maintained by a non-profit organization called the Unicode Consortium.

The Consortium first published version 1.0.0 in 1991 and continues to develop standards based on that original work. Nowadays Unicode uses more than 16 bits and hence it can represent more characters. Unicode can represent characters in almost all written languages of the world.

Data Representation of Audio, Image and Video

In most cases, we may have to represent and process data other than numbers and characters. This may include audio data, images, and videos. We can see that like numbers and characters, the audio, image, and video data also carry information.

We will see different file formats for storing sound, image, and video .

Multimedia data such as audio, image, and video are stored in different types of files. The variety of file formats is due to the fact that there are quite a few approaches to compressing the data and a number of different ways of packaging the data.

For example, an image is most popularly stored in Joint Picture Experts Group (JPEG ) file format. An image file consists of two parts – header information and image data. Information such as the name of the file, size, modified data, file format, etc. is stored in the header part.

The intensity value of all pixels is stored in the data part of the file. The data can be stored uncompressed or compressed to reduce the file size. Normally, the image data is stored in compressed form. Let us understand what compression is.

Take a simple example of a pure black image of size 400X400 pixels. We can repeat the information black, black, …, black in all 16,0000 (400X400) pixels. This is the uncompressed form, while in the compressed form black is stored only once and information to repeat it 1,60,000 times is also stored.

Numerous such techniques are used to achieve compression. Depending on the application, images are stored in various file formats such as bitmap file format (BMP), Tagged Image File Format (TIFF), Graphics Interchange Format (GIF), Portable (Public) Network Graphic (PNG).

What we said about the header file information and compression is also applicable for audio and video files. Digital audio data can be stored in different file formats like WAV, MP3, MIDI, AIFF, etc. An audio file describes a format, sometimes referred to as the ‘container format’, for storing digital audio data.

For example, WAV file format typically contains uncompressed sound and MP3 files typically contain compressed audio data. The synthesized music data is stored in MIDI(Musical Instrument Digital Interface) files.

Similarly, video is also stored in different files such as AVI (Audio Video Interleave) – a file format designed to store both audio and video data in a standard package that allows synchronous audio with video playback, MP3, JPEG-2, WMV, etc.

FAQs About Data Representation in Computer

What is number system with example.

Let us discuss some of the number systems. Computer architecture supports the following number of systems: 1. Binary Number System 2. Octal Number System 3. Decimal Number System 4. Hexadecimal Number System

Related posts:

• 10 Types of Computers | History of Computers, Advantages

What is Microprocessor? Evolution of Microprocessor, Types, Features

What is operating system functions, types, types of user interface, what is cloud computing classification, characteristics, principles, types of cloud providers.

• What is Debugging? Types of Errors
• What are Functions of Operating System? 6 Functions

What is Flowchart in Programming? Symbols, Advantages, Preparation

Advantages and disadvantages of flowcharts, what is c++ programming language c++ character set, c++ tokens, what are c++ keywords set of 59 keywords in c ++, what are data types in c++ types, what are operators in c different types of operators in c.

• What are Expressions in C? Types

What are Decision Making Statements in C? Types

Types of storage devices, advantages, examples, you might also like.

Types of Computer Memory, Characteristics, Primary Memory, Secondary Memory

What is Computer System? Definition, Characteristics, Functional Units, Components

What is Artificial Intelligence? Functions, 6 Benefits, Applications of AI

Advantages and Disadvantages of Operating System

What is Big Data? Characteristics, Tools, Types, Internet of Things (IOT)

What is Problem Solving Algorithm?, Steps, Representation

Generations of Computer First To Fifth, Classification, Characteristics, Features, Examples

Types of Computer Software: Systems Software, Application Software

Data and Information: Definition, Characteristics, Types, Channels, Approaches

• Entrepreneurship
• Organizational Behavior
• Financial Management
• Communication
• Human Resource Management
• Sales Management
• Marketing Management

SE CSE SEM 3 – DIGITAL LOGIC & COMPUTER ARCHITECTURE

Module 2- Data Representation & Arithmetic Algorithms

Username or Email Address

Remember Me

Confirm Password

 Subscribe to our Free Newsletter

Registration confirmation will be emailed to you.

• Computers & Technology

Enjoy fast, free delivery, exclusive deals, and award-winning movies & TV shows with Prime Try Prime and start saving today with fast, free delivery

Amazon Prime includes:

Fast, FREE Delivery is available to Prime members. To join, select "Try Amazon Prime and start saving today with Fast, FREE Delivery" below the Add to Cart button.

• Cardmembers earn 5% Back at Amazon.com with a Prime Credit Card.
• Unlimited Free Two-Day Delivery
• Streaming of thousands of movies and TV shows with limited ads on Prime Video.
• A Kindle book to borrow for free each month - with no due dates
• Listen to over 2 million songs and hundreds of playlists
• Unlimited photo storage with anywhere access

Important:  Your credit card will NOT be charged when you start your free trial or if you cancel during the trial period. If you're happy with Amazon Prime, do nothing. At the end of the free trial, your membership will automatically upgrade to a monthly membership.

Buy new: $43.09

Return this item for free.

Free returns are available for the shipping address you chose. You can return the item for any reason in new and unused condition: no shipping charges

• Go to your orders and start the return
• Select the return method

Download the free Kindle app and start reading Kindle books instantly on your smartphone, tablet, or computer - no Kindle device required .

Read instantly on your browser with Kindle for Web.

Using your mobile phone camera - scan the code below and download the Kindle app.

Image Unavailable

• To view this video download Flash Player

A Hand Book on Data representation and Computer Arithmetic: A Guide for data representation in computers for all beginners of computer studies

Purchase options and add-ons.

• ISBN-10 3659459585
• ISBN-13 978-3659459580
• Publication date March 18, 2015
• Language English
• Dimensions 5.91 x 0.13 x 8.66 inches
• Print length 56 pages
• See all details

Product details

• Publisher ‏ : ‎ LAP LAMBERT Academic Publishing (March 18, 2015)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 56 pages
• ISBN-10 ‏ : ‎ 3659459585
• ISBN-13 ‏ : ‎ 978-3659459580
• Item Weight ‏ : ‎ 3.42 ounces
• Dimensions ‏ : ‎ 5.91 x 0.13 x 8.66 inches

Customer reviews

Customer Reviews, including Product Star Ratings help customers to learn more about the product and decide whether it is the right product for them.

To calculate the overall star rating and percentage breakdown by star, we don’t use a simple average. Instead, our system considers things like how recent a review is and if the reviewer bought the item on Amazon. It also analyzed reviews to verify trustworthiness.

No customer reviews

• Amazon Newsletter
• About Amazon
• Accessibility
• Sustainability
• Press Center
• Investor Relations
• Amazon Devices
• Amazon Science
• Sell on Amazon
• Sell apps on Amazon
• Supply to Amazon
• Protect & Build Your Brand
• Become an Affiliate
• Become a Delivery Driver
• Start a Package Delivery Business
• Advertise Your Products
• Self-Publish with Us
• Become an Amazon Hub Partner
• › See More Ways to Make Money
• Amazon Visa
• Amazon Store Card
• Amazon Secured Card
• Amazon Business Card
• Shop with Points
• Credit Card Marketplace
• Reload Your Balance
• Amazon Currency Converter
• Your Account
• Your Orders
• Shipping Rates & Policies
• Amazon Prime
• Returns & Replacements
• Manage Your Content and Devices
• Recalls and Product Safety Alerts
• Conditions of Use
• Privacy Notice
• Consumer Health Data Privacy Disclosure
• Your Ads Privacy Choices

• Computer Fundamentals

Computer Network

Control System

• Interview Q

COA Tutorial

Basic co and design, computer instructions, digital logic circuits, map simplification, combinational circuits, flip - flops, digital components, register transfer, micro-operations, memory organization.

• Send your Feedback to [email protected]

Help Others, Please Share

Learn Latest Tutorials

Transact-SQL

Reinforcement Learning

R Programming

React Native

Python Design Patterns

Python Pillow

Python Turtle

Preparation

Verbal Ability

Interview Questions

Company Questions

Trending Technologies

Artificial Intelligence

Cloud Computing

Data Science

Machine Learning

B.Tech / MCA

Data Structures

Operating System

Compiler Design

Computer Organization

Discrete Mathematics

Ethical Hacking

Computer Graphics

Software Engineering

Web Technology

Cyber Security

C Programming

Data Mining

Data Warehouse

• Bihar Board

SRM University

Hpbose 12th result.

• HP Board Result 2024
• Punjab Board Result 2024
• JAC Board Result 2024
• WB Board Result 2024
• UK Board Result 2024
• Karnataka Board Result 2024
• Shiv Khera Special
• Education News
• Web Stories
• Current Affairs
• नए भारत का नया उत्तर प्रदेश
• School & Boards
• College Admission
• Govt Jobs Alert & Prep
• GK & Aptitude

ISC Class 11th Computer Science Syllabus 2024-25: Download PDF for ISC Class 11 Computer Science

ISC Class 11 Computer Science Syllabus 2024-25: Download the ISC Class 11 Computer Science Syllabus for the academic session 2024-25 here. Check out the complete syllabus to know the list of topics and exam pattern and evaluation scheme prescribed for ISC Class 11 Computer Science.

ISC Class 11 Computer Science Syllabus: Aims

• To understand algorithmic problem solving using data abstractions, functional and procedural abstractions, and object based and object-oriented abstractions. 
• To understand: (a) how computers represent, store and process data at different levels of abstraction that mediate between the machine and the algorithmic problem solving level and (b) how they communicate with the outside world. 
• To create awareness of ethical issues related to computing and to promote safe, ethical behavior. 
• To make students aware of future trends in computing. Aims (Skills) To devise algorithmic solutions to problems and to be able to code, validate, document, execute and debug the solution using the Java programming system.

ISC Class 11 Computer Science Syllabus Key Highlights

Isc class 11 computer science syllabus 2024-25, paper i: theory - 70 marks , paper ii: project work - 30 marks  .

Candidates will be required to have completed two projects from any topic covered in Theory. The written outcome of the project should be structured as given below: 

• Planning Session 
• Examination Session

The total time to be spent on the Planning session and the Examination session is three hours. A maximum of 90 minutes is permitted for the Planning session and 90 minutes for the Examination session. Candidates are to be permitted to proceed to the Examination Session only after the 90 minutes of the Planning Session are over.

The candidates will be required to prepare an algorithm and a hand-written Java program to solve the problem.

• Examination Session 

The program handed in at the end of the Planning session shall be returned to the candidates. The candidates will be required to key-in and execute the Java program on seen and unseen inputs individually on the Computer and show execution to the examiner. A printout of the program listing, including output results should be attached to the answer script containing the algorithm and handwritten program. This should be returned to the examiner. The program should be sufficiently documented so that the algorithm, representation and development process is clear from reading the program. Large differences between the planned program and the printout will result in loss of marks. 

EVALUATION 

Continuous evaluation , terminal evaluation, list of suggested assignments for project work: .

• Creating an expert system for road-traffic management (routing and re-routing of vehicles depending on congestion). 
• Creating an expert system for medical diagnosis on the basis of symptoms and prescribe a suitable treatment. 
• Creating a security system for age-appropriate access to social media. 
• Simulate Adders using Arduino Controllers and Components.
• Simulate a converter of Binary to Decimal number systems using Arduino Controllers and Components. 
• Develop a console-based application using Java for Movie Ticket Reservation. 
• Develop a console-based application using Java to encrypt and decrypt a message (using cipher text, Unicode-exchange, etc). 
• Develop a console-based application using Java to find name of the bank and branch location from IFSC.
• Develop a console-based application using Java to calculate taxable income (only direct tax). 
• Develop a console-based application using Java to develop a simple text editor (text typing, copy, cut, paste, delete).

Get here latest School , CBSE and Govt Jobs notification in English and Hindi for Sarkari Naukari and Sarkari Result . Download the Jagran Josh Sarkari Naukri App . Check  Board Result 2024  for Class 10 and Class 12 like  CBSE Board Result ,  UP Board Result ,  Bihar Board Result ,  MP Board Result ,  Rajasthan Board Result  and Other States Boards.

• hpbose.org 12th Result 2024
• HP Board 12th Result 2024 with Jagran Josh
• HP Board 12th Result 2024
• HPBOSE Class 12th Result 2024
• HPBOSE 12th Result 2024
• HP Board 12th Result 2024 Roll Number
• HPBOSE 12th Toppers List 2024
• एमपी बोर्ड 10 वीं टॉपर लिस्ट 2024
• AP SSC Results 2024 Manabadi by Jagran Josh
• AP SSC Topper List 2024

Latest Education News

UGC NET Exam 2024 Postponed, Check New Exam Date Here

HPBOSE 12th Toppers List 2024 OUT: कामाक्षी और छाया बनी प्रदेश टॉपर, जिलेवार छात्रों की रैंक यहां देखें

Find 3 differences between the pictures of a teacher in the classroom in 11 seconds!

IPL 2024 Stats: आईपीएल में किसने फेंकी थी पहली गेंद, किसने लिया पहला विकेट और किसने जड़ा पहला छक्का

[Official] TS SSC Results 2024 Date and Time: Check Manabadi Telangana 10th Result Notice and Official Links

HPBOSE 12th Result 2024 OUT: एसएमएस और डिजिलॉकर पर आज जारी होगी एचपी बोर्ड इंटर की मार्कशीट

HPBOSE 12th Compartment Exam 2024: फेल छात्रों को एक और मौका, कब, कैसे और कहां से करें आवेदन

HPBOSE 12th Result 2024 Scrutiny: 12वीं पुनर्मूल्यांकन के साथ कैसे बढ़ेंगे नंबर, यहां देखें विवरण

Madras University Revaluation Results 2024 OUT at unom.ac.in: Direct Link to Download UG and PG Marksheet

[Official] JAC Board 12th Result 2024 Date and Time Notification Released: Check Notice for Jharkhand Class 12 Results Here

Fastest 50s In IPL History: किसने जड़ा है आईपीएल इतिहास का सबसे तेज़ अर्द्धशतक? देखें पूरी लिस्ट

IPL 2024 Full Schedule: आईपीएल 2024 का फुल शेड्यूल,आज किस टीम का है मैच जानें यहां    

Dr MGR Medical University Result 2024 OUT at tnmgrmu.ac.in; Direct Link to Download UG and PG Marksheet

MSDSU Result 2024 OUT at msdsu.ac.in, Direct Link to Download UG and PG Odd Semester Marksheet

SRCC Gears Up For its Annual HR Fest in Partnership with Jagran Josh, Key Focus on Igniting Minds

Kerala PSC Final Answer Key 2024 Out For Assistant Professor Posts: Here's Direct Link

UPPSC Medical Officer Syllabus 2024: Check Important Topics and Exam Pattern

AP SET Answer Key 2024: Paper 1 and Paper 2 Direct Download Link to Active Soon at apset.net.in

HP Board 12th Result Analysis 2024: 73.76% Students Passed, Girls Outshine Boys, Check More Stats