data representation in computer ss3

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SSS 3 Computer Notes

Introduction to world wide web (www).

WWW is an acronym that stands for World Wide Web). WWW is part of the internet that contains linked b, images, sound, and video documents. It is also defined as an information space where documents and other web resources are identified by URLs, interlinked by hypertext links, and can be accessed via the Internet. ... CLICK HERE FOR THE FULL NOTE

Cables and Connectors

Network Cables are mediums through which information usually moves from one network to another. Types of Network Cables Network cables include the following types: a. Unshielded Twisted Pair (UTP) Cables b. Shielded Twisted Pair (STP) Cables c. Coaxial Cables d. Fibre Optics e. Telephone... CLICK HERE FOR THE FULL NOTE

BASIC Programming III (One Dimensional Array)

An array is a list of variables of the same kind. A variable is a name the computer assigns value to. To create an array, the DIM (dimension) command is used. The DIM statement has the following syntax: DIM arrayName(n). For example DIM Score (5) will reserve 6 spaces, Score (0) Score (1), Score (2), Score (3), Score (4) and Score (5) in the memory to hold value. The number inside the parentheses of the individual variables are called subscripts, and each variable is called a subscripted variable or element... CLICK HERE FOR THE FULL NOTE

High Level Language

High Level Languages are programming languages that allow for programs to be written in forms that are readable to human beings. A high-level language is a programming language that, in comparison to low-level programming languages, maybe more abstract, easier to use, or more portable across platforms... CLICK HERE FOR THE FULL NOTE

Data Representation

Data representation refers to the methods used to internally represent information stored in a computer. Computers store a lot of different types of information which include: Numbers, Text, Graphics, and Sound. At least all seem different to us. However, all types of information stored in a computer are stored internally in the same format: a sequence of 0’s and 1’s. .... CLICK HERE FOR THE FULL NOTE

A computer network consists of a collection of computers, printers and other equipment that are connected together so that they can communicate with each other. It is also defined as a group of two or more computer systems linked together. Computer Networking is the scientific and engineering discipline concerned with communication between computer systems. ... CLICK HERE FOR THE FULL NOTE

Database is a collection of related data organized for rapid search and retrieval. It can also be defined as a persistent, logically coherent collection of inherently meaningful data, relevant to some aspects of the real world. ... CLICK HERE FOR THE FULL NOTE

Introduction to CorelDraw

Overview of number system.

A number system is a collection of symbols used to represent small numbers, together with a system of rules for representing larger numbers. There are various number systems, some are examined below: Decimal Number System The decimal numeral system (also called base ten or occasionally denary) uses various symbols (digits) for no more than ten distinct values (0, 1, 3, 4, 5, 6, 7, 8, and 9) ... CLICK HERE FOR THE FULL NOTE

Security and Ethics

Data security is the practice of keeping data protected from corruption and unauthorized access. Sources of Security Breaches 1. Viruses, Worms and Trojan horses: Viruses, worms and Trojan Horses are all malicious programs that can cause damage to your computer, but there are differences among the three, and knowing those differences can help you to better protect your computer from their often damaging effects. Virus: A computer virus attaches itself to a program or files so it can spread from one computer to another, leaving infections as it travels. It is important to note that a virus cannot be spread without a human action, (such as running an infected program) to keep it going. ... CLICK HERE FOR THE FULL NOTE

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SS3 Computer Studies Lesson Notes

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Take quiz on SS3 Computer Studies 1st Term Take quiz on SS3 Computer Studies 2nd Term Take quiz on SS3 Computer Studies 3rd Term

Topics covered in the NERDC curriculum are listed below. Theme 1 Information And Communications Technology 1. Networking 2. Introduction to World Wide Web (WWW) 3. Cables & Connectors

Theme 2 Computer Applications 1. Databases 2. Graphics (Introduction to CorelDraw) Theme 3 Problems – Solving Skills 1. BASIC programme III (One dimensional array) 2. High Level Languages (H.L.L.)

Theme 4 Coding Systems In Computer 1. Overview of Number BASES 2. Data Representation Theme 5 Computer Ethics 1. Security and Ethics

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5. Data Representation

Introduction

Computers are machines that do stuff with information. They let you view, listen, create, and edit information in documents, images, videos, sound, spreadsheets and databases. They let you play games in simulated worlds that don’t really exist except as information inside the computer’s memory and displayed on the screen. They let you compute and calculate with numerical information; they let you send and receive information over networks. Fundamental to all of this is that the computer has to represent that information in some way inside the computer’s memory, as well as storing it on disk or sending it over a network.

Chapter sections

• 5.1. What's the big picture?
• 5.2. Getting started
• 5.3. Numbers
• 5.5. Images and Colours
• 5.6. Program Instructions
• 5.7. The whole story!
• 5.8. Further reading

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.

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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

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SS3 COMPUTER SCHEME OF WORK – 1st, 2nd, 3rd Term

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• Introduction to the world wide web
• Cables and connectors
• Graphics (introduction to corel draw)
• Programming iii (one dimensional array)
• High level languages (HLL)
• Overview of number bases
• Data representation
• Security and ethics

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SS1 Information and Communication Technology (ICT) lesson notes and questions & answers.

Recommended : SS3 ICT lesson notes .

• Introduction to The World Wide Web
• Uses/Benefit Of World Wide Web
• Basic Programing III
• Basic Programming IV
• High Level Language II
• Overview Of Number Bases
• Overview Of Number Bases II
• Database II
• Graphics (Introduction To Corel Draw)
• Graphics (Corel Draw) II

SECOND TERM

• Data Representation
• Security and Ethics
• Security And Ethics II
• Security And Ethics III

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Overview of Number Bases II

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Welcome to class! 

In today’s class, we will be talking more about the overview of number bases. Enjoy the class!

There are some other binary-related terms you’ll need to know. Firstly, a  bit  is a  binary digit  – i.e. a single occurrence of 0 or 1. This is the smallest unit of storage you can have inside a computer. Groups of 8 bits are called  bytes . A byte can be used to represent a number, or colour, or a character (e.g. using ASCII). You may also hear the term nibble , which is 4 bits. Finally, a  word is the largest numbers of bits that a processor can handle in one go – for example, when we say that new computers have 64-bit processors, we mean that the word length is 64-bits or 8 bytes.

The largest value that you can store using a particular number of bits can be determined quite easily. Using n  bits, the largest value you can store is  2 n  – 1 , and the number of different values you can store is  2 n  (from 1 to  2 n  – 1 , and then 0 as well). So using 8 bits, the largest number you can store is 2 8 – 1 = 255, and the number of possible values is 2 8  = 256 (i.e. 0 – 255). A 32-bit computer can therefore handle values up to 4,194,967,296 in one clock cycle – it can cope with larger numbers, but it would need to split them up first.

Octal (Base 8):

I’ve never come across anything that uses octal! I think it’s probably included on exam specifications for purely academic reasons, and because it’s easy to convert into binary (see below).

Hexadecimal (Base 16):

Hexadecimal is still used quite a lot – particularly for things like colours in HTML or programming languages. It’s also quite useful because representations of large numbers are relatively compact, but are easily converted to binary so that you can see the bit patterns.

Shifting Bits

You’ve no doubt noticed that with numbers in base 10, you can move the digits left or right one place by multiplying or dividing the number by 10. The same trick works with different number bases – you just multiply and divide by the base number (e.g. multiply by 2 in binary to shift the bits left one place).

This can be useful for things like creating hexadecimal colour values (e.g. for web pages). In a 24-bit system (such as HTML), colours are represented by 24-bit numbers from 000000 to FFFFFF (each hexadecimal digit corresponds to 4 bits – see below). The 24 bits are made up of 8 bits each for the amount of red, green and blue in the colour.

So, each component is represented by 8 bits – i.e. a number from 0 to 255. If you know how much red, green and blue you want, how do you combine them to find the complete colour? For HTML, the correct order of the bits is RRGGBB (r = red, g = green, b = blue), so what we need to do is “shift” the values of green and red components, and then add all three components together.

We can leave the blue value as it is, but we need to move the green value along two places. To move along one place in hexadecimal, we multiply by 16, so to move along two places, just do it twice – 16 x 16 = 256 – so multiply the green value by 256. For the red value, we need to move four places – 16 x 16 x 16 x 16 = 65,536 – so we multiply the value of the red component by 65,536.

If you were just trying to work out the colour yourself, you wouldn’t need to go through these steps, but if you were to create a program like my colour mixer, then this is how you’d do it.

Let’s look at base-two, or binary, numbers. How would you write, for instance, 12 10  (“twelve, base ten”) as a binary number? You would have to convert to base-two columns, the analogue of base-ten columns. In base ten, you have columns or “places” for 10 0  = 1, 10 1  = 10, 10 2  = 100, 10 3  = 1000, and so forth. Similarly in base two, you have columns or “places” for 2 0  = 1, 2 1  = 2, 2 2  = 4, 2 3  = 8, 2 4  = 16, and so forth.

The first column in base-two math is the units column. But only “0” or “1” can go in the units column. When you get to “two”, you find that there is no single solitary digit that stands for “two” in base-two math. Instead, you put a “1” in the twos column and a “0” in the units column, indicating “1 two and 0 ones”. The base-ten “two” (2 10 ) is written in binary as 10 2 .

A “three” in base two is actually “1 two and 1 one”, so it is written as 11 2 . “Four” is two-times-two, so we zero out the twos column and the units column, and put a “1” in the fours column; 4 10  is written in binary form as 100 2 . Here is a listing of the first few numbers:

Converting between binary and decimal numbers is fairly simple, as long as you remember that each digit in the binary number represents a power of two.

Convert 101100101 2  to the corresponding base-ten number.

I will list the digits in order, and count them off from the RIGHT, starting with zero:

The first row above (labelled “digits”) contains the digits from the binary number; the second row (labelled” numbering”) contains the power of 2 (the base) corresponding to each digits. I will use this listing to convert each digit to the power of two that it represents:

1×2 8  + 0×2 7  + 1×2 6  + 1×2 5  + 0×2 4  + 0×2 3  + 1×2 2  + 0×2 1  + 1×2 0

= 1×256 + 0×128 + 1×64 + 1×32 + 0×16 + 0×8 + 1×4 + 0×2 + 1×1

= 256 + 64 + 32 + 4 + 1

= 357    All Rights Reserved

Then 101100101 2  converts to 357 10

Operations with binary numbers

We can add, subtract and multiply binary numbers in much the same ways as we operate with base ten numbers. The main things to remember in base two are:

0 + 0 = 0                      1 + 0 = 1

0 + 1 = 1                        1 + 1 = 10

Multiplication:

0 X 0 = 0                       1 X 0 = 0

0 X 1 = 0                        1 X 1 = 1

Add the following

• (1010.011) 2 = (a) (10.365) 10   (b) (10.375) 10   (c) (11.365) 10    (d) (11.375) 10
• (41)⊂10 in binary is (a) 101101   (b) 101011  (c) 101001  (d) 101101
• Convert (0.6875)⊂10 to binary (a) 0.1011  (b) 0.1011  (c) 0.0101  (d) 0.0111
• Convert (153.513) 10 in octal number system is(a) 231.408517  (b) 231.407517  (c) 231.406517  (d) 231.406617
• All these are different types of number systems except (a) Arabic  (b) Babylonian  (c) Roman  (d) Nigerian

In our next class, we will be talking about Data Representation.   We hope you enjoyed the class.

Should you have any further question, feel free to ask in the comment section below and trust us to respond as soon as possible.

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Computer Science > Computation and Language

Title: reft: representation finetuning for language models.

Abstract: Parameter-efficient fine-tuning (PEFT) methods seek to adapt large models via updates to a small number of weights. However, much prior interpretability work has shown that representations encode rich semantic information, suggesting that editing representations might be a more powerful alternative. Here, we pursue this hypothesis by developing a family of $\textbf{Representation Finetuning (ReFT)}$ methods. ReFT methods operate on a frozen base model and learn task-specific interventions on hidden representations. We define a strong instance of the ReFT family, Low-rank Linear Subspace ReFT (LoReFT). LoReFT is a drop-in replacement for existing PEFTs and learns interventions that are 10x-50x more parameter-efficient than prior state-of-the-art PEFTs. We showcase LoReFT on eight commonsense reasoning tasks, four arithmetic reasoning tasks, Alpaca-Eval v1.0, and GLUE. In all these evaluations, LoReFT delivers the best balance of efficiency and performance, and almost always outperforms state-of-the-art PEFTs. We release a generic ReFT training library publicly at this https URL .

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