ADVANCED DATA REPRESENTATION Lesson Note

Learning Objectives

By the end of the lesson, students should be able to:

Data representation refers to the way data is encoded, stored and processed inside a computer system.

Although humans normally use letters, numbers, pictures, sounds and symbols, computers process information using binary digits (bits).

A bit can have only two possible values: 0 and 1

Therefore, computers represent different forms of information by using combinations of 0s and 1s.

For example:

10110101

Is a binary number containing eight bits.

The binary number system is the basic number system used by digital computers.

It has a base of 2 and uses only: 0 and 1

Each position in a binary number represents a power of 2.

Example

Consider:

1011₂

Binary digit 1 0 1 1

Place value 2³ 2² 2¹ 2⁰

Value 8 4 2 1

Therefore:

1011₂ = 8 + 0 + 2 + 1 = 11₁₀

So, 1011₂ = 11₁₀.

The decimal number system is the number system commonly used by humans.

It has a base of 10 and uses ten digits:

0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

For example:

583₁₀

Means:

5 × 10² = 500

8 × 10¹ = 80

3 × 10⁰ = 3

Therefore:

583 = 500 + 80 + 3

The octal number system has a base of 8.

It uses the digits:

0, 1, 2, 3, 4, 5, 6 and 7.

It does not use 8 or 9 as digits.

Example

347₈

Can be expanded as:

3 × 8² = 192

4 × 8¹ = 32

7 × 8⁰ = 7

Therefore:

347₈ = 231₁₀

Octal representation can provide a shorter way of writing some binary values.

The hexadecimal number system has a base of 16.

It uses 16 symbols:

0–9 and A–F

The letters represent:

Hexadecimal Decimal

A 10

B 11

C 12

D 13

E 14

F 15

Example

2F₁₆

= (2 × 16¹) + (F × 16⁰)

= 32 + 15

= 47₁₀

Hexadecimal is commonly used in computing because it provides a compact representation of binary data.

The major number systems used in computing include:

Number System.  Base

Digits/Symbols

Binary(2) 0–1

Octal (8).  0–7

Decimal (10)  0–9

Hexadecimal(16)  0–9, A–F

To convert binary to decimal, multiply each binary digit by its corresponding power of 2 and add the results.

Example

Convert 1101₂ to decimal.

1101₂

= (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰)

= 8 + 4 + 0 + 1

= 13₁₀

One common method is to repeatedly divide the decimal number by 2 and record the remainders.

Example: Convert 13₁₀ to binary

13 ÷ 2 = 6 remainder 1

6 ÷ 2 = 3 remainder 0

3 ÷ 2 = 1 remainder 1

1 ÷ 2 = 0 remainder 1

Read the remainders from bottom to top:

1101₂

Therefore:

13₁₀ = 1101₂

A bit is the smallest unit of digital data. It can have a value of 0 or 1.

A byte consists of 8 bits.

Common storage units

1 bit = 0 or 1

1 byte = 8 bits

1 kilobyte (KB) ≈ 1,024 bytes

1 megabyte (MB) ≈ 1,024 KB

1 gigabyte (GB) ≈ 1,024 MB

1 terabyte (TB) ≈ 1,024 GB

These units are used to describe the size of files and storage devices.

Computers need a method of representing letters, numbers and symbols.

Two important character-encoding systems are:

ASCII stands for American Standard Code for Information Interchange.

It is a character encoding system used to represent common letters, numbers, punctuation marks and control characters.

For example, ASCII assigns numerical codes to characters such as:

A

B

C

A

B

0

1

Punctuation marks

ASCII traditionally uses 7 bits, giving it 128 possible codes.

An extended form commonly uses 8 bits.

Unicode is a much larger character-encoding standard designed to represent characters from many writing systems around the world.

It can represent:

*English letters

*Arabic characters

*Chinese characters

*Greek characters

*Mathematical symbols

*Many other writing systems and symbols

Unicode is important because modern computers need to handle text from different languages.

REPRESENTATION OF TEXT

When a user types a word into a computer, the computer does not store the letters in the same visual form that humans see.

Instead, each character is converted into a numerical code according to an encoding system.

For example:

CAT

Can be represented internally using numerical character codes, which are ultimately stored as binary data.

The computer then uses the appropriate encoding to display the characters again.

REPRESENTATION OF IMAGES

Digital images are represented using pixels.

A pixel is the smallest addressable element of a digital image.

An image is made up of many pixels arranged in rows and columns.

For example:

Pixel + Pixel + Pixel + Pixel → Digital Image

Each pixel contains information about its colour or brightness.

IMAGE RESOLUTION

Resolution refers to the number of pixels used to represent an image.

For example:

1920 × 1080

Means the image has:

1920 pixels horizontally × 1080 pixels vertically.

Higher resolution generally means that an image contains more pixels and can contain more detail.

COLOUR REPRESENTATION

Computers can represent colours using numerical values.

One common method is RGB, which stands for:

R – Red

G – Green

B – Blue

Different combinations of red, green and blue values can produce many colours.

For example, a pixel may store different amounts of red, green and blue to determine the colour displayed on the screen.

COLOUR DEPTH

Colour depth refers to the number of bits used to represent the colour of a pixel.

Greater colour depth allows a computer to represent a larger number of possible colours.

REPRESENTATION OF SOUND

Sound is naturally an analogue signal, but computers normally store sound digitally.

To convert sound into digital form, the sound wave is sampled at regular intervals.

Two important concepts are:

Sampling Rate

The sampling rate is the number of samples taken from a sound signal per second.

It is measured in hertz (Hz).

For example, a sampling rate of 44,100 Hz means that 44,100 samples are taken every second.

Bit Depth

Bit depth describes the number of bits used to represent each audio sample.

Higher bit depth can provide more possible levels for representing each sample.

REPRESENTATION OF VIDEO

Digital video consists of a sequence of digital images called frames, usually accompanied by audio.

For example:

Frame 1 → Frame 2 → Frame 3 → Frame 4 → …

When these frames are displayed rapidly one after another, they create the appearance of motion.

Important video characteristics include:

*Resolution

*Frame rate

*Colour depth

*Audio quality

*Compression

*File size

*Frame Rate

Frame rate is the number of frames displayed per second.

It is commonly measured in frames per second (fps).

DATA COMPRESSION

Data compression is the process of reducing the amount of storage space required by data.

Compression is useful because large files can occupy considerable storage space and take longer to transmit.

There are two major types:

Lossless compression reduces file size without permanently removing information.

When the file is decompressed, the original data can be recovered.

Examples include: ZIP and PNG

Lossy compression reduces file size by removing some information that may be less noticeable to humans.

It is commonly used for:

*Images

*Audio

*Video

Examples include common JPEG images and many compressed audio/video formats.

FLOATING-POINT REPRESENTATION

Computers also need to represent numbers that contain fractions or very large values.

Floating-point representation is a method of representing real numbers using a combination of significant digits and an exponent.

For example, a number can be represented conceptually as:

6.25 × 10³

Floating-point representation is useful in:

*Scientific calculations

*Engineering

*Graphics

*Simulations

*Financial applications

BOOLEAN DATA REPRESENTATION

Boolean data represents a value that can have only two states:

True

False

In digital systems, these states are often represented as:

True = 1

False = 0

Boolean representation is important in:

*Computer programming

*Logic circuits

*Decision-making

*Database operations

*Digital electronics

IMPORTANCE OF ADVANCED DATA REPRESENTATION

Advanced data representation is important because it enables computers to:

SUMMARY

Data representation is the method used to encode information so that a computer can store and process it.

Important concepts include:

Binary, decimal, octal and hexadecimal number systems.

Bits and bytes.

ASCII and Unicode.

Pixel and colour representation.

Sampling and bit depth for audio.

Frames and frame rates for video.

Lossless and lossy compression.

Floating-point representation.

Boolean representation.

Understanding these concepts helps students understand how computers store and process different types of information.