int 2/ higher - data representation - 1 why use binary? it is a two state system (on/off) which...

Int 2/ Higher - Data Representation - 1

Why use Binary?

It is a two state system (on/off) which makes it simple to operate

Even if degradation of current occurs (ie a slight drop in voltage) it will still be detected as a 1

There are only four rules for addition in binary compared to 100 in decimal

[0+0=0 ; 0+1=1 ; 1+0=1; 1+1=10]


Number Systems - Decimal

The decimal system is a base-10 system. There are 10 distinct digits (0 to 9) to represent any quantity.

For an n-digit number, the value that each digit represents depends on its weight or position. The weights are based on powers of 10.

POSITION

WEIGHT

4TH. 3RD. 2ND. 1ST.

103 = 1000 102 = 100 101 = 10 100 = 1

For example, 491610 = 4*1000 + 9*100 + 1*10 +6*1


Number Systems - Binary

The binary system is a base-2 system. There are 2 distinct digits (0 and 1) to represent any quantity.

To express any number in base 2 we use powers much like our own decimal system.

8TH 7TH 6TH 5TH 4TH 3RD 2ND 1ST POSITION

27=128 26=64 25=32 24=16 23=8 22=4 21=2 20=1 WEIGHT

For example: 110100102 = 1*128 + 1*64 + 0*32 + 1*16 + 0*8 + 0*4 + 1*2 +0*1 = 21010


Number Systems - Binary to Decimal

Converting binary to decimal

27=128 26=64 25=32 24=16 23=8 22=4 21=2 20=1

1 0 1 1 1 0 0 1

1x128+0x64+1x32+1x16+1x8+0x4+0x2+1= 128+32+16+8+1= 185


Number Systems - Decimal to Binary

We use the same table as before

• To convert the decimal number 115

27=128 26=64 25=32 24=16 23=8 22=4 21=2 20=1

115 is less than 128 so we put a zero in the 128 column 0we need a 64 to ‘built’ up to 115 so place a ‘1’ in the 64 column 64

96+16 is 112….just short we place a ‘1’ in the 16 column 112

We just need a 3 to give 115 so a ‘1’ in the 2s column anda ‘1’ in the units column gives 115

64+32 is 96 so could use a 32 place a ‘1’ in the 32 column 96

So that 115 using 8 bit binary is 01110011So that 115 using 8 bit binary is 01110011

0 1 1 1 0 0 1 1


Storage of data

1byte = 8 bits1KiloByte = 1024 bytes1MegaByte = 1024 Kbytes1GigaByte = 1024 Mbytes1TetraBytes = 1024 GBytes

Hierarchy of storage


Numbers

Numbers may be classified as real or integer.

Real numbers include ALL numbers, whole and fractional, positive or negative. From the smallest negative fraction to the largest number imaginable.

Integers include only whole numbers but can be both positive and negative

The method to represent integer numbers is different from the method used to represent real numbers.


Integers

The size of the number that can be represented depends on the number of bytes which are available in the computers memory to store it.

If one byte is used to store the number then the numbers 0000 0000 to 1111 1111 can be stored, that is, 0 to 255 in decimal. A total of 256 numbers.

If two bytes are assigned to store numbers, then numbers from 0000 000 0000 000 to 1111 1111 1111 1111 can be stored, that is, 0 to 65 535 in decimal. A total of 65 536 numbers.


Numbers/addresses

IPv4 allows 32 bits for an Internet Protocol address, and can therefore support 232 addresses (4,294,967,296)

IPv6 uses a 128-bit address, 2 128, that’s 340 undecillion possible addresses which is: 340,000,000,000,000,000,000,000,000,000,000,000,000 addresses.


Real Numbers -Floating Point Representation

FPR is used to store real numbers, that is, fractional numbers, very large and very small numbers.

In decimal any number can be represented with a decimal point in a fixed position and a multiplier which is a power of 10.

398 = .398 * 1000 = .398 * 103



.398 * 103 can be described as m * basee where m is called the mantissa, 10 the base and e the exponent. The exponent is the number of times the point moves.

398 is the mantissa, the base is 10 and the exponent is 3

FPR works exactly the same way when used to represent real numbers in binary. As the base is always 2 it does not need to be stored. Only the mantissa and the exponent need be stored.



398 = 110001110 in binary, using FPR, only the mantissa and the exponent need be stored

110001110 = .1100011101001 (the point moves 9 places) The mantissa is .110001110 and the exponent 1001

(which is binary for 9)

If the binary number was 1101.0111 using FPR this would become .11010111100

As the point moves four places and 100 is binary for 4.


Negative numbers

In decimal negative numbers are represented by using the negative sign (-) for example, (positive) 53 becomes negative 53 by using the negative sign, -53.

In a computer system if a sign was used it would have to be stored and that would take up one of the bits used for storing the number and therefore reduce the size of the numbers.


Negative numbers

If one bit is used for the sign then only 7 bits would be available for the number 000 0000 to 111 1111, that is 0 to 127 in decimal. A total of 128 numbers.

If 1 is used for negative and 0 for positive then the range of numbers would be 1111 1111 (- 127) to 0111 111 (+127), the problem with this is you get two values for zero, 1000 000 (negative zero) and 0000 0000 (positive zero)


Two’s Compliment

To avoid the two values for zero problem a system called two’s compliment is used. To represent the negative of a number you change all the 0’s to 1s and all the 1’s to 0’s and then add 1.

Remember that in binary adding 1 and 1 gives 0.


Two’s Compliment


Floating Point Representation

A range of very large and very small numbers can be represented with only a few digits by using scientific notation. For example:

• 976,000,000,000,000 = 9.76 * 1014

• 0.0000000000000976 = 9.76 * 10-14

This same approach can be used for binary numbers. A number represented by M*B±E can be stored in a binary word with three fields:

• Mantissa

• Exponent E

• The base B is implicit and need not be stored


Typical 32-bit Floating Point Format

First 8 bits contain the exponent The remaining 24 bits contain the mantissa

The more bits we use for the exponent, the larger the range of numbers available, but at the expense of precision. We still only have a total of 232 numbers that can be represented.

Exponent Mantissa

8 bits 24 bits


Floating point representation

How to represent the binary number

11010.11011011101

This has to be converted to the form M*B±E

. 1101011011011101 Mantissa

The point has been moved 5 placed so exponent +5

.1101011011011101 x 2 101

Only the mantissa and the exponent need to be stored to represent this number

Note: this assumes that all numbers are positive


Representing negative numbers

TWO’s compliment Positive 9 0000 1001 Negative 9 1111 0111

To represent the negative Find the positive 0000 1001 Change all the ones to zeros and vice versa 1111 0110 Add 1 +1 Negative number 1111 0111


Representing Text

Alphanumeric data such as names and addresses are represented as strings of characters containing letters, numbers and symbols.

Each character has a unique code or sequence of bits to represent it. As each character is entered from a keyboard it must be converted into its binary code.


Representing Text

Character Set

The CS is the group of letters and numbers and characters that the computer can represent and manipulate

Each letter, number or character has its own unique binary value


Coding Methods ASCII

ASCII

American Standard Code for Information Interchange

strictly speaking a 7-bit code (128 characters) has an extended 8-bit version used on PC’s and non-IBM mainframes widely used to transfer data from one computer to

another codes 0 to 31 are control codes


Coding Methods ASCII continued

ASCII

Character code sets contain two types of characters: Printable (normal characters) Letters, numbers and symbols Upper and lower case letters have their own codes Numbers 0 to 9 Punctuation and other symbols, for example, % £ “ !


Coding Methods ASCII continued

ASCII

Character code sets contain two types of characters: Non printable Control Codes (characters) The first 32 codes are set aside for control characters They all have their own unique code Examples are: <return>, <tab>, <escape>, They are often used when transmitting data, there are

codes for <start of text>, <end of text> and <end of transmission>


ASCII Coding Examples

An ASCII subset

A 41 B 42 C 43 D 44 E 45 F 46 0 30 1 31 2 32 3 33 4 34 5 35 6 36 7 37

Symbol Code“BAD” = 0100 0010 0100 0001 0100 01002

“F1” = 0100 0110 0011 00012

“3415” = 0011 0011 0011 0100 0011 0001 0011 01012

Note that this is a text string and no arithmetic may be done on it. A postcode is a good example of the need to store numbers as text.


Other coding methods

ASCII is a 7 bit code giving 128 code values 96 characters and 32 control codes Extended ASCII, using an 8 bit code gives 256

characters but is still not enough to represent the major writing schemes of the world

Unicode A 16 bit code Can represent 65,536 different characters First 256 values are used to represent 8 bit ASCII, this

makes conversion between the two easy


Unicode

Advantages over ASCII Can support 65 380 more characters than 8 bit ASCII Every character base alphabet in the world can be

coded such as French, German and Finish

And others such as Arabic

The large ideographic languages can be coded such as Chinese, Japanese and Korean


Unicode

Originally 49 000 of the codes were predefined 6400 can be used by software developers 10 000 codes set aside for future developments

Now Mobile phones use Unicode for SMS text messaging More and more character sets have been added Unicode now represents 109,000 characters Even 16 bit Unicode is no longer enough


Unicode

Unicode takes up much more storage space than ASCII and it takes longer to transmit Unicode than ASCII (because there are more bits to transmit)

Both of these factors are less of a disadvantage now because storage space has increased significantly as transmission bandwidths


Representing Graphics

There are two ways of representing graphics

• Bit Mapped Graphics

• Vector Graphics


Bit Mapped Graphics

Any graphic is made up from a series of pixels (Picture Elements).

Each pixel is an individual point on the screen


Bit Map

Assuming only black and white (1 or 0) for each pixel the image below would be stored as shown

Pixel Pattern using 8x8 grid The BIT MAP of the image

0 0 1 1 1 1 0 00 1 0 0 0 0 1 00 1 0 0 0 0 1 01 0 1 0 0 1 0 10 1 0 0 0 0 1 00 1 0 1 1 0 1 00 0 1 0 0 1 0 00 0 0 1 1 0 0 0


Resolution

The quality of the image depends on the number of pixels

More pixels means higher resolution and clearer image

Pixel Pattern using 8x8 grid Pixel Pattern using 16x16 grid There is a one to one correspondence between pixels and bits


Memory Storage

The image below is 4 inches x 6 inches. The resolution is 300 d.p.i. (dots per linear inch) and the image is black and white. Calculate the memory requirements

Length: 6x300 = 1800 pixels

Breadth: 4x300 = 1200 pixels

Total no. pixels = 1800x1200

= 2160000

1 bit per pixel

Storage = 2160000 bits /8

= 270000 bytes /1024

= 263.67 Kb

= 264 Kb


Vector Graphics

Each Image is made from objects (line,rect,circle)

• Every object has ATTRIBUTES which define it

To draw the rectangle below we need to know:

Start X and Y coordinates

The length

The breadth

The thickness and colour of the lines

The type of line (dashed)

The fill colour


A Vector Image


Vector Vs Bit-Mapped

Advantages of vector graphics (draw packages)

• Images can be enlarged without losing resolution

• Objects can be edited by changing their attributes

• Objects can be layered on top/behind

• Images take up less disc space

• Ideal for drawing plans; use library of objects

Disadvantages of vector graphics

• Individual pixels cannot be altered

• Not realistic


Vector Vs Bit-Mapped

Advantages of bit-mapped graphics (paint packages)

• Each pixel can be altered

• More realistic when used for photos/real life

Disadvantages of bit mapped representation

• requires large amounts of storage space;

• image becomes course (jagged) when scaled;

• does not take advantage of resolutions that are higher than the resolution of the image.


Compression

A colour bit – mapped image with a high resolution and 24 bit colour needs a lot of storage (50MB for a smallish photo)

File compression is used to reduce storage requirements.

Different techniques – coding using and index of colours actually used and not coding differences indistinguishable to the human eye.


Compression types

Lossless means that none of the original data is lost

Lossy compression involves sacrificing some of the data in order to reduce the file size

JPEG is a file format commonly used for data compression of bit mapped files

JPEG uses lossy compression


Advantages of compression

Saves backing storage

Smaller size of file, less time to transmit over a network

Smaller files faster to load on a web page


Disadvantages of compression

Detail will be lost using lossy compression

Changes can be made to the original due to compression. These changes are known as artefacts

It takes time to compress. Larger files take longer.

Repeated compressing and decompressing of a file can lead to a reduction in image quality


Images

Bit-Mapped Vector graphic

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