1 Data Representation How integers are represented in computers? How fractional numbers are represented in computers? Calculate the decimal value represented.

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Data Representation How integers are represented in computers? How fractional numbers are represented in

computers? Calculate the decimal value represented by a

binary sequence. How to determine the binary sequence that

represent a decimal number? Explain how characters are represented in

computers? Explain how colours, images and sound are

represented in computers?

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Binary numbers Binary number is simply a number comprised of only 0's and

1's.

Computers use binary numbers because it's easy for them to communicate using electrical current -- 0 is off, 1 is on.

You can express any base 10 (our number system -- where each digit is between 0 and 9) with binary numbers.

The need to translate decimal number to binary and binary to decimal.

There are many ways in representing decimal number in binary

numbers. (later)

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How does the binary system work? It is just like any other system In decimal system the base is 10

We have 10 possible values 0-9 In binary system the base is 2

We have only two possible values 0 or 1. The same as in any base, 0 digit has non

contribution, where 1’s has contribution which depends at there position.

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Example 3040510 = 30000 + 400 + 5

= 3*104+4*102+5*100 101012 =10000+100+1

=1*24+1*22+1*20

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Examples: decimal -- binary Find the binary representation of

12910.

Find the decimal value represented by the following binary representations: 10000011 10101010

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Examples: Representing fractional numbers Find the binary representations of:

0.510

0.7510

Using only 8 binary digits find the binary representations of: 0.210

0.810

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Representation of Numbers Representing whole numbers

Unsigned notation Signed magnitude notion Excess notation Two’s complement notation.

Representing fractions Fixed point representation Floating point representation Normalisation IEEE 754 Standard.

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Integer Representations All data in computers are

represented in terms of 1s and 0s. Any integer can be represented as

a sequence of 1s and 0s. Several ways of doing this.

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Unsigned Representation Represents positive integers. Unsigned representation of 157:

Addition is simple:

1 0 0 1 + 0 1 0 1 = 1 1 1 0.

position 7 6 5 4 3 2 1 0

Bit pattern 1 0 0 1 1 1 0 1

contribution 27 24 23 22 20

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Advantages and disadvantages of unsigned notation

Advantages: One representation of zero Simple addition

Disadvantages Negative numbers can not be

represented. The need of different notation to

represent negative numbers.

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Representation of negative numbers Is a representation of negative numbers

possible? Unfortunately you can not just stick a negative

sign in from of a binary number. (it does not work like that)

There are three methods used to represent negative numbers.

Signed magnitude notation Excess notation notation Two’s complement notation

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Signed Magnitude Representation Unsigned: - and + are the same. In signed magnitude the left most

bit represents the sign of the integer.

0 for positive numbers. 1 for negative numbers.

The remaining bits represent to magnitude of the numbers.

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Example Suppose 10011101 is a signed magnitude

representation. The sign bit is 1, then the number represented is

negative

The magnitude is 0011101 with a value 24+23+22+20= 29

Then the number represented by 10011101 is –29.

position 7 6 5 4 3 2 1 0

Bit pattern 1 0 0 1 1 1 0 1

contribution - 24 23 22 20

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Exercise 1

1. 3710 has 0010 0101 in signed magnitude notation. Find the signed magnitude of –3710 ?

2. Using the signed magnitude notation find the 8-bit binary representation of the decimal value 2410 and -2410.

3. Find the signed magnitude of –63 using 8-bit binary sequence?

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Disadvantage of Signed Magnitude Addition and subtractions are difficult. Signs and magnitude, both have to carry

out the required operation. They are two representations of 0

00000000 = + 010

10000000 = - 010

To test if a number is 0 or not, the CPU will need to see whether it is 00000000 or 10000000.

0 is always performed in programs. Therefore, having two representations of 0 is

inconvenient.

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Signed-Summary In signed magnitude notation,

the most significant bit is used to represent the sign.

1 represents negative numbers 0 represents positive numbers. The unsigned value of the remaining bits represent The

magnitude. Advantages:

Represents positive and negative numbers Disadvantages:

two representations of zero, difficult operation.

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Excess Notation In excess notation:

the value represented is the unsigned value with a fixed value subtracted from it.

For n-bit binary sequences the value subtracted fixed value is 2(n-1).

Most significant bit: 0 for negative numbers 1 for positive numbers

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Excess Notation with n bits 1000…0 represent 2n-1 is the decimal

value in unsigned notation.

Therefore, in excess notation: 1000…0 will represent 0 .

Decimal valueIn unsigned

notation

Decimal valueIn excess notation

- 2n-1 =

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Example (1) - excess to decimal Find the decimal number

represented by 10011001 in excess notation. Unsigned value

100110002 = 27 + 24 + 23 + 20 = 128 + 16 +8 +1 = 15310

Excess value: excess value = 153 – 27 = 152 – 128 = 25.

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Example (2) - decimal to excess Represent the decimal value 24 in 8-

bit excess notation. We first add, 28-1, the fixed value

24 + 28-1 = 24 + 128= 152

then, find the unsigned value of 152 15210 = 10011000 (unsigned notation). 2410 = 10011000 (excess notation)

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example (3) Represent the decimal value -24 in 8-bit

excess notation. We first add, 28-1, the fixed value

-24 + 28-1 = -24 + 128= 104

then, find the unsigned value of 104 10410 = 01101000 (unsigned notation). -2410 = 01101000 (excess notation)

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Example (4) -- 10101 Unsigned

101012 = 16+4+1 = 2110 The value represented in unsigned notation is 21

Sign Magnitude The sign bit is 1, so the sign is negative The magnitude is the unsigned value 01012 = 510

So the value represented in signed magnitude is -510

Excess notation As an unsigned binary integer 101012 = 2110

subtracting 25-1 = 16, we get 21-16 = 510. So the value represented in excess notation is 510.

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Advantages of Excess Notation It can represent positive and negative integers. There is only one representation for 0. It is easy to compare two numbers. When comparing the bits can be treated as

unsigned integers. Excess notation is not normally used to

represent integers. It is mainly used in floating point

representation for representing fractions (later floating point rep.).

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Exercise 21. Find 10011001 is an 8-bit binary sequence.

Find the decimal value it represents if it was in unsigned and signed magnitude.

Suppose this representation is excess notation, find the decimal value it represents?

2. Using 8-bit binary sequence notation, find the unsigned, signed magnitude and excess notation of the decimal value 1110 ?

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Excess notation - Summary In excess notation, the value represented is the

unsigned value with a fixed value subtracted from it.

i.e. for n-bit binary sequences the value subtracted is 2(n-1).

Most significant bit:

0 for negative numbers . 1 positive numbers.

Advantages: Only one representation of zero. Easy for comparison.

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Two’s Complement Notation The most used representation for

integers. All positive numbers begin with 0. All negative numbers begin with 1. One representation of zero

i.e. 0 is represented as 0000 using 4-bit binary sequence.

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Two’s Complement Notation with 4-bits

Binary pattern Value in 2’s coml.not.

0 1 1 1 70 1 1 0 60 1 0 1 50 1 0 0 40 0 1 1 30 0 1 0 20 0 0 1 10 0 0 0 01 1 1 1 -11 1 1 0 -21 1 0 1 -31 1 0 0 -41 0 1 1 -51 0 1 0 -61 0 0 1 -71 0 0 0 -8

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Properties of Two’s Complement Notation

Positive numbers begin with 0 Negative numbers begin with 1 Only one representation of 0, i.e.

0000 Relationship between +n and –n.

0 1 0 0 +4 0 0 0 1 0 0 1 0 +18

1 1 0 0 -4 1 1 1 0 1 1 1 0 -18

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Advantages of Two’s Complement Notation

It is easy to add two numbers.0 0 0 1 +1 1 0 0 0 -80 1 0 1 +5 0 1 0 1 +5

+ +

0 1 1 0 +6 1 1 0 1 -3

• Subtraction can be easily performed.

• Multiplication is just a repeated addition.• Division is just a repeated subtraction• Two’s complement is widely used in ALU

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Evaluating numbers in two’s complement notation

Sign bit = 0, the number is positive. The value is determined in the usual way.

Sign bit = 1, the number is negative. three methods can be used:

Method 1 decimal value of (n-1) bits, then subtract 2n-1

Method 2 - 2n-1 is the contribution of the sign bit.

Method 3 Binary rep. of the corresponding positive number. Let V be its decimal value. - V is the required value.

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Example- 10101 in Two’s Complement The most significant bit is 1, hence it is a

negative number. Method 1

0101 = +5 (+5 – 25-1 = 5 – 24 = 5-16 = -11) Method 2

4 3 2 1 0 1 0 1 0 1 -24 22 20 -11

Method 3 Corresponding + number is 01011 = 8 + 2+1 = 11 the result is then –11.

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Two’s complement-summary In two’s complement the most significant for an n-bit

number has a contribution of –2(n-1). One representation of zero All arithmetic operations can be performed by using

addition and inversion. The most significant bit: 0 for positive and 1 for

negative. Three methods can the decimal value of a negative

number:Method 1 decimal value of (n-1) bits, then subtract 2n-1

Method 2 - 2n-1 is the contribution of the sign bit.

Method 3 Binary rep. of the corresponding positive number. Let V be its decimal value. - V is the required value.

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Exercise - 10001011 Determine the decimal value

represented by 10001011 in each of the following four systems.

1. Unsigned notation? 2. Signed magnitude notation?3. Excess notation?4. Tow’s complements?

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Fraction Representation To represent fraction we need

other representations: Fixed point representation Floating point representation.

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Fixed-Point Representation

old position 7 6 5 4 3 2 1 0New position 4 3 2 1 0 -1 -2 -3Bit pattern 1 0 0 1 1 1 0 1Contribution 24 21 20 2-1 2-3 =19.625

Radix-point

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Limitation of Fixed-Point Representation To represent large number of very

small numbers we need a very long sequences of bits.

This is because we have to give bits to both the integer part and the fraction part.

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Floating Point Representation

In decimal notation we can get around thisproblem using scientific notation or floatingpoint notation.

Number Scientific notation Floating-point notation

1,245,000,000,000

1.245 1012 0.1245 1013

0.0000001245 1.245 10-7 0.1245 10-6

-0.0000001245 -1.245 10-7 -0.1245 10-6

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M B E

Sign mantissa or base exponent significand

Floating point format

Sign Exponent Mantissa

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Floating Point Representation format

The exponent is biased by a fixed

value b, called the bias. The mantissa should be normalised,

e.g. if the real mantissa if of the form 1.f then the normalised mantissa should be f, where f is a binary sequence.

Sign Exponent Mantissa

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IEEE 745 Standard

The most important floating point representation is defined in IEEE Standard 754.

It was developed to facilitate the portability from one processor to another.

The IEEE Standard defines both a 32-bit and 64-bit format.

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Representation in IEEE 754 single precision sign bit:

0 for positive and, 1 for negative numbers

8 biased exponent by 127 23 bit normalised mantissa   Sign Exponent Mantissa

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The exponent bit needs to represent both negative and positive numbers.

Excess notation is used to represent negative and positive exponent.

The exponent is represented with an excess of a fixed value b

The value b is called to bias. And we say that the exponent is biased by the value b

In IEEE single-precision format the the bias is 127. We say the exponent is biased by 127

Biased Exponent

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Mantissa in IEEE single precision format

The mantissa in this case has 23 bits. It should be normalised. For example: if the real mantissa is: 1.01101 then the normalised mantissa

should be 01101 000 000 000 000 000 000

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Mantissa in IEEE single precision format

The mantissa in this case has 23 bits. It should be normalised. For example: if the real mantissa is: 1.01101 then the normalised mantissa

should be 01101 000 000 000 000 000 000

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Example (1)

5.7510 in IEEE single precision 5.75 is a positive number then the sign bit is 0. 5.75 = 101.11 * 20

= 10.111 * 21

= 1.0111 * 22

The real mantissa is 1.0111, then the normalised mantissa is

0111 0000 0000 0000 0000 000 The real exponent is 2 and the bias is 127, then the

exponent is 2 + 127=12910 = 1000 00012. The representation of 5.75 in IEE sing-precision is:

0 1000 0001 0111 0000 0000 0000 0000 000

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Example (2)

which number does the following IEEE single precision notation represent?

The sign bit is 1, hence it is a negative number. The exponent is 1000 0000 = 12810

It is biased by 127, hence the real exponent is 128 –127 = 1.

The mantissa: 0100 0000 0000 0000 0000 000. It is normalised, hence the true mantissa is

1.01 = 1.2510 Finally, the number represented is: -1.25 x 21 = -2.50

1 1000 0000 0100 0000 0000 0000 0000 000

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Representation in IEEE 754 double precision format

It uses 64 bits 1 bit sign 11 bit biased exponent 52 bit mantissa

  Sign Exponent Mantissa

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IEEE 754 double precision Biased = 1023 11-bit exponent with an excess of 1023. For example:

If the exponent is -1 we then add 1023 to it. 1+1023 = 1022 We then find the binary representation of 1022

Which is 0111 1111 110 The exponent field will now hold 0111 1111 110

This means that we just represent -1 with an excess of 1023.

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Floating Point Representation format (summary)

the sign bit represents the sign

0 for positive numbers 1 for negative numbers

The exponent is biased by a fixed value b, called the bias.

The mantissa should be normalised, e.g. if the real mantissa if of the form 1.f then the normalised mantissa should be f, where f is a binary sequence.

Sign Exponent Mantissa

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Character representation- ASCII ASCII (American Standard Code for Information

Interchange)

It is the scheme used to represent characters.

Each character is represented using 7-bit binary code.

If 8-bits are used, the first bit is always set to 0

See (table 5.1 p56, study guide) for character representation in ASCII.

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ASCII – exampleSymbol decimal Binary

7 55 00110111

8 56 00111000

9 57 00111001

: 58 00111010

; 59 00111011

< 60 00111100

= 61 00111101

> 62 00111110

? 63 00111111

@ 64 01000000

A 65 01000001

B 66 01000010

C 67 01000011

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Character strings How to represent character strings? A collection of adjacent “words” (bit-string

units) can store a sequence of letters

Notation: enclose strings in double quotes "Hello world"

Representation convention: null character defines end of string

Null is sometimes written as '\0' Its binary representation is the number 0

'H' 'e' 'l' 'l' o' ' ' 'W' 'o' 'r' 'l' 'd' '\0'

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Layered View of Representation

Textstring

Sequence ofcharacters

Character

Bit string

Information

Data

Information

Data

Information

Data

Information

Data

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Working With A Layered View of Representation Represent “SI” at the two layers shown

on the previous slide. Representation schemes:

Top layer - Character string to character sequence: Write each letter separately, enclosed in quotes. End string with ‘\0’.

Bottom layer - Character to bit-string:Represent a character using the binary equivalent according to the ASCII table provided.

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Solution SI ‘S’ ‘I’ ‘\0’ 010100110100100000000000

The colors are intended to help you read it; computers don’t care that all the bits run together.

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exercise Use the ASCII table to write the

ASCII code for the following: CIS110 6=2*3

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Unicode - representation ASCII code can represent only 128 = 27

characters. It only represents the English Alphabet plus

some control characters. Unicode is designed to represent the

worldwide interchange. It uses 16 bits and can represents 32,768

characters. For compatibility, the first 128 Unicode are the

same as the one of the ASCII.

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Colour representation Colours can represented using a sequence of bits.

256 colours – how many bits? Hint for calculating

To figure out how many bits are needed to represent a range of values, figure out the smallest power of 2 that is equal to or bigger than the size of the range.

That is, find x for 2 x => 256

24-bit colour – how many possible colors can be represented?

Hints 16 million possible colours (why 16 millions?)

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24-bits -- the True colour 24-bit color is often referred to as the

true colour. Any real-life shade, detected by the

naked eye, will be among the 16 million possible colours.

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Example: 2-bit per pixel

4=22 choices 00 (off,

off)=white 01 (off, on)=light

grey 10 (on, off)=dark

grey 11 (on, on)=black

0 0

= (white)

1 0

= (dark grey)

10

= (light grey)

11

= (black)

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Image representation An image can be divided into many tiny squares,

called pixels. Each pixel has a particular colour. The quality of the picture depends on two factors:

the density of pixels. The length of the word representing colours.

The resolution of an image is the density of pixels.

The higher the resolution the more information information the image contains.

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Bitmap Images Each individual pixel (pi(x)cture element) in

a graphic stored as a binary number Pixel: A small area with associated coordinate

location Example: each point below is represented by a 4-

bit code corresponding to 1 of 16 shades of gray

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Representing Sound Graphically

X axis: time Y axis: pressure A: amplitude (volume) : wavelength (inverse of frequency = 1/)

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Sampling Sampling is a method used to digitise

sound waves. A sample is the measurement of the

amplitude at a point in time. The quality of the sound depends on:

The sampling rate, the faster the better The size of the word used to represent a

sample.

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Digitizing Sound

Zoomed Low Frequency Signal

Capture amplitude at these points

Lose all variation between data points

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Summary Integer representation

Unsigned, Signed, Excess notation, and Two’s complement.

Fraction representation Floating point (IEEE 754 format )

Single and double precision Character representation Colour representation Sound representation

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Exercise1. Represent +0.8 in the following

floating-point representation: 1-bit sign 4-bit exponent 6-bit normalised mantissa (significand).

2. Convert the value represented back to decimal.

3. Calculate the relative error of the representation.