csc159-chapter 2 part1

7/28/2019 Csc159-Chapter 2 Part1

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MACHINE LEVEL REPRESENTATION OF DATA

(Part 1)

Prepared by: Nor Fauziah Binti Abu Bakar, FSKM


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Contents1. Bits, bytes, and words

2. Numeric data representation and number bases

Binary, Octal, Hexadecimal3. Conversion between bases


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Bits, Bytes, and Words

Bits The basic unit of information in computing and

telecommunication

In computing, a bit is defined as a variable or computed

quantity that can have only two possible These two values are often interpreted as binary digits and

are usually denoted by 0 and 1

In several popular programming languages, numeric 0 isequivalent (or convertible) to logicalfalse, and 1 to true.

The correspondence between these values and the physicalstates of the underlying storage or device is a matter ofconvention, and different assignments may be used even

within the same device or program


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Bytes a unit of digital information in computing and

telecommunications, that most commonly consists ofeight bits

a byte was the number of bits used to encode a singlecharacter of text in a computer and it is for this reasonthe basic addressable element in many computerarchitectures.

The byte size and byte addressing are often used inplace of longer integers for size or speed optimizationsin microcontrollers and CPUs

Bits, Bytes, and Words - cont


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Words In computing,word is a term for the natural unit of data

used by a particular computer design A word is simply a fixed sized group of bits that are

handled together by the system The number of bits in a word (theword size orwordlength) is an important characteristic of computerarchitecture.

The size of a word is reflected in many aspects of a

computer's structure and operation; the majority of theregisters in the computer are usually word sized and theamount of data transferred between the processing partcomputer and the memory system, in a single operation, ismost often a word

Bits, Bytes, and Words - cont


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NUMBERING SYSTEM :BASE

PLACE

5TH

PLACE

4TH

PLACE

3RD

PLACE

2ND

PLACE

1ST

PLACE

SINGLE

UNIT

1ST

PLACE

2ND

PLACE

3RD

PLACE

DECIMAL

105 104 103 102 101 100 10-1 10-2 10-3

100,000 10,000 1,000 100 10 1 0.1 0.01 0.001

1/10 1/100 1/1000

BINARY

25 24 23 22 21 20 2-1 2-2 2-3

32 16 8 4 2 1 0.5 0.25 0.125

1/2 1/4 1/8

OCTAL

85 84 83 82 81 80 8-1 8-2 8-3

32,768 4,096 512 64 8 1 0.125 0.0156251.953125

X 103

1/8 1/64 1/512

HEXA-

DECIMAL

165 164 163 162 161 160 16-1 16-2 16-3

1,048,576

65,536 4,096 256 16 1 0.06253.906 X

1032.4414062 X 104

1/16 1/256 1/4096


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Radix :

when referring to binary, octal, decimal,

hexadecimal, a single lowercase letter appended tothe end of each number to identify its type.

E.g.

hexadecimal 45will be written as 45h

octal 76will be written as 76o or 76q binary 11010011will be written as 11010011b

Number Bases


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Number System

The following table shows the equivalent values for decimal numbers in binary,octal and hexadecimal:


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Number System

Decimal system: system ofpositional notation basedon powers of 10.

Binary system: system ofpositional notation basedpowers of 2

Octal system: system ofpositional notation based on

powers of 8

Hexadecimal system: system ofpositional notationbased powers of 16


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Base: The number of different symbols required to represent any

given number

The larger the base, the more numerals are required

Base 10: 0,1, 2,3,4,5,6,7,8,9

Base 2: 0,1

Base 8: 0,1,2, 3,4,5,6,7

Base 16 : 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

For a given number, the larger the base

the more symbols required

but the fewer digits needed

Number System


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EXAMPLE:

Example #1:

6516 10110 1458 110 01012

Example #2:

11C16 28410 4348 1 0001 11002

Number System


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Binary SystemWHY??

Early computer design was decimal

Mark I and ENIAC

John von Neumann proposed binary data processing (1945)

Simplified computer design

Used for both instructions and data

Natural relationship betweenon/off switches andcalculation using Boolean logic

On Off

True False

Yes No

1 0


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A computer stores both instructions and data asindividual electronic charges.

represent these entities with numbers requires a systemgeared to the concept ofon and offor true and false

Binary is a base 2 numbering system

each digit is either a 0 (off) or a 1 (on)

Computers store all instructions and data as

sequences of binary digit e.g. 010000010100001001001000011 = ABC

Binary System


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Binary System

Each digit in binary number has a value depending on itsposition

Example:

The number 1002 means

(1 X 2) + ( 0 X 2) + (0 X 2)

4 + 0 + 0

4


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Base 2

7/21/2013 15


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Decimal System

Decimal is a base 10 numbering system

We use a system based on decimal digits to representnumbers

Each digit in the number is multiplied by 10 raised to apower corresponding to that digit position.

Example :

The number 472810 means :

(4 X 1000) + (7 X 100) + (2 X 10) + 84000 + 700 + 20 +8

4728


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Base 10

7/21/2013 17


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Octal System

As known as base 8 numbering system

There are only eight different digits available (0, 1, 2, 3,

4, 5, 6, 7) Example :

The number 7238 means

(7 X 8) + (2 X 8) + (3 X 8)

448 + 16 + 3467


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Base 8

7/21/2013 19


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Hexadecimal System

Hexadecimal is a base 10 numbering system

Used not only to represent integers

Also used to represent sequence of binary digits Example :

The number 2C16 means:

(2 X 16) + (C X 16)

32+ 12

44


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Base 16

7/21/2013 21


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Conversion Between Bases

1. Conversion of decimal to binary

2. Conversion of decimal to octal

3. Conversion of decimal to hexadecimal

4. Conversion of binary to decimal5. Conversion of binary to octal

6.Conversion of binary to hexadecimal

7. Conversion of octal to decimal

8.Conversion of octal to binary9.Conversion of hexadecimal to binary

10.Conversion of hexadecimal to decimal


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CONVERSION :

DECIMAL OTHER BASES


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Base 10 42

2 ) 42 ( 0 Least significant bit2 ) 21 ( 1

2 ) 10 ( 0

2 ) 5 ( 1

2 ) 2 ( 0

2 ) 1 ( 1 Most significant bit

0

Base 2 101010

Remainder

Quotient

From Base 10 to Base 2


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Convert 3710 to binary.

* MSB (most-significant-bit) : left most bit

LSB (least-significant-bit) : right most bit

37 2 = 18 balance 1 (LSB)18 2 = 9 balance 09 2 = 4 balance 14 2 = 2 balance 02 2 = 1 balance 01 2 = 0 balance 1 (MSB)

Therefore , 3710 = 1001012



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What is the value of 37.687510 in binary?Steps :

1. Convert the integer to binary by using method shown in previous

slide.2. Convert the decimal point to binary by using the followingmethod.

So, (0.6875)10 = (0.1011)2 ; Therefore, 37.687510 = 100101.10112

0.6875X 2

(MSB) 1 1.3750X 2

0 0.7500

X 21 1.5000X 2

(LSB) 1 1.0000

The 1 is saved asresult, then droppedand the processrepeated

From Base 10 (decimal point) to Base 2


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Base 10 135

8) 135 ( 7 Least significant bit

8) 16 ( 0

8) 2 ( 2 Most significant bit

0

Base 8 207

Quotient

Remainder



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8 21 58 2 2

0

0.25x 8

2 2.00

From Base 10 (decimal point) to Base 8Convert 21.2510 to octal.

Therefore,

21.2510 = 25.28


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Convert 21.2510 to octal. (OTHER METHOD)

21

2 = 10 balance 1 (LSB)10 2 = 5 balance 0

5 2 = 2 balance 1

2 2 = 1 balance 0

1 2 = 0 balance 1 (MSB)

So, 2110 = 101012

Now, 0.25X 2

(MSB) 0 0.50

X 2

(LSB) 1 1.00

So, 0.2510 = 0.012


Therefore,

Refer to conversion of binary to hexadecimal

21.2510 = 010 101 . 0102

= 25.28


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Base 10 5,735

16 ) 5,735 ( 7 Least significant bit

16 ) 358 ( 6

16 ) 22 ( 6


0

Base 16 1667

Quotient

Remainder



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Base 10 8,039

16 ) 8,039 ( 7 Least significant bit

16 ) 502 ( 6

16 ) 31 ( 15


0

Base 16 1F67

Quotient

Remainder



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Convert 2110 to hexadecimal.

21 16 = 1 balance 5 (LSB)


Therefore , 2110 = 1516



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16 21 516 1 1

0

0.25x 16

4 4.00

From Base 10 (decimal point) to Base 16Convert 21.2510 to hexadecimal.

Therefore,

21.2510 = 15.416


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Convert 21.2510 to hexadecimal. (OTHER METHOD)

21

2 = 10 balance 1 (LSB)10 2 = 5 balance 0

5 2 = 2 balance 1

2 2 = 1 balance 0


So, 2110 = 101012

Now, 0.25X 2

(MSB) 0 0.50

X 2

(LSB) 1 1.00

So, 0.2510 = 0.012


Therefore,

Refer to conversion of binary to hexadecimal

21.2510 = 10101 . 01002

= 15.416


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EXERCISES1. Find the binary representation for 31.62510.Show

your work.

1. Convert the following numbers to the respectivenumbering system.

5610

base 8 (until 5 binary point)

2. Convert 95.2510 to hexadecimal.

csc159-chapter 2 part1

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