Download - 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
From Base 10 to Base 2
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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
From Base 10 to Base 8
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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
From Base 10 (decimal point) to Base 8
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
16 ) 1 ( 1 Most significant bit
0
Base 16 1667
Quotient
Remainder
From Base 10 to Base 16
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Base 10 8,039
16 ) 8,039 ( 7 Least significant bit
16 ) 502 ( 6
16 ) 31 ( 15
16 ) 1 ( 1 Most significant bit
0
Base 16 1F67
Quotient
Remainder
From Base 10 to Base 16
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Convert 2110 to hexadecimal.
21 16 = 1 balance 5 (LSB)
1 16 = 0 balance 1 (MSB)
Therefore , 2110 = 1516
From Base 10 to Base 16
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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
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
From Base 10 (decimal point) to Base 16
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.