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Information SystemsFundamentals II
1IS 121: Information Systems Fundamentals II
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Data representations and encryptions;
Basic logic used in programming.
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Objectives
Understanding a computers basic data unitssuch as binary numbers, bits, bytes, words, etc.and their conversions from and to octal, decimal,and hexadecimal digits
Understanding basic concepts of computerinternal data representation, focusing onnumeric data, character codes etc
Understanding proposition calculus and logical
operations
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Some Terminology
Data representation unit and processing unit
1. Binary Digits (Bits) Two levels of status in computers electronic circuits
Whether the electric current passes through it or not Whether the voltage is high or low
1 digit of the binary system represented by 1 or 0
Smallest unit that represents data inside the computer
1 bit can represent 2 values of data,0
or1
2 bits can represent 4 different values
00, 01, 10, 11
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(or Column)
(or Row)
(or
Table)
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Bit representation
Switches Open (0) or closed (1)
Current Not flowing (0) or flowing (1)
Lights Off (0) or on (1)
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2. Bytes A byte is a unit that represents with 8 bits 1
character or number, 1 byte = 8 bits
E.g. 00000000, 00000010, etc. 1 bit can be represented in 2 ways, i.e.
combination of 8 bit patterns into 1 byte enablesthe representation of 28 = 256 types ofinformation
Using a 1-byte word, 256 different characters canbe represented sufficient for most Westerncharacter sets
Numeric Conversion
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2. Bytes However, the number of kanji (Chinese
characters) amounts to thousands of differentcharacters, hence a 1-byte word system is
insufficient Two bytes are connected to obtain 16 bits, 216 =
65,536 A 2-byte word
Numeric Conversion
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3. Word The smallest unit that represents data inside a
computer
Increase operation speed
Numeric Conversion
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4. Number systems Binary system is used to simplify the structure of
electronic circuits that make up a computer
Hexadecimal number is a numeric value
represented by 16 numerals from 0 to 15 toease the representation of binary numbers for
humans computers are capable of only usingbinary numbers
Numeric Conversion
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Numeric Systems
Also known as Base Systems or RadixSystems
Available digits: Decimal system (base 10) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Binary system (base 2) 0, 1
Octal system (base 8)
0, 1, 2, 3, 4, 5, 6, 7 Hexadecimal (base 16) 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F where A=10,B=11,C=12,D=13,E=14,F=15
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Numeric DataRepresentation
The true value of numbers arethe same
The representation of numbersvary
Decimal
Binary
Octal
Hexadecimal
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Numeric data representation
DECIMAL number
(Radix/Base = 10)
2 1 9 9 8
Weight 104 103 102 101 100
Value 2*104 2*103 2*102 9*101 8*100
Final (true) value 20000 + 1000 + 900 + 90 + 8 = 2199810
BINARY number
(Radix/Base = 2)
1 1 0 0 1
Weight 24 23 22 21 20
Value 1*24 1*23 0*22 0*21 0*20
Final (true) value 16 + 8 + 0 + 0 + 1 = 252
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Numeric data representation
OCTAL number
(Radix/Base = 8)
2 1 7 7 2
Weight
Value
Final (true) value
HEXA number
(Radix/Base = 16)
A 2 5 7 C
Weight
Value
Final (true) value
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Binary Arithmetic
Addition and subtraction of binary numbers Addition
0 + 0 = 0 (or 010) 0 + 1 = 1 (or 110)
1 + 0 = 1 (or 110) 1 + 1 = 10 (or 210)
Subtraction 0 0 = 0 0 1 = -1
1 0 = 1 1 1 = 0
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Binary Addition
Result = 1001102
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Binary Subtraction
Result = 10102
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4. Addition and subtraction of hexadecimalnumbers Addition
Performed starting at the lowest (first from the
right) digit A carry to the upper digit is performed when the
result is higher than 16
Subtraction
Performed starting at the lowest (first from theright) digit
A borrow from the upper digit is performed whenthe result is negative
Hexadecimal arithmetic
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Hexadecimal Addition
First column from rightD + 7 = (In the decimal system: 13 + 7 = 20) = 16 (carried 1) + 4The sum of the first column is 4 and 1 is carried to the second column.
Second column from right1 + 8 + 1 = (In the decimal system: 10) = A
Carried from the first column Third column from right
A + B = (In the decimal system: 10 + 11 = 21) = 16 (carried 1) + 5The sum of the third column is 5 and 1 is carried to the fourth column.
The result is (15A4)16.
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Hexadecimal Subtraction
First column from rightSince 3 4 =1, a borrow is performed from D in the second digit
(D becomes C).
16 (borrowed 1) + 3 4 = F (In the decimal system: 19 4 = 15)
Second column from right
C 7 = 5 (In the decimal system: 12 7 = 5) Third column
6 1 = 5
The result is (55F)16.
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Exercises
Compute the following
a) 2710 + 1510b) 110112 + 11112c) 338 + 178
d) 1B16 + F16
Compute the following
a) 5010 2210
b) 1100102 - 101102c) 628 268d) 3216 - 1616
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Representation of numeric data
1. Radix and weight
Decimal numbersweight and its meaning
10 is called Radix
upper right of 10 (in this example, 4) is called exponent
Binary digits weight and its meaning
2. Auxiliary units and power representation
Used to represent big, small amounts, and exponent towhich the radix is raised
Numeric data representation
300010 = 3 * 103
Radix/Base
Exponent
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In order to process numeric values in a computer, decimalnumbers are converted into binary or hexadecimal numbers
However, since we ordinarily use decimal numbers, it would bedifficult to understand the meaning of the result of a process if itwere represented by binary or hexadecimal numbers.
This operation is called radix conversion The following radix/base conversion techniques will be
discussed:1. Decimal to Binary2. Binary to Decimal
3. Binary to Hexadecimal4. Hexadecimal to Binary5. Octal to Binary6. Binary to Octal
Radix/Base Conversion
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1. Decimal to Binary (Integer)
1. Decimal integer is divided into 2
2. The quotient and remainder are obtained
3. The quotient is divided into 2 again until thequotient becomes 0
4. The binary value is obtained by placing theremainder(s) in reverse order
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1. Decimal to Binary (Integer)
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1. Decimal to Binary (Fraction)
Decimal fraction is multiplied by 2
Resulting integer portion is extracted (always be 0 or 1)
Resulting fraction portion is multiplied by 2
Operation is repeated until the fraction portion becomes 0
When decimal fractions are converted into binaryfractions, most of the times, the conversion is notfinished, since no matter how many times the fractionportion is multiplied by 2, it will not become 0. Most
decimal fractions become infinite binary fractions.
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1. Decimal to Binary (Fraction)
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2. Binary to Decimal (Integer)
Performed by adding up the weights of each ofthe digits of the binary bit string
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2. Binary to Decimal (Fraction)
Same technique as for binary integers.
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3. Binary to Hexadecimal
4-bit binary strings are equivalent to 1hexadecimal digit
The binary number is divided into groups of 4
digits starting from the decimal point In the event that there is a bit string with less
than 4 digits, the necessary number of 0s isadded and the string is considered as a 4-bitstring
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3. Binary to Hexadecimal(Integer)
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3. Binary to Hexadecimal(Fraction)
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4. Hexadecimal to Binary(Integer)
1 digit of the hexadecimal number isrepresented with a 4-digit binary number
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4. Hexadecimal to Binary(Fraction)
Same technique as per integer
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5. Octal to Binary
Convert 1038 to its binary form
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6. Binary to Octal
Convert 10000112 to Octal
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Exercises
Convert into binary, octal and hexa
a) 2710b) 1510
c) 50.2210 Convert into decimal
a) 110112b) 338c) 1B.F16
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Octal-Binary Conversions
Binary to/from Octal conversion Conversion of binary to/from octal (whole numbers) Conversion of octal fractions
In decimal, 26.9210 = (2 * 101) + (6 * 100) + (9 * 10-1) + (2 * 10-2)
0.48 means 4 * 8-1 = (4/8) 10 = 10 = 0.510
0.2118 means (2 * 8-1) + (1 * 8-2) + (1 * 8-3)
Conversion of binary fractions Binary fractions can be converted in a similar manner to octal as that
of octal fractions The number can then be converted to decimal by adding up the
whole numbers and convert the fractions to decimals
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Try thisA. What number does the next digit position
represent in the hexadecimal system?
B. Use the answer to evaluate the decimalequivalent of 2A9D16
C. What is the highest decimal number which maybe represented by four hexadecimal digits?
D. What is the highest decimal number which maybe represented by four octal digits?
Quiz
? ? 256 16 1
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Numeric Presentation
Data
Decimal
Numbers
Unpacked Decimal
Fixed Point (Integers)
CharacterData
Packed Decimal
Floating Point (Real Numbers)Numeric
Data
BinaryNumbers
Representedusing decimalarithmetic
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Decimal digit representation
Binary coded decimal
Unpacked decimal format
Packed decimal format
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Decimal digit representation
o Binary-coded decimal (BCD) code Uses 4-bit binary digits (correspond to numbers 0 to 9 of
decimal system)
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Decimal digit representation
BCD code
Example:
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Decimal digit representation
Unpacked decimal format
Uses 1 byte for each digit of decimal number
Represents values from 0 to 9 in least significant 4 bits of 1byte and in most significant 4 bits (zone bits)
Half of a byte is used (excepting the least significant byte)where the least significant half-byte is used to store the sign
1100 = +ve
1101 = -ve
Waste of resources (eliminated by packed decimal format)
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Decimal digit representation Unpacked decimal format
+78910 = F7F8C916
-78910 = F7F8D916
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Decimal digit representation
o Packed decimal format 1 byte represents a numeric value of 2 digits
the least significant 4 bits represent the sign
bit pattern for the sign is the same as per unpackeddecimal format
+78910 = 789C16
-78910 = 789D16IT 121: Information Technology Fundamentals 2
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Questions
A) Represent 7089310 in Unpacked Decimal Format in Packed Decimal Format
B) Represent 789310 in Unpacked Decimal Format in Packed Decimal Format
C) F3F9C116 is represented in standard UnpackedDecimal Format What is its equivalent in decimal? Possible solution?
D) 3F9C16 is represented in standard Packed DecimalFormat What is its equivalent in decimal? Possible solution?
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Decimal digit representation
o Packed decimal format versus Unpacked decimalformat
A numeric value can be represented by fewer bytes
The conversion into the binary system is easy
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Representation of negative integers Absolute value representation
0 for positive, 1 for negative
Complement representation Decimal complement
9s complement
10s complement
Binary complement
1s complement 2s complement
Binary Representation
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Binary Representation
Absolute value representation Examples
(00001100)2 = (+12)10
(10001100)2 = (-12)10 Issues
(00000000)2 = +0
(10000000)2 = -0
Range of values (assumption: 7-bit absolute value representationused) -63 to +63 equivalent to(26-1) to +(26-1)
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Binary Representation
Complementrepresentation ofnegative numbers
Decimal complement The subtraction of
each of the digits of anumeric value from the
complement
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Binary Representation
Binary complement
1s complement of a given numeric value is the result of thesubtraction of each of the digits of this numeric value from 1,
as a result, all the 0 and 1 bits of the original bit string areswitched.
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Binary Representation
Binary complement
2s complement is 1s complement + 1
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Binary Representation
1s complement and 2s complementrepresentation of negative numbers
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Binary Representation
Advantages of 2s complement Less complicated (only one zero value)
Range of values to be represented is wider Subtractions can be performed with addition circuits, simplifying
hardware structure
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Binary Representation
1s complement and 2s complementrepresentation of negative integers
range of represented numeric values when n-bit binarynumber is represented by adopting the 1s complement
method:-(2n-1 1) to (2n-1 1)
range of represented numeric values when n-bit binary
number is represented by adopting the 2s complementmethod:
-(2n-1) to (2n-1 1)
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Binary Representation
Addition circuits only
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Binary Representation (Fixed
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Binary Representation (FixedPoint)
Fixed point
Integer representation
Fixed point is a data representation format used mainlywhen integer type data is processed
One word is represented in a fixed length (e.g. 16 bits and32 bits)
Overflow problem when attempt is made to represent anumeric value that exceeds the fixed length allocated
Fraction representation Decimal point is considered to be immediately preceded by
the sign bit
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Binary Representation (Fixed
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Binary Representation (FixedPoint)
Fixed point
Integer representation
Range of values
-(2n-1) to (2n-1 1)
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Binary Presentation (Fixed
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Binary Presentation (FixedPoint)
Fixed point
Fraction representation
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Binary Representation (Floating
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Binary Representation (FloatingPoint)
Floating point
Used to represent realnumber type data
Used to represent
extremely large orsmall size of data
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Bit Shift Operations
Using bit shifts, the multiplication and division of numericvalues can be easily performed
Shifting a binary digit 1 bit to the left, its value is doubled.
When a binary number is shifted nbits to the left, its former valueis increased 2n times
When a binary number is shifted nbits to the right, its formervalue decreases 2-n times (divided by 2n)
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Arithmetic Shift
To calculate numeric values in the fixed point format using 2scomplement representation
Rules Sign bit is not shifted Bit shifted out is lost Bit to be filled into the bit position is vacated as a result of the shift
is For left shifts, insert 0 For right shifts, insert the same bit as the sign bit
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Logical Shift
To change the bit position
Rules
Sign bit is also shifted (moved)
Bit shifted out is lost
Bit to be filled into the bit position vacated as aresult of the shift is 0.
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Bit Shifts
(-16)2 to be shifted 2 bits to the right Arithmetic Shift
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Bit Shifts
Logical Shift
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Operation and Precision
Precision of the numeric valuepresentationo The precision of a number is the range of its
erroro High precision = small error
o Single precision Range of numeric values presentable with 16 bits
(in the case of an integer without a sign) Minimum value = (0000 0000 0000 0000)2 = 0
Maximum value = (1111 1111 1111 1111)2 = 65,535
(values higher than 65,535 cannot be represented)
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Range of numeric values presentable with 16 bits(in the case of a fraction without a sign)
Minimum value = (0000 0000 0000 0001)2 = 2-16 =
0.0000152587890625000 Maximum value = (1111 1111 1111 1111)2 = 1 2
16 =0.9999847412109370000
(values lower than 0.00001525878, and values higherthan 0.99984741210937 cannot be represented)
Operation and Precision
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o Double precision
Number of digits is increased to widen the rangeof represented numeric values
Represent 1 numeric value with 2 words
1 numeric value presentable with 32 bits (in the
case of an integer without a sign)
Minimum value = (0000 0000 0000 0000 0000 00000000 0000)2 = 0
Maximum value = (1111 1111 1111 1111 1111 11111111 1111)2 = 4,294,967,295
(values up to 4,294,967,295 can be represented)
Operation and Precision
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Range of numeric values presentable with 16 bits (inthe case of a fraction without a sign)
Minimum value = (0000 0000 0000 0000 0000 0000 00000001)2 = 2
-32 = 0.00000000023283064365387
Maximum value = (1111 1111 1111 1111 1111 11111111 1111)2 = 1 232 = 0.99999999976716900000000
Operation and Precision
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Operation precisiono Precision of fixed point representation
Range of presentable numeric values depends on the computerhardware (number of bits in one word)
Range of represented numeric values differs depending on the numberof bits in one word
Step size of the integer part is always 1 (regardless of number of bits),and only the maximum value changes
In the fraction part, the smaller the step size becomes, the error is alsoreduced
o Precision and underflow Overflow and underflow
Overflow occurs when product is higher than the maximum value that can berepresented with the exponent portion (Maximum absolute value < Overflow)
Underflow occurs when product is lower than the minimum absolute value (0