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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.

2IT 121: Information Technology Fundamentals 2ICS 121: Introduction to Computer Science 2


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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

IT 121: Information Technology Fundamentals 2ICS 121: Introduction to Computer Science 2 3


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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

IT 121: Information Technology Fundamentals 2ICS 121: Introduction to Computer Science 2


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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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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


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)

IT 121: Information Technology Fundamentals 2

ICS 121: Introduction to Computer Science 2

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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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o Binary-coded decimal (BCD) code Uses 4-bit binary digits (correspond to numbers 0 to 9 of

decimal system)



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BCD code

Example:



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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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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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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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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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Complementrepresentation ofnegative numbers

Decimal complement The subtraction of

each of the digits of anumeric value from the

complement



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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 complement

2s complement is 1s complement + 1



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1s complement and 2s complementrepresentation of negative numbers



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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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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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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)




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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)




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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




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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

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