1. basic theories of information

Part 1:
Computer System

Chapter 1: Basic Theories of Information

Learning Objectives
• To design and develop programs based on internal-
design documents, and to play the following roles
in a system development project:
– Use basic knowledge and skills related to information
technology in general and contributing to system
development as a member of the project
– Prepare program design documents based on supplied
internal design documents, under the direction of higher-
level engineers such as software design and development
engineers
– Develop programs by using knowledge related to basic-
level algorithm and data structure
– Making tests of the developed program

Textbooks
• Textbook for Fundamental Information
Technology Engineers
– No.1 Introduction to Computer Systems

Overview

“ In order to make a computer
work, information needs to
be converted into a format that
can be understood by the
computer.”

Part 1: Computer System

Basic Theories of Information
(Text No. 1 Chapter 1)

Objectives
• Understanding a computer’s basic data units such
as binary numbers, bits, bytes, words, etc. and
their conversions from and to octal, decimal, and
hexadecimal digits
• Understanding basic concepts of computer internal
data representation, focusing on numeric data,
character codes etc
• Understanding proposition calculus and logical
operations

Some Terminology
• Data representation unit and processing unit
1. Binary Digits (Bits)
 Two levels of status in computer’s 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” or “1”
 2 bits can represent 4 different values
 “00”, “01”, “10”, “11”

(or
Table)

(or Row)

(or Column)

Bit representation

Switches Open (0) or closed (1)
Current Not flowing (0) or flowing (1)

Lights Off (0) or on (1)

Numeric Conversion
3. 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 enables the representation of 2 8 = 256
types of information
 Using a 1-byte word, 256 different characters can be
represented – sufficient for most Western character sets
 However, the number of kanji (Chinese characters) amounts
to thousands of different characters, hence a 1-byte word
system is insufficient
 Two bytes are connected to obtain 16 bits, 2 16 = 65,536
 A 2-byte word

Numeric Conversion

4. Word
 The smallest unit that represents data inside a computer
 Increase operation speed
5. 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” to ease the representation of binary
numbers for humans – computers are capable of only using
binary numbers

Numeric Systems
Also known as Base Systems or Radix
Systems
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

Numeric Data
Representation
• The true value of numbers are the same
• The representation of numbers vary
– Decimal
– Binary
– Octal
– Hexadecimal

Numeric data representation
DECIMAL number 2 1 9 9 8
(Radix/Base = 10)
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 1 1 0 0 1
(Radix/Base = 2)
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

OCTAL number 2 1 7 7 2
(Radix/Base = 8)
Weight

Value

Final (true) value

HEXA number A 2 5 7 C
(Radix/Base = 16)
Weight

Value

Final (true) value

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

Binary Addition

Result = 1001102

Binary Subtraction

Result = 10102

Hexadecimal arithmetic
4. Addition and subtraction of hexadecimal
numbers
 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 the
right) digit
 A borrow from the upper digit is performed when
the result is negative

Hexadecimal Addition

• First column from right
D + 7 = (In the decimal system: 13 + 7 = 20) = 16 (carried 1) + 4
The sum of the first column is 4 and 1 is carried to the second column.
• Second column from right
1 + 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) + 5
The sum of the third column is 5 and 1 is carried to the fourth column.
• The result is (15A4)16.

Hexadecimal Subtraction

• First column from right
Since 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.

Exercises
• Compute the following
a) 2710 + 1510
b) 110112 + 11112
c) 338 + 178
d) 1B16 + F16

• Compute the following
a) 5010 – 2210
b) 1100102 - 101102
c) 628 – 268
d) 3216 - 1616

Radix/Base

300010 = 3 * 10 3 Exponent

• Representation of numeric data
1. Radix and “weight”
 Decimal numbers’ “weight” and its meaning
 “10” is called “Radix”
 upper right of 10 (in this example, 4) is called “exponent”
 Binary digit’s “weight” and its meaning
2. Auxiliary units and power representation
 Used to represent big, small amounts, and exponent to which the
radix is raised

Radix/Base Conversion
• In order to process numeric values in a computer, decimal
numbers are converted into binary or hexadecimal numbers
• However, since we ordinarily use decimal numbers, it
would be difficult to understand the meaning of the result of
a process if it were represented by binary or hexadecimal
numbers.
• This operation is called radix conversion
• The following radix/base conversion techniques will be
discussed:
1. Decimal to Binary
2. Binary to Decimal
3. Binary to Hexadecimal
4. Hexadecimal to Binary
5. Octal to Binary
6. Binary to Octal

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
the quotient becomes 0
4. The binary value is obtained by placing
the remainder(s) in reverse order

1. Decimal to Binary (Integer)

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 binary
fractions, most of the times, the conversion is not finished,
since no matter how many times the fraction portion is
multiplied by 2, it will not become 0. Most decimal
fractions become infinite binary fractions.

1. Decimal to Binary (Fraction)

2. Binary to Decimal (Integer)
• Performed by adding up the weights of each of the
digits of the binary bit string

2. Binary to Decimal (Fraction)
• Same technique as for binary integers.

3. Binary to Hexadecimal
• 4-bit binary strings are equivalent to 1
hexadecimal 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
0’s is added and the string is considered as a
4-bit string

3. Binary to Hexadecimal (Integer)

3. Binary to Hexadecimal (Fraction)

4. Hexadecimal to Binary (Integer)
• 1 digit of the hexadecimal number is represented
with a 4-digit binary number

4. Hexadecimal to Binary (Fraction)
• Same technique as per integer

5. Octal to Binary
• Convert 1038 to its binary form

6. Binary to Octal
• Convert 10000112 to Octal

Exercises
• Convert into binary, octal and hexa
a) 2710
b) 1510
c) 50.2210
• Convert into decimal
a) 110112
b) 338
c) 1B.F16

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 = ½10 = 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

Quiz
• Try this…
A. What number does the next digit position represent in
the hexadecimal system?

? ? 256 16 1
B. Use the answer to evaluate the decimal equivalent of
2A9D16
C. What is the highest decimal number which may be
represented by four hexadecimal digits?
D. What is the highest decimal number which may be
represented by four octal digits?

Numeric Presentation

Fixed Point (Integers)
Binary
Numeric Numbers
Floating Point (Real Numbers)
Data
Data
Character Unpacked Decimal
Data Decimal Represented
Numbers using
Packed Decimal decimal
arithmetic

Decimal digit representation
• Binary coded decimal
• Unpacked decimal format
• Packed decimal format

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

• BCD code
• Example:

– Unpacked decimal format
• Uses 1 byte for each digit of decimal number
• Represents values from 0 to 9 in least significant 4 bits of 1
byte 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)

• Unpacked decimal format

+78910 = F7F8C916

-78910 = F7F8D916

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 unpacked decimal
format

+78910 = 789C16

-78910 = 789D16

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 Unpacked Decimal
Format
– What is its equivalent in decimal?
– Possible solution?
• D) 3F9C16 is represented in standard Packed Decimal
Format
– What is its equivalent in decimal?
– Possible solution?

o Packed decimal format versus Unpacked
decimal format
A numeric value can be represented by fewer bytes
The conversion into the binary system is easy

Binary Representation
• Representation of negative integers
Absolute value representation
 “0” for positive, “1” for negative
Complement representation
 Decimal complement
9’s complement
10’s complement
 Binary complement
1’s complement
2’s complement

• 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 representation used)
• -63 to +63 equivalent to –(26-1) to +(26-1)

• Complement
representation of
negative numbers
– Decimal complement
– The subtraction of each
of the digits of a
numeric value from the
complement

• Binary complement
– 1’s complement of a given numeric value is the result of the
subtraction 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 are switched.

• Binary complement
– 2’s complement is “1’s complement” + 1

• 1’s complement and 2’s complement
representation of negative numbers

• Advantages of 2’s 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

• “1’s complement” and “2’s complement”
representation of negative integers
– range of represented numeric values when n-bit binary
number is represented by adopting the “1’s complement”
method:
-(2n-1 – 1) to (2n-1 – 1)
– range of represented numeric values when n-bit binary
number is represented by adopting the “2’s complement”
method:
-(2n-1) to (2n-1 – 1)

• Addition circuits only

Binary Representation (Fixed Point)
– Fixed point
• Integer representation
– Fixed point is a data representation format used mainly
when integer type data is processed
– One word is represented in a fixed length (e.g. 16 bits and
32 bits)
– Overflow problem when attempt is made to represent a
numeric value that exceeds the fixed length allocated
• Fraction representation
– Decimal point is considered to be immediately preceded
by the sign bit

Binary Representation (Fixed Point)
– Fixed point
• Integer representation
• Range of values
-(2n-1) to (2n-1 – 1)

Binary Presentation (Fixed Point)
– Fixed point
• Fraction representation

Binary Representation (Floating Point)
– Floating point
• Used to represent real
number type data
• Used to represent
extremely large or small
size of data

Bit Shift Operations
• Using bit shifts, the multiplication and division of numeric
values can be easily performed

• Shifting a binary digit 1 bit to the left, its value is doubled.
When a binary number is shifted n bits to the left, its former value is
increased 2n times
When a binary number is shifted n bits to the right, its former value
decreases 2-n times (divided by 2n)

Arithmetic Shift
• To calculate numeric values in the fixed point format using 2’s
complement 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

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 a
result of the shift is 0.

Bit Shifts
• (-16)2 to be shifted 2 bits to the right
• Arithmetic Shift

Operation and Precision
• Precision of the numeric value presentation
o The precision of a number is the range of its
error
o “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)

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 higher than
0.99984741210937 cannot be represented)

o Double precision
Number of digits is increased to widen the range
of 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 0000
0000 0000)2 = 0
 Maximum value = (1111 1111 1111 1111 1111 1111
1111 1111)2 = 4,294,967,295
(values up to 4,294,967,295 can be represented)

Range of numeric values presentable with 16 bits (in
the case of a fraction without a sign)
 Minimum value = (0000 0000 0000 0000 0000 0000 0000
0001)2 = 2-32 = 0.00000000023283064365387
 Maximum value = (1111 1111 1111 1111 1111 1111 1111
1111)2 = 1 – 2 –32 = 0.99999999976716900000000

• Operation precision
o Precision of fixed point representation
 Range of presentable numeric values depends on the computer
hardware (number of bits in one word)
 Range of represented numeric values differs depending on the number
of 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 also
reduced
o Precision and underflow
 Overflow and underflow
 Overflow occurs when product is higher than the maximum value that
can be represented with the exponent portion (Maximum absolute value
< Overflow)
 Underflow occurs when product is lower than the minimum absolute
value (0 < Underflow < Minimum absolute value)

Cancellation
 When subtraction of 2 floating point numbers of almost
equal values is performed
 Result becomes extremely small, it is left out of the range
of numeric values which can be represented
Loss of information
 Addition of extremely small value and extremely large
value is performed
 Exponents adjusted to the exponent of the largest value
(mantissa portion of the small value is shifted largely to
the right), leading to the loss of information that should
have been presented

Non-numeric Value Representation
• In order to represent characters using binary digits, codes are
used
– ASCII, ANSI, UNICode
• Character Representation
– Numeric keys: 0 to 9  10 types
– Character keys: Uppercase: A to Z
Lower case: a to z  52 types
– Symbolic keys: 40 types
– Control character keys: 34 types (Space key etc)
• To assign a unique bit pattern corresponding to these 136
types of characters and symbols, 256 types of bit patterns i.e.
8 bits are used.

Non-numeric Value
Representation
• Character codes
o ASCII (American Standard Code for Information
Interchange) code
 Character code of 8 bits (alphabet, numeric characters, etc.)
 Used in PCs and data transmission
o ISO (International Organization for Standardization)
code
 7-bit character code
 Base of the character codes used in all countries of the world
o JIS (Japanese Industrial Standard) code
 Represents 1 character with 2 bytes (16 bits)

Non-numeric Value
Representation
o EBCDIC (Extended Binary Coded Decimal Interchange
Code)
 Established to be used as standards
o Shift JIS (Japanese Industrial Standards) code
 Represents 1 character with 2 bytes
o Unicode
 2-byte code system unified to all countries
 To smooth the exchange of data amongst PCs

Non-numeric Value Representation
• Audio representation
o Multimedia audio
o Audio analysis is performed using a numeric formula and once it is
converted into digital codes it is processed in the computer.
o Word processors that accept audio input and speaker recognition are
examples of its recent applications.
• Image representation
o Image data must be processed to support current multimedia
o Image data is processed as a set of dots
o Example
o 1 bit is used to register the information of each dot (black, white)
o The representation method that combines the basic colors in each dot
is used. Systems that combine the three primary colors (Red, Green
and Blue) in 256 levels respectively and represent approximately
16,000,000 colors. In this case, since 8 bits are needed for 1 color, in
order to register the information of 1 dot, 24 bits are used.

Information and Logic

• Proposition Logic
• Logical operation

Proposition Logic
• A proposition is an assertion that something is the
case. We use sentences to express propositions.
• Examples:
(i) The following sentences express the same
proposition:
- “It is raining”
(ii) The following sentences express the same
proposition:
- “John loves Mary”
- “Mary is loved by John”

Proposition Logic
• Proposition will always be either “true” or “false”
• Philosophers argue a lot about what constitutes
truth. For now, we'll keep it simple:
o "P" is true if and only if P.
o "P" is false if and only if not P.
• Examples:
(i) The proposition "Snow is white" is true if and
only if snow is white.
(ii) The proposition "Snow is white" is false if
and only if snow is not white.

Proposition Logic
• Truth table
Proposition 1 Proposition 2 “If the wind
“The wind is blowing” “It is raining” blows it rains”
True True True
True False False
False True False
False False False

Proposition Logic
• Examples
p q
T T
T F p q p and q
F T T T T
F F T F F
F T F
F F F

Logical Operation
• A logical operator joins two propositions to
form a new, complex, proposition.
• The truth value of the new proposition is
determined by the truth values of the two
propositions being joined and by the
operator that joins them.

Logical Operation

• Negation
o Any proposition p can be converted into its
negation with a negation operator, producing
the new, complex, proposition:
¬p means Not p
The proposition Not p is true if and only if p is false
It is false only if p is true

Logical Operation
o Truth tables for Negation
p ¬p
T F
F T p q Not p
T T F
T F F
F T T
F F T

Logical Operation
• Logical Product
o Any two propositions p and q can be connected
with the conjunction “AND”, producing the
new, complex, proposition:
p and q (p ٨ q)
The proposition p and q is true if and only if both p
and q are true
It is false otherwise

Logical Operation

o Truth tables for Logical product

p q p٨ q
T T T
T F F
F T F
F F F

Logical Operation
• Logical Sum
o Any two propositions p and q can be connected
with the conjunction “OR”, producing the new,
complex, proposition:
p or q (p ۷ q)
The proposition p and q is true if and only if either
p or q are true
It is false only if both p and q are false

Logical Operation
o Truth tables for Logical sum
p q p۷q
T T T
T F T
F T T
F F F

Logical Operation
• Exclusive OR
o Any two propositions p and q can be
connected with the conjunction “EOR”,
producing the new, complex, proposition:
p eor q (p q)
The proposition p eor q is true only if when p or q
is true
It is false when both p and q are true or false

Logical Operation
o Truth tables for Exclusive OR
p q p q
T T F
T F T
F T T
F F F

Logical Operation
• Negative AND (NAND)
o Any proposition p can be converted into its
negation with a negation operator, producing
the new, complex, proposition:
Not p
The proposition Not p is true if and only if p is false
It is false only if p is true

Logical Operation

o Truth tables for Negative AND (NAND)

P Q P.Q Not (P.Q)
T T T F
T F F T
F T F T
F F F T

Logical Operation
• Negative logical sum (NOR)
o Negation of the logical sum
o ¬(p ۷ q)
• Summary of the truth table for the logical operations
P Q NOT p P AND q P OR q P EOR q P NAND q P NOR q
T T F T T F F F
T F F F T T T F
F T T F T T T F
F F T F F F T T

Logical Operation
• Logical expression laws
o Logical symbols

Meaning Symbols Notation example
Negation NOT ¬ ¯
Logical product AND ٨ · X·Y
Logical sum OR ۷ + X+Y
Exclusive OR EOR X Y

Logical Operation
- Laws of logical expressions
Logical product law X · X = X, X · = 0, X · 0 = 0, X · 1 = X
Logical sum law X + X = X, X + = 1, X + 0 = X, X + 1 = 1
Exclusive OR law X X = 0, X = 1, X 0 = X, X 1=
Commutative law X + Y = Y + X, X · Y = Y · X
Associative law X + (Y + Z) = (X + Y) + Z, X· (Y · Z) = (X · Y) · Z
Distributive law X + (Y · Z) = (X + Y) · (X + Z)
X · (Y + Z) = (X · Y) + (X · Z)
Absorption law X + (X · Y) = X, X · (X + Y) = X
Restoring law =X
De Morgan’s law = · , = +

Exercises
• Use the Laws of Logical Propositions to
simplify each of the propositions below to
one of the propositions F, T, p, q, p.q, p+q
a) p + q + -p
b) p + (q + p) + -q

Where to Get More Information
• Truth table practice:
http://www.math.csusb.edu/notes/quizzes/tablequi
z/tablepractice.html
• Digital Logic : Operation and
Analysis
by Jefferson C. Boyce
• Digital Logic and Switching
Circuits : Operation and Analysis
by Jefferson C. Boyce

Topics
• Floating point representation format
– Excess 64
– IEEE
• Data Structures
– Basic data structure
• Basic data type
• Structured data type
• Abstract data type
– Problem-oriented data structure
• List
• Stack
• Queue
• Tree
• Hash

Floating point representation
format in mainframe computers
• This format was adopted in the first general-purpose
computer in the world the "IBM System/360" and it was
called Excess 64.

• Exponent portion
– 7 bits
– Range: (0000000)2 to (1111111)2 , which in the decimal
system is 0 to 127. However, a numeric value 64 times
larger than the real exponent is represented. For that
reason, the real exponent is equivalent to −64 to +63.
– Likewise, since the radix is considered to be 16, the
numeric values that can be represented with the exponent
portion range between 16-64 to 1663
– Then, including the sign bit, the range of numeric values
that can be represented with the exponent portion is
further increased (see next slide)

• Mantissa portion
– When the decimal fraction 0.05 is converted
into a binary fraction, it becomes a repeating
binary fraction.
– (0.0000110011001100110011001100...)2

• Representing 0.0510

• Normalisation
– Since the mantissa portion has 24 bits, in this case, the decimal fraction 0.05
will not be represented correctly. (The error that occurs in this case is called a
rounding error)
– If we look at the bit pattern of the mantissa portion, we can see that the 4 top
bits are 0, if we then extract these 4 bits and shift the remaining bits to the left,
4 rounded bits can be represented.
– As a result of shifting the mantissa portion 4 bits to the left, the original value
of the mantissa portion was increased by 24 = 16. In order to cancel this
increase it is necessary to divide it into 16 (×16-1).
– Since the radix is 16, the value of the exponent portion can be set to -1 (63 –
64 = -1).
– Used to reduce the rounding error to its minimum as well as to maximize
precision. Also known as normalization. Furthermore, as a result of this
normalization technique, the bit strings that represent a value are standardized.
This operation is performed automatically by the hardware.

• http://www.cis.usouthal.edu/faculty/feinstein/50
2/chap2.htm

Binary Representation (Floating Point)
 IEEE Floating point representation format

S Exponent portion (8 bits) E Mantissa portion (23 bits) F
Radix: 2 Only binary fraction lower than 1
can be represented
Mantissa sign (1 bit)
The position of the decimal point is
0: Positive represented using the floating point format: (-1)S x 2E-127 x (1
 Value considered to be here
1: Negative
+ F)
 A value resulting from the addition of 127 to the value of the
original exponent portion is represented (this addition is called
bias)

1. basic theories of information

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