1. basic theories of information

Part 1: Computer System

Chapter 1: Basic Theories of Information

Digital Divide?

Digital Divide?

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

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

Overview

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 Column)

(or Row)

(or Table)

Bit representation

Switches Open (0) or closed (1)

Current Not flowing (0) or flowing (1)

Lights Off (0) or on (1)

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 28 = 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, 216 = 65,536 A 2-byte word

Numeric Conversion

4. WordThe smallest unit that represents data inside a computerIncrease operation speed

5. Number systemsBinary system is used to simplify the structure of electronic

circuits that make up a computerHexadecimal 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 Conversion

Numeric SystemsAlso known as Base Systems or Radix

SystemsAvailable 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, Fwhere A=10,B=11,C=12,D=13,E=14,F=15

Numeric DataRepresentation

• The true value of numbers are the same• The representation of numbers vary

– Decimal

– Binary

– Octal

– Hexadecimal

Numeric data representationDECIMAL 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

Numeric data representationOCTAL 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

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

4. Addition and subtraction of hexadecimal numbersAddition

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 arithmetic

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 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 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 rightC – 7 = 5 (In the decimal system: 12 – 7 = 5)

• Third column6 – 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

• Representation of numeric data1. 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 representationUsed to represent big, small amounts, and exponent to which the

radix is raised

Numeric data representation

300010 = 3 * 10 3

Radix/Base

Exponent

• 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 Binary2. Binary to Decimal3. Binary to Hexadecimal4. Hexadecimal to Binary5. Octal to Binary6. Binary to Octal

Radix/Base Conversion

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 decimala) 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

• Try this…A. What number does the next digit position represent in

the hexadecimal system?

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?

Quiz

? ? 256 16 1

Numeric Presentation

Data

Decimal Numbers

Unpacked Decimal

Fixed Point (Integers)

Character Data

Packed Decimal

Floating Point (Real Numbers)Numeric

Data

Binary Numbers

Represented using decimal arithmetic

Decimal digit representation

• Binary coded decimal

• Unpacked decimal format

• Packed decimal format

Decimal digit representation 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)

Decimal digit representation• Unpacked decimal format

+78910 = F7F8C916

-78910 = F7F8D916

Decimal digit representationo Packed decimal format

1 byte represents a numeric value of 2 digitsthe least significant 4 bits represent the signbit 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 formatA numeric value can be represented by fewer bytesThe conversion into the binary system is easy

• Representation of negative integersAbsolute value representation

“0” for positive, “1” for negativeComplement representation

Decimal complement9’s complement10’s complement

Binary complement1’s complement2’s complement

Binary Representation

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 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

Binary Representation• 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)

Binary Representation• 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 timesWhen 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

Bit Shifts

• Logical Shift

Operation and Precision

• Precision of the numeric value presentationo 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 = 0Maximum 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 precisionNumber of digits is increased to widen the range

of represented numeric valuesRepresent 1 numeric value with 2 words1 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 precisiono 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 underflowOverflow 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)


CancellationWhen 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 representedLoss 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.

• Character codeso ASCII (American Standard Code for Information

Interchange) codeCharacter code of 8 bits (alphabet, numeric characters, etc.)Used in PCs and data transmission

o ISO (International Organization for Standardization) code7-bit character codeBase of the character codes used in all countries of the world

o JIS (Japanese Industrial Standard) codeRepresents 1 character with 2 bytes (16 bits)


o EBCDIC (Extended Binary Coded Decimal Interchange Code)Established to be used as standards

o Shift JIS (Japanese Industrial Standards) codeRepresents 1 character with 2 bytes

o Unicode2-byte code system unified to all countriesTo smooth the exchange of data amongst PCs


• Audio representationo Multimedia audioo 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 representationo Image data must be processed to support current multimediao Image data is processed as a set of dotso 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 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

“The wind is blowing”

Proposition 2

“It is raining”

“If the wind blows it rains”

True True True

True False False

False True False

False False False

Proposition Logic

• Examples

p q

T T

T F

F T

F F

p q p and q

T T T

T F F

F T F

F F F

Proposition Logic

• 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

• Negationo Any proposition p can be converted into its

negation with a negation operator, producing the new, complex, proposition:¬p means Not pThe proposition Not p is true if and only if p is falseIt 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 Producto 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 trueIt is false otherwise

Logical Operation

o Truth tables for Logical product

p q p ٨ qT T T

T F F

F T F

F F F

Logical Operation

• Logical Sumo 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 trueIt is false only if both p and q are false

Logical Operation

o Truth tables for Logical sum

p q p ۷ q

T T T

T F T

F T T

F F F

Logical Operation

• Exclusive ORo 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 trueIt 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 pThe proposition Not p is true if and only if p is falseIt 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 lawso 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

= · , = + De Morgan’s law

= XRestoring law

X + (X · Y) = X, X · (X + Y) = XAbsorption law

X + (Y · Z) = (X + Y) · (X + Z)

X · (Y + Z) = (X · Y) + (X · Z)

Distributive law

X + (Y + Z) = (X + Y) + Z, X· (Y · Z) = (X · Y) · ZAssociative law

X + Y = Y + X, X · Y = Y · XCommutative law

X X = 0, X = 1, X 0 = X, X 1 = Exclusive OR law

X + X = X, X + = 1, X + 0 = X, X + 1 = 1 Logical sum law

X · X = X, X · = 0, X · 0 = 0, X · 1 = X Logical product law

Logical Operation - Laws of logical expressions

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+qa) p + q + -p

b) p + (q + p) + -q

Where to Get More Information

• Truth table practice: http://www.math.csusb.edu/notes/quizzes/tablequiz/tablepractice.html

• Digital Logic : Operation and Analysisby Jefferson C. Boyce

• Digital Logic and Switching Circuits : Operation and Analysisby 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/502/chap2.htm

IEEE Floating point representation format

Value represented using the floating point format: (-1)S x 2E-127 x (1 + F)

A value resulting from the addition of 127 to the value of the original exponent portion is represented (this addition is called bias)

Binary Representation (Floating Point)

S Exponent portion (8 bits) E Mantissa portion (23 bits) F

Mantissa sign (1 bit)0: Positive1: Negative

The position of the decimal point is considered to be here

Only binary fraction lower than 1 can be represented

Radix: 2