02. Machine Level Representation OF DATA PDF

Preview

Summary

02. Machine Level Representation OF DATA...

Description

LESSON OUTCOMES Upon completion of this chapter, students should understand about:

CHAPT Machine Level Representation Of Data

•  Numeric(conversion(between(number(bases( o  Decimal&↔&Binary&↔&Octal&↔&Hexadecimal& •  Frac2onal(Conversions( •  Signed(Numbers(Representa2on( o  Sign-magnitude,&1s&complement,&2s&complement& •  Arithme2c(opera2ons( o  AddiAon&and&SubtracAon& •  Floa2ng(point(format( o  IEEE&754&standard&–&single&precision&

Edited&on&Mac2020&

Data Representation

Number System

Bytes Bits Each digit 0 or 1 is called a bit Smallest unit of data that the computer can represent

Eight bits grouped together is called a byte Each character such as a letter or a number is represented in a byte

Words 16 bits grouped together is called a word

Doubleword 32 bits grouped together is called a doubleword Equivalent to 4 bytes

CSC159&|&CHAPTER&2&|ZAZALEENA&ZAKARIAH&

system&of& posiAonal¬aAon& based&on&powers& of&10.&

DECIMAL

BINARY

system&of& posiAonal¬aAon& based&on&powers& of&8.&

OCTAL

HEXADEC IMAL

system&of& posiAonal& notaAon&based& on&powers&of&2.&

system&of& posiAonal& notaAon&based& on&powers&of&16&

BINARY NUMBERS

Only 2 voltage levels present in a digital circuit – logic High (1) and logic Low (0).

2.1 Numeric Conversion

Converting Base 10 to Base 2 A decimal number 4210 can be converted to a binary number by dividing the number by 2:

Base 2: 0, 1

Converting Base 10 to Base 2 A decimal number 8710 can be converted to a binary number by dividing the number by 2:

Converting Base 2 to Base 10 A binary number 11001112 is converted to a decimal numbers by summing the weights of various positions in the binary number which contains a 1.

BINARY NUMBERS : exercises

BINARY NUMBERS : exercises Convert the following number: a)  Decimal 23010 to binary b)  Binary 101101112 to decimal

OCTAL NUMBERS

Convert the following number: a)  Decimal 31110 to binary b)  Binary 11011102 to decimal

The octal number uses base 8.

It uses the digits 0, 1, 2, 3, 4, 5, 6, 7

Converting Base 10 to Base 8

Converting Base 8 to Base 10

A decimal number 26610 can be converted to an octal number by dividing the number by 8:

To convert an octal number 72638 to a decimal number, multiply each octal value by the weight of the digit and sum the results.

Converting Base 8 to Base 10

OCTAL NUMBERS

To convert an octal number 3248 to a decimal number, multiply each octal value by the weight of the digit and sum the results.

Each octal digit can be represented by a 3-bit binary number as shown below: Octal Digits

3-bit Binary number

0

000

1

001

2

010

3

011

4

100

5

101

6

110

7

111

OCTAL NUMBERS

OCTAL NUMBERS

Each octal digit can be represented by a 3-bit binary number as shown below:

Each octal digit can be represented by a 3-bit binary number as shown below: A binary number is converted into an octal number by taking groups of 3 bits, starting from LSB, and replacing them with an octal digit.

OCTAL NUMBERS

OCTAL NUMBERS : exercises

Each octal digit can be represented by a 3-bit binary number as shown below: A binary number is converted into an octal number by taking groups of 3 bits, starting from LSB, and replacing them with an octal digit.

Convert the following number:

a) Decimal 24510 to octal b) Octal 12538 to decimal

OCTAL NUMBERS : exercises

OCTAL NUMBERS : exercises

Convert the following number:

c)  Octal 7648 to binary d)  Binary 110101112 to octal

HEXADECIMAL NUMBERS

Convert the following number: a)  Decimal 31110 to octal b)  Binary 11011102 to hexa c)  Octal 7348 to hexa

HEXADECIMAL NUMBERS Please refer to the table below:

1 2

The hexadecimal number uses base 16. It uses the digits 0 through 9 plus the letters A, B, C, D, E and F.

The letter A stands for decimal 10, B for 11, C for 12, D for 13, E for 14 and F for 15.

Hexadecimal

Decimal

Binary

0 1 2 3 4 5 6 7 8 9 A B C D E F

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

Converting Base 10 to Base 16

Converting Base 10 to Base 16

A decimal number can be converted to hex number by successively dividing the number by 16 as follows:

A decimal number can be converted to hex number by successively dividing the number by 16 as follows:

HEXADECIMAL NUMBERS

HEXADECIMAL NUMBERS

To convert a hex number to a decimal number, multiply each hex value by the weight of the digit and sum the results.

Conversion from hex to binary is very straightforward. Each hex digit is replaced by 4-bit binary number.

HEXADECIMAL NUMBERS Another example of the conversion: Hexadecimal à Binary

HEXADECIMAL NUMBERS A binary number is converted into a hex number by taking grou bits, starting from LSB, and replacing them with a hex digit

HEXADECIMAL NUMBERS : exercises

HEXADECIMAL NUMBERS : exercises

Convert the following number:

Convert the following number:

a) Decimal 31510 to hexadecimal b) Hexadecimal 6E916 to decimal

c)  Hexadecimal 24F16 to binary d)  Binary 10110100111012 to hexadecimal

Exercises Convert the following number:

574 8 = AF2 16 = 652 8 =

2 = 2 = 2 =

16 = 8 = 16 =

10 10 10

2.2 Fractional Conversions

Fractions

•  Number&point&or&radix&point& &♦Decimal&point&in&base&10& &♦Binary&point&in&base&2& •  No&exact&relaAonship&between&fracAonal& numbers&in&different&number&bases& ♦Exact&conversion&may&be&impossible&

Decimal Fractions

•  Move&the&number&point&one&place&to&the&right& &♦&Effect:&mulAplies&the&number&by&the&base&number& &♦&Example:&139.010& &139010& & •  Move&the&number&point&one&place&to&the&le\& &♦&Effect:÷s&the&number&by&the&base&number& &♦&Example:&139.010& &13.910&

Decimal Fraction ConverAng&Base&10&and&Base&2&

Decimal Fractions Base&10&and&Base&2&

•  No&general&relaAonship&between&fracAons&of&types&1/10k& and&1/2k& &♦&Therefore&a&number&representable&in&base&10&may&& &&&&&&&&&&¬&be&representable&in&base&2& &♦&But:&the&converse&is&true:&all&fracAons&of&the&form&& &&&&&&&&&&&1/2k&can&be&represented&in&base&10& •  FracAonal&conversions&from&one&base&to&another&are& stopped& &♦&If&there&is&a&raAonal&soluAon&or& &♦&When&the&desired&accuracy&is&aaained&

Decimal Fractions Mixed&number&conversion&

•  Integer&and&fracAon&parts&must(be(converted( separately( •  Radix&point:&fixed&reference&for&the&conversion& &♦Digit&to&the&le\&is&a&unit&digit&in&every&base& &♦B0&is&always&1®ardless&of&the&base&

2.3 Representations for signed integers

Complementary Representation

•  We&normally&represent&signed&integers&by&a&plus& or&minus&sign&and&a&value& •  In&computers,&we&cannot&use&a&sign&symbol& &♦&Must&restrict&to&0’s&and&1’s& •  RepresentaAon&of&signed&numbers:& &♦&Sign-and-magnitude& &♦&1s&complement& &♦&2s&complement&

One’s Complement

Sign and Magnitude Representation •  Represent&signed&integers&by&a&plus&or&minus&sign& and&a&value& •  In&computer,&we&cannot&use&a&sign& &♦Must&restrict&to&0’s&and&1’s& •  Example:&

Two’s Compliment Find&the&1’s&complement&and&add&1&to&the&result.&For&example,&2’s&complement&of&+2&in&binary.&&

•  Numbers&that&begin&with&0&are&posiAve& •  Numbers&that&begin&with&1&are&negaAve& •  Performed&by&changing&every&0&to&1&and& every&1&to&0&(inversion)&à&FLIP!( •  Example:&

Addition AddiAon&in&Octal&

Carry&

2.4 Arithmetic Operations (addition and subtraction)

Carry&

1&&&1&

&&&&4568& +&&1238& &&&&6018&

1&&&1&&&&&1&&&&1&&&1&

&&&&&&777148& +&&&&&&&&&&&768& &&&&1000128& Remember!!& Digits&in&Octal&are&0& to&7.&

Addition

Addition

AddiAon&in&Hexadecimal&

AddiAon&in&Binary&

0& Carry& Carry&

1&&&1&

&&&&7A616& +&&2BA16& &&&&A6016&

1&&&1&

9& 10& A& 11& B& 12& C& 13& D& 14& E& 15& F&

&&&&1112& +&&&&102& &&10012&

Carry& 0&&&&1&&&&1&&&&1&

&&&&7& &&&&001112& +&&2& +&&101012& &&&&9& &&&&111002&

&&&&7& +21& &&28&

Subtraction

Subtraction

SubtracAon&in&Octal&

SubtracAon&in&Hexadecimal&

0&

Borrow& +8& &&&2&&&&4&&&+8&

Borrow&

If&you&borrow & in&Octal,& the&value&& borrowed& is&8.&

&&&&3568& &&&& 68& -&&&1578& -&&&1578& &&&&1778& &&&&1778&

&B&&+16&&9&&&+16&

&&&&C9AD16& &&&& 9 D16& -&&&5A8F16& -&&&5A8F16& &&&&6F1E16& &&&&6F1E16&

If&you&borrow& in&Octal,& the&value&& borrowed& Is&16.&

9& 10& A& 11& B& 12& C& 13& D& 14& E& 15& F&

Subtraction SubtracAon&in&Binary&

Borrow&

2.5 Floating Point format (IEEE 754 standard single precision)

1&&&1& 0&&&

&&10&

&&&&110002& &&&&11 02& -&&&&&&&1112& -&&&&&&&1112& &&&&100012& &&&&100012&

If&you&borrow & in&Octal,& the&value&& Borrowed& is&210&or&10 .& 2

Floating Point ScienAfic&NotaAon&&

Floating Point ScienAfic&NotaAon&&

•  In&many&calculaAons&the&range&of&numbers& used&is&very&large.& •  A&number&can&be&represented&by&using& scienAfic&representaAon&commonly&used&in& physics,&chemistry,&and&engineering.& •  The&computer&version&of&the&scienAfic& notaAon&is&called&floaAng&point.&

Floating Point ScienAfic&NotaAon&&

Floating Point ScienAfic&NotaAon&&

EXCERCISE

Floating Point Example:&Convert&-23.675&to&floaAng&point&numbers&in&IEEE.&Give&the&result&in&hexadecimal.& Step&1:&Convert&to&binary&

-23.675&=&10111.1012&

Step&2:&Put&into&1.XXXX&x&2y&

1.01111012&x&24&

Step&3:&Get&the&sign&

Sign&–ve&=&1&

Step&4:&Get&the&biased&exponent&

Exponent&=&4& Biased&exponent&=&127&+&4&=&131& &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&=&100000112&

Step&5:&Get&&the&manAssa&and& significand&

ManAssa&=&1.01111012& Significand&=&0111&1010&0000&0000&0000&

Step&6:&Put&into&IEEE&format&

sign& Biased& exponent&

ManAssa&/&significand&

1&

0111&1010&0000&0000&0000&

Step&7:&Convert&to&hexadecimal&

1100&0001&1011&1101&0000&0000&0000&0000& &&&C&&&&&&&&1&&&&&&&&B&&&&&&&D&&&&&&&0&&&&&&&&0&&&&&&&&0&&&&&&&&0&

10000011&

EXCERCISE& Convert&the&following&numbers&to&IEEE&single&precision&format&in&hexadecimal&form:&...