Lecture 2 - Data Representation & Storage.pdf

8/14/2019 Lecture 2 - Data Representation & Storage.pdf

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1992-2012 by Pearson Education, Inc. & John Wiley & SonsSome portions are adopted from C++ f or Everyone by Horstmann

ENGR 1200U Introduc tion to Programming

Lecture 2Data Representation & Storage

Dr. Eyhab Al-Masri

ENGR 1200UWinter 2013 - UOIT

https://piazza.com/uoit.ca/spring2013/engr1200u


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Who is the inventor of the Analytical Engine?What is Programming?What are the main components inside acomputer? What does ALU stands for? Its purpose?What are the main software categories? List application software sub-categories (i.e. types)Why machine language is important?

Can a high-level program run without itsmachine language?


While performing arithmetic, one usually gives littlethought on the question: How many decimal digits it takes to represent a number?

Physicists can calculate that there are 10 78 electrons inthe universe without being bothered by the fact that itrequires 79 decimal digits to write that number out in full

Example

Calculating value of a function to six/seven/eightsignificant digits (using pencil and paper) The problem of the paper not being wide enough for seven-

eight digit numbers never arises!


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With computers, its quite different! On most computers, amount of memory available for

storing a number is fixed at the time that the computeris designed

With a certain amount of effort, programmer canrepresent numbers two, or three or even many timeslarger than this fixed amount

Challenge: fixed amount of memory


Finite nature of the computer forces us to dealonly with numbers that can be represented in afixed number of digits finite-precisionnumbers

Example : set of positive integers representableby three decimal digits (no decimal or sign)?

Set has exactly 1000 members: 000, 001, 002, , 999


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Since computers have finite memories (then itmust perform arithmetic on finite-precisionnumbers), the results of certain calculations willbe just plain wrong (from a mathematicalperspective)

Algebra of finite-precision numbers is differentfrom normal algebra


Example Associative law (can group numbers in any way without

changing the answer)

a + (b + c) = (a + b) c

for a = 700, b = 400, c = 300

Left side: 800 right side: overflow

associative law does not hold (overflow in the infinitearithmetic of three-digit integers)


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Example Distribution law (distribute pieces as you separate or break it

into parts)

a x (b c) = a x b a x c

for a = 5, b = 210, c = 195

Left side: 5 x 15 75 right side: overflow

A x b causes an overflow: 5 x 210 1050

distributive law does not hold (overflow in the infinitearithmetic of three-digit integers)


Conclusion:

Are computers suitable for doing arithmetic?

Yes


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Original computers were mainly designed toperform calculations (i.e. calculator)

Designers used electronic components available at thetime

Designers soon realized they could use a simple codingsystem (the binary system) to represent numbers


All types of information inside a computer canbe represented using binary system

Numbers Letters of alphabet Punctuation marks Microprocessor instructions Graphics/Video Sound


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Arithmetic used by computers differs in someways from the arithmetic used by people

Two main differences Most important difference is that computers perform

operations on numbers whose precision is finite andfixed

Most computers use the binary rather than the decimal

system for representing numbers


binary digit is a single numeral in a binarynumber

Example: 1 0 0 1 0 Each 0 and 1 in this example is a binary digit

Term bi nary digi t is commonly called a it

Eight bits grouped together form a yt


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Ordinary decimal number consists of a string ofdecimal digits and, possibly, a decimal pointThe general form and its usual representationare shown below

radix k number system requires k different symbols torepresent the digits 0 to k -1


Choice of 10 as the base for exponentiation (called the radix ) ismade because we are using decimal, or base 10 numbers

When dealing with computers, it is frequently convenient to useradices other than 10

Zero is always the first digit of any base When 1 is added to the largest digit, a sum of zero and a carry of one

results

Most important radices are 2 (binary) 8 (octal) 16 (hexadecimal)


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Binary numbers are built up from two binarydigits 0 1

Octal numbers are built up from the eight octaldigits 0 1 2 3 4 5 6 7

Hexadecimal numbers are built up from 16

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


Decimal number 2001 expressed in binary,octal, and hexadecimal


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Decimal numbers andtheir binary, octal, and

hexadecimal equivalents


Decimal numbers andtheir binary, octal, and

hexadecimal equivalents


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Fundamental aspect of number systems is thatthe rules governing are the same in any system

Counting Arithmetic operations Conversions from one base to another

The rules for performing an operation in onenumber system should be applied to anothernumber system i.e. addition rules in decimal should be able to apply

them in binary


Any number to the 0 (zero) power is 1

4 0 =1 5 0 = 1 1,800 0 = 1

Any number to the first power is the numberitself

6 1 = 6 87 1 = 87 1828 1=1828


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Prefix deci stands for tenthDecimal number system is a based 10 numbersystem

k = 10 k-1 = 9

There are 10 different symbols (called digits)

0 1 2 3 4 5 6 7 8 9

Each place value in a decimal number is a power of 10


10 3 10 2 10 1 10 0

1000 100 10 1

1 8 2 8


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1 x 1000 = 1000

8 x 100 = 800

2 x 10 = 20

8 x 1 = 8

1828

+


Prefix bi stands for two

Binary number system is a base 2 numbersystem

There are 2 symbols0 1

Each place value in a binary number is a power of 2


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2 3 2 2 2 1 2 0

8 4 2 1

1 1 0 1


1 x 8 = 8

1 x 4 = 4

0 x 2 = 0

0 x 1 = 0

12

+


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2 7 2 6 2 5 2 4 2 3 2 2 2 1 2 0

128 64 32 16 8 4 2 1

1 0 1 1 0 1 0 1


Each octal digit can be represented as a group of3-bit binary numberEach hexadecimal digit can be represented as agroup of 3-bit binary number


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Decimal numbers can be converted into binary intwo different ways

Repeated Subtraction Repeated Division

Both methods produce the same result and youshould use whichever method you are mostcomfortable with


Method A Repeated Subtraction) Largest power of 2 smaller than the number is

subtracted from the number

STEP 1:Write down all of the binary place values in order until youget to the first binary place value that is GREATER THANthe decimal number you are trying to convert

Example

512 256 128 64 32 16 8 4 2 1

485


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Method A Repeated Subtraction) contd)STEP 2 Mark out the largest place value (this will tell us how

many place values we will need)

485

512 256 128 64 32 16 8 4 2 1


Method A Repeated Subtraction) contd)STEP 3 Subtract the largest place value from the decimal

number. Then, place a 1 under the place valueSTEP 4 Repeat step 3 until all of the place values are

processed

485 256 = 229256 128 64 32 16 8 4 2 11


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229 128 = 101256 128 64 32 16 8 4 2 1

1 1101 64 = 37

256 128 64 32 16 8 4 2 11 1 1 37 32 = 5

256 128 64 32 16 8 4 2 11 1 1 1

5 16 = X256 128 65 32 16 8 4 2 11 1 1 1 0

256 128 65 32 16 8 4 2 11 1 1 1 0 0 1 0 1


Method B Repeated Division - for integers only) Dividing the number by 2

The quotient is written directly beneath the originalnumber and the remainder, 0 or 1, is written next tothe quotient

The quotient is then considered and the process is

repeated until the number 0 has been reached


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Method B Repeated Division)STEP 1 Divide the decimal by 2 in a regular long division until

you have a final remainder

STEP 2 Use the remainder as the LEAST SIGNIFICANT DIGIT of

the binary number

STEP 3 Divide the quotient from the first division until you

have final remainder


Method B Repeated Division)STEP 4 Use the remainder as the next digit of the binary

number

STEP 5 Repeat steps 3 and 4 as many times as necessary until

you have a quotient that can not be divided by 2

STEP 6 Use the remainder (one you cant divide by 2) as the

MOST SIGNIFICANT DIGIT


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Performing the same example of 485

485 / 2 = 242 rem 1242 / 2 = 121 rem 0121 / 2 = 60 rem 160 / 2 = 30 rem 030 / 2 = 15 rem 015 / 2 = 7 rem 1

7 / 2 = 3 rem 1

3 / 2 = 1 rem 11 / 2 = X rem 1

Least Significant Digit

Most Significant Digit

111100101


Prefix oct (octo) stands for eight

Octal number system is a base 8 number system

There are 8 symbols0 1 2 3 4 5 6 7

Each digit multiples a power of 8

Each group of 3 binary digits can be represented by asingle octal digit


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STEP 1Start with the LEAST SIGNIFICANT digit, mark outthe digits in groups of 3

Example

1 0 1 1 1 0 1 1 1


STEP 2Convert each group of three digits to is octaldigit (or symbol)

Example

1 0 1 1 1 0 1 1 1

5 6 7

The last groupon the left can

have anywherefrom 1 to 3

binary digitgroup


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Converting octal numbers to binary is just thereverse operation of converting binary to octal

Simply convert each octal digit to its three-bitbinary pattern. The resulting set of 1s and 0s isthe binary equivalent of the octal number


5 6 7

1 0 1 1 1 0 1 1 1


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8 3 8 2 8 1 8 0

512 64 8 1

7 5 0 1


Multiply each digit of the octal number by itsplace value and add the results

Example: 7501

7 x 512 = 3584 5 x 64 = 320

0 x 8 = 0 1 x 1 = 1

3905

+


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Prefix hexa stands for 6 and the prefix decistands for 10

Hexadecimal number system is a base 16 numbersystem

There are 16 symbols0 1 2 3 4 5 6 7 8 9 A B C D E F

Each digit multiples a power of 16

Each group of 4 binary digits can be represented by asingle hexadecimal digit


STEP 1Start with the LEAST SIGNIFICANT digit, mark outthe digits in groups of 4

Example

1 0 1 1 1 0 1 1 1


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STEP 2Convert each group of four digits to ishexadecimal digit (or symbol)

Example

1 0 1 1 1 0 1 1 1

1 7 7

The last groupon the left can

have anywherefrom 1 to 4binary digit

group


Converting hexadecimal numbers to binary is just the reverse operation of converting binary tohexadecimal

Simply convert each hexadecimal digit to itsfour-bit binary pattern. The resulting set of 1s

and 0s is the binary equivalent of thehexadecimal number


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16 3 16 2 16 1 16 0

4096 256 16 1

2 E C 3


Multiply each digit of the hexadecimal numberby its place value and add the results

Example: AB87

10 x 4096 = 40960 11 x 256 = 2816

8 x 16 = 128 7 x 1 = 7

43911

+


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There are a set of terms used in electronicswhich are used to represent different powers often

There is also a set of terms used to representlarge whole numbers and a set of terms used torepresent small fractional numbers


Prefix Value Abbreviation______ ______ ____________

Kilo 1,000 K

Mega 1,000,000 M

Giga 1,000,000,000 G

Tera 1,000,000,000,000 T


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INFR 2810 Computer Architecture

Data Types and Storage

Negative Binary Numbers


When data is represented in memory, it isrepresented as a sequence of bits

This sequence may represent: an instruction, a numeric value, a character,

a portion of an image, or digital signal, or some other type of data


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Computers use a standard binary code torepresent letters of the alphabet, numerals,punctuation marks, and other special characters

The code is referred to as American NationalStandards Institute (ANSI) Based on a previous encoding scheme called (ASCII)

American Standard Code for Information Interchange

Currently, there are 256 code combinations inthe ANSI format ASCII had 128 code combinations



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Character Code HEX_________ _____ ____

B 0100 0010 42

b 0110 0010 62

& 0010 0110 26

[BACKSPACE] 0000 1000 8


Two basic data types

integer floating

point


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Integer data is often stored in memory using 4bytes, or 32 bits

Left most bit is reserved for the sign of thenumber

Remaining 31 bits represent the magnitude ofthe number


Representation of data affects the efficiency ofarithmetic and logic operations

For efficiency, negative integers are oftenrepresented in their 2s complement form

The 2s complement of an integer is formed bynegating all of the bits and adding one


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Ones ComplementHas a sign bit with 0 used for plus and 1 forminus. To negate a number, replace each 1 bya 0 and each 0 by a 1. This holds for the signbit as well. Ones complete is obsolete

Switch all 0s to 1s and 1s to 0sBinary # 101100111s complement 01001100


Twos ComplementHas a sign bit that is 0 for plus and 1 for minus. Tonegate a number, it is done in a two step process:

1. Each 1 is replaced by a 0 and each 0 is replaced by a 1 (justas in the ones complement)

2. 1 is added to the result.Binary addition is the same as decimal addition except thata carry is generated if the sum is greater than 1 rather thangreater than 9

Example: 0 0 0 0 0 1 1 0 (+6)1 1 1 1 1 0 0 1 (-6 in ones complement)1 1 1 1 1 0 1 0 (-6 in twos complement)


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A signed number has the range -2 n-1 to 2 n-1 -1

An unsigned number has the range 0 to 2 n-1


Floating point types represent real numbers,such as 12.25, that include a decimal point

Digits to the right of the decimal point form thefractional part of the number

Digits to the left of the decimal point form theintegral part of the number


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Convert 12.25 10 to binary

First convert the integer part: 12 10 =1100 2Then repeatedly multiply the fractional part by 2: .25*2=0.5 C0 .50*2=1.0 C1Therefore:

12.2510 =1100.01 2


Textbook: Page 20: Practice exercises 1-10

Textbook: Page 22: Practice exercises 1-10

Textbook: Page 32

Practice exercises 10-13

Lecture 2 - Data Representation & Storage.pdf

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