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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 Introduction to Programming

Lecture 3

Data Representation & Storage (contd)

Dr. Eyhab Al-Masri

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Prefix oct (octo) stands for eight

Octal number system is a base 8 number system

There are 8 symbols

0 1 2 3 4 5 6 7

Each place value in an octal number is a power of 8

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

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STEP 1 Start with the LEAST SIGNIFICANT digit, mark out

the digits in groups of 3

Example

1 0 1 1 1 0 1 1 1

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STEP 2 Convert each group of three digits to its octal

digit (or symbol)

Example

1 0 1 1 1 0 1 1 1

5 6 7

The last group

on the left can

have anywhere

from 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

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5 6 7

1 0 1 1 1 0 1 1 1

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There are two methods to choose from:

Do a decimal-to-binary conversion, and then abinary-to-octal conversion

Do a direct conversion using the repeated divisionmethod

Since this is a conversion to octal, 8 is the divisor eachtime

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Performing the same example of 485

485 / 8 = 60 rem 0.625 0.625 x 8 = 560 / 8 = 7 rem 47 / 8 = X rem 7

Least Significant Digit

Most Significant Digit

7 4 5

Since we are dividing by 8 in the repeated division, you

may end up with remainders of anywhere from 0 to 7

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Example:

Given octal number 7501, what is the equivalentdecimal value?

83 82 81 80

512 64 8 1

7 5 0 1

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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 deci standsfor 10

Hexadecimal number system is a base 16 numbersystem

There are 16 symbols 0 1 2 3 4 5 6 7 8 9 A B C D E F

Each place value in a hexadecimal number is a power of16

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

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STEP 1 Start with the LEAST SIGNIFICANT digit, mark out

the digits in groups of 4

Example

1 0 1 1 1 0 1 1 1

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STEP 2 Convert each group of four digits to is

hexadecimal digit (or symbol)

Example

1 0 1 1 1 0 1 1 1

1 7 7

The last group

on the left can

have anywhere

from 1 to 4

binary digit

group

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Converting hexadecimal numbers to binary isjust 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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Example:

Given hexadecimal number 2EC3, what is theequivalent decimal value?

163 162 161 160

4096 256 16 1

2 E C 3

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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

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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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Prefix Value Abbreviation______ ______ ____________

milli 1/1,000 m

micro 1/1,000,000

nano 1/1,000,000,000 n

pico 1/1,000,000,000,000 p

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kilobyte (KB) is 1,024 bytes

megabyte (MB) is 1,048,576 bytes

These values are derived from the nearest binaryplace values 1,000 and 1,000,000

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Data Types and Storage

Negative Binary Numbers

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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

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Two basic data types

integerfloating

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

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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 by

negating all of the bits and adding one

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Ones Complement

Has 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 complement is obsolete

Switch all 0s to 1s and 1s to 0s

Binary # 10110011

1s complement 01001100

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Twos Complement Has a sign bit that is 0 for plus and 1 for minus. To

negate 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 (just

as 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 -2n-1 to 2n-1 -1

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

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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 the

integral part of the number

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Convert 12.2510 to binary

First convert the integer part: 1210=11002

Then repeatedly multiply the fractional part by 2: .25 2=0.5 C0 .50 2=1.0 C1

Therefore:

12.2510 =1100.012

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What does the decimal value of 124 stand for?

What is the decimal value of the binary value of110?

1 2 4 rightmost place

102 101 100

x100 + x10 + x1

124

base

1 1 0 rightmost place

22 21 20

x4 + x2 + x1

110

base

124answer

6answer

1 21

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What is the decimal value of the octal value 176?

What is the decimal value of the hexadecimalvalue of B29?

1 7 6 rightmost place

82 81 80

x64 + x8 + x1

176

base

B 2 9 rightmost place

162 161 160

x256 + x16 + x1

B29

base

2857answer

126answer

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Convert the following:

1012 = ?10

Begin with rightmost bit (ones place).

The next bit is the twos place, then the fours place,and so on

1 0 1

22 21 20

x4 + x2 + x1

5

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Convert the following:

10102 = ?10

Begin with rightmost bit (ones place).

The next bit is the twos place, then the fours place,and so on

1 0 1 0

23 22 21 20

x8 + x4 + x2 + x1

10

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Convert the following:

2708 = ?10

Begin with rightmost bit (ones place).

The next bit is the eights place, then the sixty-foursplace, and so on

2 7 0

82 81 80

x64+ x8 + x1

184

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Convert the following:

112A16 = ?10

Begin with rightmost bit (ones place).

The next bit is the 16s place, then the 256s place,and so on

1 1 2 A

163 162 161 160

x4096 + x256 + x16 + x1

4394