Computer Notes MFS Booklet Class 9th.compressed

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Now converting this decimal to equivalent octal form:

8 421
8 52 5
8 64

06

Thus the octal equivalent o f(421)10 is (645)8 (110100101)2 = (421)10= (645)8
(110100101)2=(421)10 = (645)8
Converting the same number by the second method that is by grouping the

binary number.

Binary Number. 110 100 101
(grouped in 3 bits)

In decimal form 4 21 4 21 4 21

In actual from 645 4 x 1 + 2 x 1 + 1 x O 4xl+2xO+1xO 4xl+2xO+1x1

645

Therefore the octal equivalent of (110100101)2 is (645)8
3. Fraction

Example:

(110111001.100001)2

Let us first convert the above given binary number to the decimal equivalent:

Integer part:
(110111001)2 = 1x28+ 1x27 + Ox26 + lx25 + Ix24 + 1x23 + Ox22 + Ox21 + 1x2°

= 256 + 128 + 0 + 32 + 16 + 8 + 0 + 0 + 1

= (442)10
Now converting this decimal form into octal form:

8 422 2
8 55 7
87
7
0

Thus octal equivalent of integer part (442)10 is (772)8
Fraction Part:
(0.100001)2 =1/2 + 1/64
= 33/64
= (0.515625)

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Now converting this decimal form into octal form:

Fraction = 0.515625

Fraction Integer Fraction Integer

0.515625 x8 = 4.125 1

0.125 x8 = 1.000 1

Thus the octal equivalent 0f(0.514625)10 = (0.41)8
Therefore (110111001.100001) 2 = (442.515626)10= (772.41)8
Binary into Hexadecimal

Integer:

For converting a binary number into its Hexadecimal equivalent, it should be
first converted into groups of four bits and then these groups can directly be
converted into their hexadecimal equivalents.

Example:

(lll 0 1 000 10111100)2

1110 1000 1011 1100

Ix8+1x4+ lX2+0X1 1X8+0X4+0X2+0Xl lX8+OX4+ 1X2+1X1 1X8+ 1x4+0x2+0x

13=D 8=8 11=B 12=C

Thus the hexadecimal equivalent of (1110100010111100)2 is (D8BC)16

Fraction:

Even the fraction part is converted into a group of four bits and then converted
to corresponding hexadecimal number. The grouping starts from the very next
bit after the decimal and proceeds towards the right

Example:

(0.11001101 )2 1101
1100

12 = C 13 = D

Therefore the hexadecimal equivalent of (0.11001101)2 = (CD)16
Octal Number System to Other Systems:

Octal into Decimal':

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Integer:

For converting the octal number to its Decimal equivalent, each digit

is multiplied with its respective position weight and then added to get the

desired octal number

Example:- 3 4 9
82 =64 81=8 80 =1
(349)8
129 32 9
=192 + 32 + 9

(233)10

Fraction:

For converting the fractional octal number to its decimal equivalent,

each digit is multiplied with its respective position weights (i.e. the negative

powers of the radix 8) and then added together to get the result.

Example:
(0.265)8

= 2 X 8' + 6 X 8-2+ 5 X 8-3
= 2/8 + 6/64 + 5/ 512
= (0.353515625)10
Therefore (0.265)8 = (0.35315625)10
Integer:
For converting the Octal number to Binary number, it has to be first converted
into Decimal form and then Binary.
Example:
(243)8
Converting the above given Octal number to Decimal form first:
(243)8 = 2 X 82 + 4 x 81 + 3 x 80
= 2 x6 4 + 4 x 8 + 3 x 1
= 128 + 32 + 3
=(168)10

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Now converting this Decimal equivalent to the Binary form:

2 168 1
2 81 1
2 40 0
2 20 0
2 10 0
25 1
22 0
21 1

0
Thus the binary equivalent of (163) 10 = (10100011 )2

Therefore (256)8 = (163)10 = (l0100011)2

Fraction:

For converting the Octal number to its Binary equivalent, it has to be converted
first into its Decimal form

Example:

(0.25)8

(0.25)8 = 2 X 8-1 + 5 X 8-2
= 2/8 + 5/64

= 21/64

= (0.328125)10
Now converting this decimal number to its binary equivalent:

0.328125 x 2 = 1.3125 Fraction Integer
0.3125 1

0.3125 x 2 = 1.25 0.25 1

0.25 x 2 = 0.50 0.50 0
0.50 X21 = 1.00 0.00 1

Thus the binary equivalent of (0.328125)10 is (0.1101)2

Therefore (0.25)8 = (0.328125)10 = (0.1101)2

Hexadecimal System to other Systems:

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Hexadecimal to Decimal:-
Each digit of the Hex Decimal number is multiplied with its respective position
weight and then added to get the number in the proposed system.

Example:

(E28B)16

E = 14 2 8 B = 11
163 = 4096 162 = 256 161 = 16 160 = 1

14 x 4096 2 x 256 8 x 16 11x 1
128 11
57344 512

=57344+512+ 128+ 11

= (57995)10
Therefore (E28B)16 = (57995)10

1st Method:

The hexadecimal number can be converted to its binary Equivalent by first
converting it to the equivalent decimal number and then converting the
resultant decimal number to the binary number.

Example:

(C3B 1 )16

Converting (C3B1)16 to its equivalent decimal form:

1x 16° = 1
Bx161= 176
3 x 162 = 768
C x 163 = 49152

50097

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Thus (C3B1)16 = (50097)10
Now converting this decimal from to its equivalent binary form:

2 50097 I
2 25048 0
2 12524 0
2 6262

2 3131 0
2 1565 I
2 782 I
2 391 0
2 195 I
2 97 I
2 48 1

2 24 0
2 12 0
2 60
2 30

11

Thus the binary equivalent of (50097)10 is (1100001110110001)2
Therefore (C3B1)16 = (50097)10 = (1 10000 1110 1 1000l)2
2nd Method:

The Hexadecimal number is first converted into its Decimal equivalent and then

the Decimal number so obtained is converted to the group of binary digits

(BIT) It uses the group of four bits to represent a single number.

Hexadecimal C 3 B 1

Decimal 12 3 11 1
8421 8421 8421
Binary Number 8421 0011 10ll 0001
1100

5.6 ARITHMETIC OPERATIONS IN BINARY SYSTEMS:

All the arithmetic operations (addition, subtraction, multiplication and

division) in binary system are performed in the same way as in decimal number

system. All the four arithmetic operations are described as below:

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1. Addition:

Rules for carrying out binary additions are:
0+0=0
0+ 1 = 1
1+0=1
1 + 1 = 0 with one (1) carry over.

Examples:

1. For adding 101 1 102 and 11110 2
Binary Decimal
101110 46
111101 61
1101011 107

Thus (1101011)2 in binary system is equivalent to (107) 10 in Decimal system.
1. For adding 1000002 and 10112:

Binary Decimal
100000 32
1011 11
101011 43
The binary equivalent to (43) 10 in Decimal system is (101011)2 in Binary
system.
2. Subtraction:
Rules for carrying out binary subtractions are:
1.0-0=0
2. 0 - 1 = I with one borrow
3.1-0 = 1
4. 1 - 1 = 0

Examples:

1. For subtracting 1011102 from 1111012
Binary Decimal
111101 61
101110 -46
001111 15

Thus 001111 in binary system is equivalent to 15 in decimal system.
1. For subtracting 1011 from 100000:
Binary Decimal
100000 32
1.0ll 11
10101 21

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The binary equivalent to 21 in decimal system is 10101 in binary system:
2. Multiplication:
The rules for the multiplication are:
OxO=O
Ox1=0
1xO=O
1 x 1=1

Examples are:
Multiplying 111101 with 1110:
Binary. Decimal
111101 61
x 1110 x14
000000 244
111101x 61x
111101xx 854
1 11101 x xx
110101011 0
Multiply 1 10 with 010 Binary:
110
x 010

000
110x
OOOxx

01100
4.Division:

The rules for division are same in binary system as those in decimal
number system.

110
110 )1OO 100
-110
1 1O
-110

This is same as division of decimal 36 with 6.Fixed And floating Number

Representation:

Position of the decimal point in a mixed number can be represented in two

ways. Fixed point represented and the floating point Representation. Fixed

Point Representation:

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The decimal is fixed point Representation. For Example: 8.9,28.015,0.52

Floating Point" Representation: Floating point representation has the three

components. The mantissa, radix or base and the exponent. Mantissa and the

exponents are stored in the memory location but not the Radix which is 2 for

binary. To indicate whether a number is positive or negative sign bit 0 or 1 is

used respectively.

Coding scheme:

A single binary digit is called a "bit". The number of bits in the data

sequence processed by a given computer is called its world size coding is a

process of representing numeric or non-numeric data and instructions in

binary. An 8-bit microprocessor can retrieve, process, store and transmit data

or instructions in 8-bit or one byte word. Similarly the word size of a 16-bit

microprocessor is two bytes long.

5.7 CODES USED IN COMPUTER:

Digital computers use binary number for performing arithmetic

operations on data. Computers read input number data and print answers in

decimal number different types of codes are used to represent numerical data,

one of which is binary-coded decimal (BCD).

 Binary-Coded Decimal (BCD):

In BCD code, each decimal digit is represented by its four-bit binary

equivalent instead of 'converting the entire decimal number into its pure

binary equivalent. For example the BCD code Tora decimal number 896 will be

as follows.

Decimal 8 96

BCD 1000 1001 0110

Alphanumeric Codes:

Codes such as EBCIDIC, ASCII, ANSI or UNICODE have been developed when the

data contains letters and symbols in addition to numbers Modem computers

use 8-bit codes, which can accommodate 36 alphanumeric characters and 220

special characters.

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In this way computers can recognize 28 = 256 different characters.

There are two most popular coding schemes developed for this purpose.

i) ASCII Code:

ASCII code is a 7-bit code used to handle alphanumeric data having 128

different characters. It stands for American Standard Code for Information

Interchange. It is the most popular coding system for PC's and data

communication. This code allows manufacturers to standardize input/output

devices such as keyboard, printer, visual display units etc. An extension of ASCII

code uses 8-bits called as ASClI-8 code with an extra eight bit as a parity bit

ii) EBCDIC Code:

EBCDIC (Extended Binary Coded Decimal Interchange Code) is an 8-bit

code and can provide 256 different characters. It is primarily used by

international Business Machine on IBM mainframes and other large computers

5.8 EXERCISE

Q No.1 what is Data?

Data:

The word Data is derived from Latin language. It is plural of Datum. (But

Data is usually used as a singular term) Data is the collection of raw facts and

figure, therefore, we need symbols for their representation. These symbols

may be letters, words or figures such as 3 books, 6 students, 20 March, 100

watts, Pk-429 etc. Even pictures, photographs, drawings, charts and maps can

be treated as Data. In computers, data can be classified into the following two

types:

(1) Numeric Data (2) Character Data

Q No.2 Define various types of Data?

Ans. Types of Data:

In computers data items can be classified in to the following two

types:

1. Numeric Data 2. Character Data

1 Numeric Data:·

Numeric Data contains discrete numbers only such as

20,345, 8.25, -45, -16.2 etc.

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Numeric data can be classified into two types:

i. Integer Data ii. Real Data

(i) Integer Data:

Integer Data consists of positive or negative whole numbers including

zero.

For example: +5, +8, -32 are integers.

(ii) Real Data:

Real data contains numbers which may be fractions or incremental

including integer.

For example: 2/7, 13.4, 0.14 etc. are representatives of real numbers.

Real data is further divided into two types.

(a) Fixed-point data (b) Floating-point data

(a) Fixed Point Data:

Fixed point data may include digits (0-9), decimal point, and +/-. sign. For
example, percentage of marks, weight etc.
(-42.002,0.05, + 426.8 etc).

(b) Floating Point Data:

Floating point data may include digits (0-9), decimal point, +/- sign and

letters "D", "d", "E", "e" etc. For example, speed of light, mass of
atomic particle etc. (1.602 x .10-19 (charge of electron in coulomb)

2. Character Data:

Character data falls into two groups.

i) String Data ii) Graphical Data

i) String Data:

String data consists of the sequence of characters. Characters may be
English alphabets, numbers or spaces the space, which separates two words, is

also a character. The string data is further divided into two types:
a) Alphanumeric Data b) Alphabetic Data

(a) Alpha-Numeric Data:

Alphanumeric data contains a combination of numerals and letters of

alphabets including special characters such as #, %, ?, *, + etc. For example:

1432, B/29. PK102, B-14 Block# 1 etc

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(b) Alphabetic Data:

Alphabetic data includes all the. uppercase and lowercase letters of
alphabets and combination such as A, B. c. Z, a, b, c, .. z. MCB, AASHIR,
PAKISTAN etc.

(iii) Graphical Data:

It is possible that pictures, charts and maps can be treated as data. The

scanner is 0 normally used to enter this type of data. The common use of

Graphical data is found in the National Identity card. The photographs and

thumb impression are scanned 'and in the computer to identity a person.

Q No.3 Differentiate between Data and Information.

Data Information

1) UN- processed information is 1)Processed form of data is called

called Data. information.

2) Raw facts and figures are called 2) Organized and meaningful data

data. is called information.

3) We can not take decision on 3) We can take a decision on

data. information.

4) Example, Each student is test 4) The class average score or the

score is one piece of Data. school's average score is the.

information that can be concluded

from the give data.

5} Data is the computer's language. 5) Information is our translation of

this language.

Q No.4 What is a Number System?

Ans. Number System:-· since early times different number system

have been used number system, we mean using a base for

counting.

Digital computers are the machines which respond numbers. All

the 'data and instructions .must therefore be represented in a numeral format.

The binary number system has been found to be the most natural and efficient

system for modem digital machine. The number of digits a system uses is called

its base or radix. There are four types of number systems used in computer

operations.

i. Decimal (Base 10) ii. Binary (Base 2)

iii. Octal (Base 8) iv. Hexadecimal (Base 16)

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Q No.5 How many types of number systems do you know?
There are four types of number system used in computer operations.

(i) Decimal (ii) Binary octal
(Iii) Octal (iv) Hexadecimal

1. Decimal number system (Deci means 10):

The decimal number system consists of 10 digits (from 0 to 9). We use

this number system in our daily life for counting and calculations.

2. Binary Number System (Bi means 2):

In binary number system any number can be expressed by the digits 0

and I only. The binary system is ideal for internal working of electronic

computers.

3. Octal Number System (Oct means Eight):

The Octal number system consists Of 8 digits (0, 1, 2....7). In this

number system; the base is 8. Each digit position in octal number system

represents a power of eight.

4. Hexa-Decimal Number System (Hexa means 6 and Decimal
means 10):

The Hexa-Decimal number system consists of 16 digits (0, 1,2 3 ..... 9,

A, B, C, D, E, F). The alphabets A, B, C, D, E and F are used to represent decimal

numbers 10, 11,12,13; 14 and 15 respectively. In this number system, the base

is 16.

Q No.6 Which number system computer uses for processing of data
Ans. and why?
Digital computers are the ma hi es hi h " espo d u e s
All the data and instructions must therefore be represented in a
numeral format. Thus number systems are very important to
understand because a computer understands numbers only. The
binary number system has been found to be the most natural
and efficient system for modem digital machines.

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Q No.7 how many types of "codes" are used in computer system?
Ans. Digital computers use binary number for performing arithmetic
operations on data. Computers read input number data and
print the answers in decimal number different types of codes are
used to represent numerical data. The codes include Binary
codes, Binary- coded-decimal (BCD) and alphanumeric codes.

Alphanumeric codes include ASCII and EBCDIC.

Q No.8 Define the various coding schemes used in the computer

system?

Ans. Codes Used In Computer:

Digital computers use binary number for performing arithmetic

operations on data. Computers read input number data and

print answers in decimal number different types of codes are

used to represent numerical data, one of which is binary-coded

decimal (BCD).

(1) Binary-Coded Decimal (BCD):

In BCD code, each decimal digital is represented by its four-bit

binary equivalent instead of converting the entire decimal number into its

pure binary equivalent. For example the BCD code for a decimal number 896

will be as follows:

Decimal 8 96
1001 0110
BCD 1000

(2) Alphanumeric Codes:

Codes such as EBCIDIC, ASCII, ANSI or UNICODE have been developed
when the data contains letters and symbols in addition to numbers. Modem
computers use 8-bit codes, which can accommodate 36 alphanumeric
characters and 220 special characters. In this way computers can recognize
28 = 256 different characters. There are two most popular coding schemes

developed for this purpose.

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TYPES OF ALPHANUMERIC CODES:
(i) ASCII Code:

ASCII code is a 7-bit code used to handle alphanumeric data having 128
different characters. It stands for American standard code for information
i te ha ge. It is the ost popula odi g s ste fo PC s a d data
o u i atio . This ode allo s a ufa tu e s to standardize input/output
devices such as keyboard, printer, visual display units etc. An extension of
ASCII code uses 8-bits called as ASCII code with an extra eight bit as a parity
bit.

(ii) EBCDIC Code:

EBCDIC (Extended Binary Coded Decimal Interchange Code) is an 8-bit

code and can provide 256 different characters. It is primarily used by

International Business Machine on IBM mainframes and other large

computers.

Q No.9 How a floating-point number is represented in the computer?

Ans. Floating point representation has three components.

The mantissa, radix or base and the exponent. Floating-point

data may include. digits (0 - 9), decimal point, ( +, -, /)sign and

letters "D", "d", "E", OR "e". The data, which is in the

exponential form, can be represented in the floating-point

rotation.

For example: Speed of light, mass of atomic particles etc.
1.602 x 10-19

Above value can be feed into the-computer as:
1.602E - 19

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Q No.10 What is complement of a number? How 1's and 2's

complements are represented in computers?

Ans. Complement (invert) all the bits the result at this stage is known

as one's complement. In binary number system 2's complement

and 1 's complement of a number N is obtained by subtracting
the binary number N from 2n and 2" -1 respectively, where

exponent is the number of digits in the number N. To find its
complement of a binary u e , si pl ha ge all its s into O's

and O's into 1's. To find 2's complement of a binary number add

1 in its 1 's complement..A computer performs subtraction using

the principle-of additive subtraction or complementing.

Q No.11 Convert the following Decimal numbers into Binary.
(2014,2013,2012,2011,2010,2008)

1. (45) 2. (126)

2 45 2 126
2 22-1 2 63-0
2 11-0 2 31-1
2 5-1 2 15-1
2 2-1 2 7-1
2 3-1
1-0
(101101)2 Ans 1-1
(101101)2 Ans
3. (425)
2 425 2. (628)
2 212-1
2 106-0 2 628
2 53-0 2 314-0
2 26-1 2 157-0
2 13-0 2 78-1
2 6-1 2 39-0
2 3-0 2 19-1
1-1 2 9-1
(110101001)2 Ans 2 4-1
2 2-0

1-0
(1001110100)2 Ans

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5. (814) 6. (1089)
2 814 2 1089
2 407-0 2 544-1
2 203-1 2 272-0
2 101-1 2 136-0
2 50-1 2 68-0
2 25-0 2 34-0
2 12-1 2 17-0
2 6-0 2 8-1
2 3-0 2 4-0
1-1 2 2-0
(1100101110)2 Ans 1-0
(10001000001)2 Ans
7. (1706)
2 1706 8. (5138)
2 853-0
2 426-1 2 5138
2 213-0 2 2569-0
2 106-1 2 1284-1
2 53-0 2 642-0
2 26-1 2 321-0
2 13-0 2 160-1
2 6-1 2 80-0
2 3-0 2 40-01
1-1 2 20-0
(1100101110)2 Ans 2 10-0
2 5-0
2 2-1

1-0
(10100000010010)2 Ans

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9. (3578) 10. (4653)

2 3578 2 4653
2 1789-0 2 2326-1
2 894-1 2 1163-0
2 447-0 2 581-1
2 223-1 2 290-1
2 111-1 2 145-0
2 55-1 2 72-1
2 27-1 2 36-0
2 13-1 2 18-0
2 6-1 2 9-0
2 3-0 2 4-1
2 2-0
1-1 2 1-0
2 2-1
(110111111010)2 Ans
1-0
(1001000101101)2 Ans

Q No. 12 Convert the following Binary numbers into Decimal.
(2014,2013,2012,2011,2010,2009,2008)

(i) 101 (ii) 1101
=> 1 X22 + Ox21 + 1x2° => 1x23 + 1x22+ Ox21 + 1x2°
=> 1 x8 + 1 x4 +0x2 +1 x 1
=> 1x4+0x2 + 1x1
=>8+4+0+1
=>(5)1O Ans. => (13)10 Ans

(iii) 111 00 (iv) 11001
=> I X24 + Ix23 + Ix22 + Ox21 + Ox2° => lx24+ lx23 + 0x22 + Ox21 + lx20
=>lxl6 + lx8 + lx4 + 0x2 + 0xl => lx16 + lx8 + 0x4 + Ox2 + l x l
=>6+8+4+0+0 =>16+8+0+0+1
=> 1(28)10 Ans. => (25)10 Ans.

(v) 100011 (vi) 100111
=> 1x25+Ox24+0X23+Ox21+ 14xll+ lx2°
=> 1x32 + Oxl6 + 0) (,8 + Ox4 + lx2 + l x l => 1x25 + 0x24 + 0x23 + 1 X22 + 1 x21 + lx2°
=> 32 + 0 + 0 +0 + 2 + 1
=> (35)10 Ans. => lx32+0xl6+0x8+ lx4+ lx2+ l x l
=> 32 + 0 + 0+ 4 + 2'+ I
=> 1(39)10 Ans

(vii) 1001010
=> 1 x26 + 0x25 + 0x24 + 1 x23 + 0x22 + 1 x2' + 0x2°

=> lx64 +0x32 + 0x16 + lx8 + 0x4 + Ix2 + 0xl

=>64 + 0 + 0 + 8 + a + 2 + 0

=> 1(74)10 Ans.

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(viii) 1111000
=> lx26 + lx25 + 1x24 + lx23 + 0x22 + 0x2' + 0x2°
=> lx64 + lx32 +lx16 + lx8 + 0x4 + 0x2 + 0xl

=> 64 + 32 + 16+ 8+ 0+ 0 + a
=> (120)10 Ans

(ix) 10001100
=> 1x27 + 0x26 + 0x25 + 0x24 + 0x2) + lx22 + 0x2' + 0x2°
=> lx128 + 0x64 + 0x32 + 0x16 + lx8 + lx4 + 0x2 +0xl
=> 128 + 0 + 0 + 0 + 8 + 4 + 0 + a
=> (140)10 Ans

(x) 1001110011
=> 1 x29 + 0x28 + 0x27 + lx26 + 1x25 + lx24 + 0x23 + 0x22 +lx21 + lx2°

=> 1x512 + 0x256 + 0xl28 + Ix64 + lx32 + lx16 + 0x8 + 0x4 + lx2 + 1x1
=> 512 + 0 + 0 + 64 + 32 + 16 + 0 + 0 + 2 + 1
=> (627)10 Ans

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Q No.13 Convert the following Decimal numbers into equivalent octal
numbers.

(i) 482 (ii) 232 232 (iii) 750 750
8 29-5 93-6
8 482 8 3-5 8 11-5
8 60-2 8 1-3
8
7-4

(742)8 Ans. (350) 8 Ans. (1356) 8 Ans.

(iv) 1200 (v) 854 (vi) 2560
8 1200 8 854
8 150-0 8 106-6 8 2560
8 18-6 8 13-2 8 320-0
2-2 8 40-0
1-5
(2260)8 Ans. 5-0
(1526) 8 Ans. (5000) 8 Ans.
(vii) 1802
8 1802 (viii) 9543 (ix) 8086
8 225-2 8 9543
8 28-1 8 1192-7 8 8086
3-4 8 194-0 8 1010-6
8 18-5 8 12-2
(3412)8 Ans. 8 15-6
2-2
(x) 11456 1-7
(22507) 8 Ans. (17626) 8 Ans.
8 1146
8 1432-0
8 179-0
8 22-3

2-6
(26300)8 Ans.

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Q No.14 Convert the following octal numbers into equivalent decimal
numbers.

(i) 65 (ii) 132 (iii) 655

=> 6x81 + 5x8° => lx82+ 3x81 +2x8° => 6x82 + 5xSI + 5x8°

=> 6x8+ 5xl => lx64 + 3x8 + 2xl => 6x64 + 5x8 + 5x I

=> 48 + 5 => 64 + 24 + 2 => 384 + 40 + 5
=> (53)10 Ans. => (90)10 Ans. => (429)10 Ans.

(iv)763 (v) 454 (vi)4550
=> 7x82 + 6x81 + 3x80 => 4x82 + 5x81 + 4x80 => 4x83 +5x82 +5x81 +0x80
=> 7x64 + 6x8 + 3x1 => 4x64 + 5x8 + 4x1 => 4x512 +5x64+5x8+0x1
=> 448 + 48 +3 => 256 + 40 +4 => 4096+320+40+0
=> (499)10 Ans. => (300)10 Ans. => (4456)10 Ans.

(vii) 1006 (viii) 7540
=> 1x83 + 0x82 + 0x80 + 6x80 => 7x83 + 5x82 + 4x81 +0x80
=> 1x512 + 0x64 + 0x8 + 6x1 => 7x512 + 5x64 + 4x8 + 0x1
=> 512 + 0 + 0+ 6 => 3584 + 320 + 32 + 0
=> (518)10 Ans. => (3936)10 Ans.

(ix) 4463 (x) 10654
=> 4x83 + 4x82 + 6x81 + 3x80 => 1x84 + 0x83 + 6x82 + 5x81 + 4x80
=> 4x512 + 4x65 + 6x8 + 3x1
=> 2048 + 256 + 48+ 3 => 1x4096 + 0x512 + 6x64 + 5x8 + 4x1
=> (2355)10 Ans.
=> 4096 + 0 + 384 + 40 + 4
=> (4524)10 Ans.

Q No.15 Convert the following Decimal numbers into Hexadecimal numbers

systems.

(i) 425 (ii) 1780 (iii) 3250

16 425 16 232 16 3250
16 26-9 16 1779-4 16 203-2
16 111-0
1-10 12-11
6-15

(1A9)16 Ans. (6F04)16 Ans. (CB2)16 Ans.

(iv) 11809 (v) 22583 (vi) 55887
16 11809 16 22583 16 55887
16 738-1 16 1411-7 16 3492-15
16 46-2 16 88-3 16 218-4
2-14
5-8 13-10
(2e21)16 Ans. (DA4F) 16 Ans.
(5837)16 Ans.

Page 171

IX- Computer Chapter # 5 Page # 172

(vii)90082 (viii)12443 (ix) 329600
16 90082 16 12443 16 329600
16 5630-2 16 777-11 16 20600-0
16 351-14 16 48-9 16 1287-8
16 21-15 3-0 16 80-7
1-5 5-0
(309B)16Ans.
(15FE2)16Ans. (50780)16Ans.

16 200455

16 12528-7
16 783-0
16 48-15

3-0

Q No.16 Convert the following Hexadecimal numbers into equivalent

decimal numbers. '(2012)

(i) 1420 (ii) 2210
=> Ix163+ 4x 162+ 2x 161+ Oxl6u => 2xI63+2xI62+ IxI6!+OxI6°

=> Ix4096 + 4x256 + 2x16 + 0 + 1 => 2x4096+2x256+ lxl6+0xl

=> 4096 + 1024+ 32 + 0 => 8192+512+ 16+0

=> (5152)10 Ans. => (8720)1O Ans

(iii) AI09 (iv) 5D60
=> IOx163+ lx162+ Ox161 + 9x16° => 5x163+ 13xI62+6x16i+OxI6°
=> IOx4096 + 1 x256 + Ox 16 + 9x 1 => 5x4096 + 13x256 + 6x 16 + Ox 1
=> 40960 + 256 + 0+ 9 => 20475 + 3328 + 96 + 0
=> (41225)10 Ans. => (23899)10 Ans.

(v) 60805 (vi) CD550
=> 12x164+
=>6x164+Ox163+8x162+ Oxl6i + 5x16° 13xI63+5xI62+5x16i+OxI6°

=>6x65536 +' Ox4096 + 8x256 + => 12x65536 + 13x4096 + 5x256 + 5x r6 + Ox 1
Oxl6 + 5xl =>393216+0+2048+0+5
=> 786432 + 53248+ 1280 + 80 + 0
=> (395269)lOAns.1
=> (841040)1O Ans.

(vii) ABCD9 (viii) 126A2
=>10x164 + l1x163 + 12x162 + 13x161 + 9x16° => lx164+ 2x163+ 6x162+ 10x161 + 2x16°
=>IOx65536 + 11x4096 + 12x256 + =>lx65536 + 2x4096 + 6x256 + 1 Ox16 + 2x
13)(16 + 9x 1 1
=>655360 + 45056 + 3072 + 208 + 9 => 65536 + 8192 +1536 + 160 + 2
=> (703705)10 Ans. => (75426)10 Ans;

(ix) 6FB9E (x)5C464F
=>6x164+ 15x163+ llxI62+9x.16i+ 15x16° =>5x165+ 12x164+4xI63+6xI62+4xI6i+ 15x16°
=>6x65536+ 15x4096+ l1x256+9x16+ 15xl =>5~1048576 + 12x65536 4x4096 + 6x256 + 4x16 + 15xl
=>393216 + 61440+ 2816 + 144 + 15 => 5242880 + 786432 + 16384 + 1536 + 64 + 15
=>(457631)10 Ans => (604 7311)10 Ans·

Page 172

IX- Computer Chapter # 5 Page # 173

Q:17: Convert the following Hexa-Decimal numbers into equivalent Binary

numbers.

(2014)

i) A35 ii) 8066

=> =>

=>(101000110101)2Ans =>(1000000001100110)2Ans

iii) 5ODC iv) 15B8
=> =>
(0101000011011100)2Ans
(0001010110111000)2Ans

(v) 44F6E (vi) 22EE
=> =>
(01000100111101101110)2Ans
(0010001011101110)2Ans

(vii) 80A5C (viii) 33C5D
=> =>
(10000000101001011101) 2Ans (00110011110001011101)2Ans

(ix) 78AA0 (x) ADF64
=> =>
(01111000101010100000)2Ans (10101101111001100100)2Ans

Q.18: Convert the-following octal numbers into Binary numbers.

(i) 453
Octal
Binary
(100101011)2Ans.

(ii) 126
Octal
Binary
(001010110)2Ans.

(iii) 165
Octal
Binary
(001110101)2Ans.

Page 173

IX- Computer Chapter # 5 Page # 174

(iv) 2274
Octal
Binary
(010010111100)2Ans

(v) 11 04
Octal
Binary
(1001000100)2Ans

(vi) 3420
Octal
Binary
(011100010000)2Ans

(vii) 4060
Octal
Binary
(100000110000)2Ans

(viii) 2647
Octal
Binary
(010110100111)2Ans

(ix) 231560
Octal
Binary
(011011001101110000)2Ans

(x) 147235
Octal
Binary
(001100111010011101)2Ans

Q.19: Convert the following Binary numbers into Octal numbers.
(i) 1110
Binary
Octal
(16)8Ans

Page 174

IX- Computer Chapter # 5 Page # 175

(ii)101010 Page 175

Binary
Octal
(52)8Ans

(iii)1001001

Binary

Octal

(111)8 Ans

(iv)11110011
Binary
Octal
(363)8Ans

(v)100101010
Binary
Octal
(452)8Ans

(vi)101001
Binary
Octal
(51)8Ans

(vii)1000110
Binary
Octal
(106)8Ans

(viii)1111000
Binary
Octal
(170)8Ans.

(ix)10000111
Binary
Octal
(207)8Ans.

IX- Computer Chapter # 5 Page # 176

(x)101011110
Binary
Octal
(536)8 Ans

Q.20: Convert the following Binary numbers into Hexa-Decimal numbers.
(i) 11011

Binary
, Hexa-Decimal
(1B)16 Ans

(ii) 1110100

Binary
Hexa-Decimal
(74)16 Ans

(iii) 1101101
Binary

, Hexa-Decimal
(6D)16Ans

(iv) 11000111
Binary

, Hexa-Decimal
(c7)16Ans

(v) 1001011110
Binary

, Hexa-Decimal
(25E)16Ans

(vi) 110011
Binary

, Hexa-Decimal
(33)16Ans

Page 176

IX- Computer Chapter # 5 Page # 177

(vii) 1010110
Binary

, Hexa-Decimal
(56)16Ans

(viii) 1011001
Binary

, Hexa-Decimal
(59)16Ans

(ix) 11110111
Binary

, Hexa-Decimal
(F7)16Ans

(x) 101011110011
Binary

Hexa-Decimal
(AF3)16Ans

Q.21: Convert the following Octal number into Hexa-Decimal numbers, (2014)
Formula:
Octal_____________Binary______ ______Hexa-Decimal

(i) 117
Octal
Binary

Binary
, Hexa-Decimal
(04F)16Ans

(ii) 205
Octal
Binary

Binary
, Hexa-Decimal
(085)16Ans

Page 177

IX- Computer Chapter # 5 Page # 178

(iii) 4235 Page 178
Octal
Binary

Binary
, Hexa-Decimal
(89D)16Ans

(iv) 66442
Octal
Binary

Binary
Hexa-Decimal
(6D22)16Ans

(v) 112534
Octal
Binary
Binary

, Hexa-Decimal
(0955C)16Ans

(vi) 1035
Octal
Binary

Binary
, Hexa-Decimal
(21D)16Ans

(vii) 3344
Octal
Binary

Binary
Hexa-Decimal
(6E4)16Ans

IX- Computer Chapter # 5 Page # 179

(viii) 20062
Octal
Binary

Binary
, Hexa-Decimal
(2032)16Ans

(ix) 55100
Octal
Binary

Binary
, Hexa-Decimal
(5A40)16Ans

(x) 2670123
Octal
Binary

Binary
, Hexa-Decimal
(0B7053)16Ans

Q.22: Convert the following Hexa-Decimal number into Octal-Decimal
numbers. Formula:
Octal_____________Binary______ ______Hexa-Decimal

(i) 4A2
Hexa Decimal
Binary

Binary

, Octal-Decimal
(2212)8Ans

Page 179

IX- Computer Chapter # 5 Page # 180

(ii) 124 Page 180
Hexa Decimal
Binary
Binary

Octal-Decimal
(011102)8Ans

(iii) 60FC
Hexa Decimal
Binary

Binary
Octal-Decimal
(060374)8Ans

(iv) 7350
Hexa Decimal
Binary
Binary

, Octal-Decimal
(071520)8Ans

(v) AA100
Hexa Decimal
Binary

Binary
Octal-Decimal
(2520400)8Ans

(vi) 806C
Hexa Decimal
Binary
Binary

Octal-Decimal
(100154)8Ans

IX- Computer Chapter # 5 Page # 181

(vii) 44CD Page 181
Hexa Decimal
Binary

Binary
Octal-Decimal
(42315)8Ans

(viii) EE600
Hexa Decimal
Binary

Binary
Octal-Decimal
(3563000)8Ans

(ix) 556BB
Hexa Decimal
Binary

Binary
Octal-Decimal
(053273)8Ans

(x) 698DB2
Hexa Decimal
Binary

Binary
Octal-Decimal
(32306662)8Ans

IX- Computer Chapter # 5 Page # 182

Q.23: Determine the s complement of each binary numbers. (2008)

(i) 1010

Binary number 1010

=> 1's complement 0 1 0 1

(ii) 110

Binary number 1100

=> 1's complement 0 0 1 1

(iii) 10111

Binary number 10111

=> 1's complement 0 1 0 0 0

(vi) 100011

Binary number 100011

=> 1's complement 0 1 1 1 0 0

(v) 00111010

Binary number 0011101

0

=> 1's complement 1 1 0 0 0 1 0

1

Q.24: Determine the 2's complement of each binary number. (2014, 2012,
2011, 2008)

(i) 1110

Binary number 1110

1's complement 0 0 0 1

Add(1) + 1

s o ple e t 0 0 1 0

(ii) 1000

Binary number 1000

1's complement 0 1 1 1

Add(1) + 1

s o ple e t 1 0 0 0

Page 182

IX- Computer Chapter # 5 Page # 183

(iii) 01110

Binary number 01110

1's complement 1 0 0 0 1

Add(1) + 1

s o ple e t 1 0 0 1 0

(vi) 1100011

Binary number 1100011

1's complement 0 0 1 1 1 0 0

Add(1) + 1

s o ple e t 0 0 1 1 1 0 1

(v) 011000101

Binary number 0 1 1 0 0 0 1 0 1

1's complement 1 0 0 1 1 1 0 1 0

Add(1) + 1

s o ple e t 1 0 0 1 1 1 0 1 1

Q.25: Express each decimal number as a 8-bit number. in the l's
complement system.

(i)-24 00011000
8-bits representation of (+24) 11100111Ans.
1'S compliment of (+24)

(ii)+64

8-bits representation of (+67) 01000011

The number is already +ve therefore, its 1's complement remains the same.

01000011Ans.

(iii)-88 01011000
8-bits representation of (+88) 10100111Ans.
1'S compliment of (+88)

Page 183

IX- Computer Chapter # 5 Page # 184

(iv)+112
8-bits representation of (+112) 01110000
The number is already +ve therefore, its 1's complement remains the same.

01110000 Ans.

(v)+225 11100001
8-bits representation of (+225) 00011110Ans.
1'S compliment of (+225)

Q.26: Express each decimal number as a 8-bit number in the 2's complement
system.

(i) + 15

8-bit representation of (+ I 5) 00001111

The number is already +ve therefore, its 2's Complements remain the same.

00001111 Ans.

(ii) -56

8-bit representation of (+56) 00111000

l's complement of (+56) 11000111

add 1 +1

2's complement of (+56) 11001000

(iii) + 103
8-blt rep. of(+103) 0 1 100 111
The number is already +ve therefore, its 2's complements remain the same.

101100111 Ans.1

(iv) -145 100 1 000 1
8-bit representation of (+ 145) 01101110
1' s complement of (+ 145) +1
add 1 01101111
2's complement of (+ 145)

(v) +160
8-bit representation of (+ 160) 10100000
The number is already +ve therefore, its 2's Completes remain the same:

110100000 Ans.1

Page 184

IX- Computer Chapter # 5 Page # 185

Q.27: Add the following Binary numbers. (2014, 2011, 2010~ 2009, 2008)

(i) 101 + 011 (i) 101 +0ll 0 1
1001 1 1 1
0 0
+1010 +0
10011 10

(iii) 101011 + 110011
101011

+ 10011
1011110

(iv) 111100 + 10110
111100

+ 10110
1010010

(v) 10001111 + 110010
10001111
+ 110010
11000001

Q.28: Perform subtraction on the following numbers; (2012, 2011, 2010,
2009,2008)

(i) 111 – 100 1 (ii) 1101 - 1011
11 0 1101
1
- 10 - 1011
01 0010

(iii) 111001 – 10111
111001

- 10111
100010

(iv) 101100 - 11110
101100
- 11110

01110

Page 185

IX- Computer Chapter # 5 Page # 186

(v) 10011 01 – 110110
1001101

- 110110
010111

Q.29: Multiply the following Binary numbers. (2014, 2013, 2012, 2011,
2010, 2009, 2008)

(i)1110xl0 (ii) 1001 x 101

1110 1001

X 10 x101

0000 1001

1110x0000x

111001001xx

101101

(iii) 1010011 x 1001
1010011
X 1001
1010011
0000000x
0000000xx
1010011xxx
1011101011

(iv) 111001 x 111
111001
X 111
111001
111001x

111001xx
110001111

(v) 11 011010 x 100
11011010
X 100
00000000
00000000x
11011010xx
1101101000

Page 186

IX- Computer Chapter # 5 Page # 187

Q.30: Divide the binary number as indicated? (2014, 2013, 2012, 2010, 2008)

(i)1100÷10 110 (ii) 1001÷ 11 (iii) 110011 ÷ 100
√ √ 11 1100.11
√
10 100
10 11 100
10 011 100
00 11 110
00

(iv) 11001÷101 (iv) 1011110÷1001
1010 1010

√ √
101 1001

0101 1011

101 1001

00 100

Q 31: Convert each of the following decimal numbers to 8421 BCD.

(i) 15 (iii) 25
Solution: Solution:
15 25

0001 0101 0010 0101

(ii) 38 (iv) 74 (v) 197
Solution: Solution: Solution:
38 74 19
7

0011 1000 0111 0100 0001 1001 0111

Q 32 Convert each of the BCD numbers to Decimal.

(i) 10 0 0 0 0 (ii) 0 0 0 1 0 0 0 1 1 0 0 1 (iii) 1 0 1 0 1 1 1 1 0 0
00
20 1 19 10 15 0
000100011001=119 101011110000=10150
100000=20

(iii) 0 1 0 0 0 0 1 1 1 1 0 0 1 0 1 0 1 (iv) 1 0 1 1 1 0 1 0 1 1 1 1

4 79 5 11 10 15

Page 187

IX- Computer Chapter # 5 Page # 188

Q.33: Fill in the blanks.
(i) The data, which consists of alphabets as well as numbers, is known as

Alphanumeric Data.
(ii) The number, which is in the exponential form, is called Floating Point data.
(iii) The Information is a meaningful, useful and processed form of data.
(iv) 8 is a base of octal system.
(iv) The maximum digit of hexadecimal number system is 16.
(v) ASCII stands for American Standard Code of Information Interchange
(vi) A floating-point number consists of two parts known as Mantissa and

exponent.
(vii) The method of 2's complement arithmetic is commonly used in computers to

handle Negative numbers.
(viii) In BCD code each decimal digit is represented by a binary code of four bits.
(ix) Computers can recognize a total of 256 different characters.

Q.34: Tick the correct answer.

(i) The data, which consists of whole number, is known as: (2014)

(a) Real (b) integer (c) fixed-point (d) string

(ii) . The number which has a decimal point, is:

(a) integer (b) character (c) fixed point (d)String

(iii) The number, which is in the exponential form, is:

(a) real (b) fixed-point (c) floating point (d) integer

(iv) The data which can be a picture, drawing, map is:

(a) alphabetic (b) alphanumeric (c) graphical (d) string

(v) The processed organized from of data is known as: (2014)

(a) string (b) information (c) graphics (d) binary

(vi) How many types of number systems are used in computers?

(a) 2 (b) 3 (c) 41 (d) 5

(vii)' Which number system is ideal for the internal working of electronic

computers?

(a) binary (b) decimal (c) octal (d) hexadecimal

(viii) The base of hexadecimal number system is:

(a) 10 (b) 8 (c) 161 (d)2

(ix) The equivalent of decimal number lOin binary is:

(a) 1100 (b) 1010 (c) 1011 (d)10

(x) The complement of 100 I 10 is:

(a) 110011 (b) 100010 (c)011100 (d) 011001

Page 188

IX- Computer Chapter # 5 Page # 189

QUESTIONS FROM,PAST PAPERS:
YEAR 2014:

Section "A" (Multiple Choice Questions) - (MCQs)
(ii) Organized form of data is called:
* DATUM * Information *System * None of these

(iv) (456)10 = (?)g:
* 710 * 5EB * 3A9 * None of these

(vii) 'The data consisting of whole numbers is known as:
* Real * Integer * Fixed point *String

(xi) Organization of computer depends on:
* Hardware * Software * Number system * None of these

Section "B" (Short -Answer Questions)

Q.5 Solve the following:

·(i)- 111001x111 (ii) 111001+111001

(iii) 110110 -7- 100 (iv) 11110(2's complement)

Q.12 Differentiate between Data and Information,

Q.13 Convert the following:
(i) (555)10 = (?)2 (ii) (78BC)16 =(?)2
(iii)(745)8 = (?)16 (iv) (1110001)2= (?)1O

Section "C" (Detailed-Answer Questions) (No Questions)
YEAR 2013:

Section "A" (Multiple Choice Questions) - (MCQs)

(iii) The number which has no decimal point is called:

* Integer * Character * Fixed point * string

Section "8" (Short =Answer Questions)

Q.3 Define data. What is Data Processing Cycle? Explain Information.

Q.5 What are the functions of Numeric Key pad in keyboard? Explain.

Q.6 Convert the following:

(i) (1010)2 = (?)8 (ii) (E34)16 = (?)102
(iii) (777)8 = (7)10 (iv) (333)10 = (?)2
Q.7 Convert the following:

(i) (1001101) - (110 I 10) (ii) (1010011) x (100 I)

(iii)(110011) / (100) . (iv) (01011) + 100111)

Page 189

IX- Computer Chapter # 5 Page # 190

Section "C" (Detailed-Answer Questions) (No Questions)
YEAR 2012:

Section "A" (Multiple Choice Questions) - (MCQs)
(No Questions)

Section "B" (Short -Answer Questions)

Q.12 Solve the following:

(i)ABC5)16=(------------)10 (ii) (11101)2=(-------)10

(iii) (337)8 = (---------)2 (iv) (555)10 = (-------)2

Q.13 Solve the following: (ii) 11102x1112 2
(i) 1001102 ÷102 s Co ple e t of
(iii) 1000012 -111112 i

Section "C" (Detailed-Answer Questions). {No Questions)
YEAR 2011:

Section ."A" (Multiple Choice Questions) - (MCQs)

(vi) The number which has no decimal point is:
* Integer *Character * fixed point * string

Section "B" (Short -Answer Questions)

Q.9 Convert the following:

(i)(8010)10= (----------)2 (ii)(111001)2=2s Complement

(iii) (11110101)2=(-------)10 (iv)(5671)8= (---------)10

Q .11 Solve the following 19 Binary numbers:

(i) 111011+101011 (ii) 110110 +1110000

(iii)11011 -10011 (iv)110111 x 11

Section "C" (Detailed-Answer Questions)
Q.16(a) What IS a number system? Describe different types of number systems:

YEAR 2010:
Section" A" (Multiple Choice Questions) - (MCQs)
(xi) The data which consist of whole numbers is known as:
*fixed point * floating point * real * integer

Section ".8" (Short -Answer Questions)
Q.2(vii) Solve the following numbers:

(a) (754)8 = (?)10 (b) (487)l0 = (?)8
(c)(1010001)2= (?)10 (d) (201)10 = (?)2

Page 190

IX- Computer Chapter # 5 Page # 191

(viii) Solve the following binary numbers:

(a)(001101)2 + (110l011)2 (b) (010101)2-(101011)2

(c) (0011)2 x (lOl)2 (d) (lOO1)2 ÷ (ll)2

Section "C" (Detailed-Answer Questions)
Q.4 (a) What is meant by Data? Define different types of data.

YEAR 2009:'

Section "A" (Multiple Choice Questions) - (MCQs)

(9) The data which consist of alphabet as well as numbers is known

as__________.

* Numeric * Alphanumeric * Alphabet * None of the above

(10) The number is the exponential form is called________.

*Floating point. * Fixed point * Mid point .* None of the above

(11) The___________is meaningful, useful and the processed form of data.

* Alphanumeric * Numeric * Alphabet * None of the above

(13) In ________code each decimal digit is represented by a binary code of four

bits.

* DCB *LCD * BCD * None of the above.

(14) Computer can recognize a total of ___________different characters.

*275 *255 *235 *256

Section "B" (Short -Answer Questions)

Q.2(viii) Solve the following:

(a)1100112 = (?)10 (b) 823610 = (?)8
(c) 3528 = (?)2 (d) 2900110= (?)16

(ix) Solve the following:

(a)1100112 + 10110112 (b) 1011012 – 1110012
(c)10012x 1112 d s o ple e t of
2

(x) Define data and briefly state its kinds.

Section "C" (Detailed-Answer Questions) (No Questions)

MULTIPLE CHOICE QUESTIONS (MCQs)

(1) The Binary number system uses ________and_______digits.

(a) 0,1 (b) 2, 4 (c) Even, odd (d)

None of the above

(2) The base of Hexa decimal number system is______.

(a) 15 (b) 16 (c) 2 (d) 8

(3) Greatest digit of octal number system is___________.

(a) 16 (b) 7 (c) 8 (d) 4

(4) Smallest digit of the octal number system is_______________.

(a) 1 (b) 2 (c) 0 (d) 7

Page 191

IX- Computer Chapter # 5 Page # 192

(5) The Base of binary number system is_______________.

(a) 2 (b) 16 (c) 8. (d) None of the above

(6) A graphical representation used to plan a program is

called__________.

(a) Software (b) Flowchart (c) Softcopy (d) System Flowchart

(7) Data is the. collection of_____________.

(a) Information (b) Software (c) Program (d) Facts and Figures

(8) The number system that has proved the most natural and efficient

system of machine use is:

(a) Decimal (b) Binary (c) Hexadecimal (d) Octal

(9) Which of the following code is used IBM compatible computer

system?

(a) ASCII (b) BCD. (c) EBCDIC (d) ANSI

(10) In which of the following code each decimal digit is represented by

its four bit binary equivalent?

(a) ASCII (b) EBCDIC (c) ANSI (d) BCD

(11) Hexadecimal number E is equal_________in decimal number

system.

(a) 15 (b) 10 (c) 14 (d) 12

(12) __________data consists of positive or negative whole numbers

including zero.

(a) Fixed point (b) Floating point (c) Integer (d) String.

(13) The range of Octal number system is____________.

(a) 0 to 9 (b) 0 to 10 .(c) O to 7 (d) 0 to 8

(14) The data which consists of whole numbers is known as

____________ (2010)

(a) Fixed point (b) Floating point (c) Real (d) Integer

(15) The data which consist of alphabet as well as numbers is known

as_______.

(a) Numeric (b) Alpha numeric (c) Alphabet (d) None of the above

(16) The number in the exponential form is called___________.

(a) Floating point (b) Fixed point (c) Mid point (d) None of

the above

(17) The _____________is meaning full, useful and the processed form

of data.

(a) Alpha numeric , (b) Numeric (c) Alphabet (d) Information

(18) The number which has no decimal point is: .(2011)

(a) Integer (b) Character (c) Fixed Point (d) String

(19) In____________code each digit is represented decimal by a binary

code of four bits. (2009)

(a) DCB (b) LeB (c) BCD (d) None of the above

(20) Organization of computer depends on _____________. (2014)

(a) Hardware (b) Software (c) Number System (d) None of these

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CHAPTER-6
BOOLEAN ALGEBRA

CLASS-IX

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BOOLEAN ALGEBRA

6.1 INTRODUCUION TO BOOLEAN ALGEBRA.
6.2 BOOLEAN ALGEBRA TERMS.
6.3 LAWS OF BOOLEAN ALGEBRA.
6.4 RULES OF BOOLEAN ALGEBRA.
6.5 DEMO‘GAM “ THEO‘M“.
6.6 SIMPLEFICATION IN BOOLEAN ALGEBRA.
6.7 STANDARD FORMS OF BOOLEAN ALGEBRA.
6.8 KARNAUGH MAP.
6.9 EXERCISE.
6.10 MULTIPLE CHOICE QUE“TION“ MCQ “

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6.1INTRODUCTION:
The English mathematicians felt there was a connection between mathematics
and logic but no one before George Boole could find this missing link in 1850s
he invented symbolic logic , Known today as Boolean Algebra. Each variable in
Boolean algebra has either of two values true or false. Is an action right or
wrong? A motive good or bad A conclusion true or false much of our thinking
involves trying to find the answer to two valued question the original purpose
of these two states algebra was to solve logic problems in 1938 Shannon
applied the e alge a to telepho e s it hi g i uits Be ause of “ha o s
work engineer soon realized that Boolean algebra could be used to analyze and
design computer circuits.

Logic Hardware:
The processing of data is done by lots of complex electrical circuits in
the CPU of the computer the hardware devices which do this are
transistors or integrated circuits.
1. Transistors:
These can function as amplifier or logic elements All types of transistors
are made from semiconductor material usually silicon which is carefully
doped with impurities e.g phosphorus and born so that electrical
current can be controlled
2. Integrated Circuits:

These are also known as chip and small circuits consisting of a
number of transistors Large Scale Integrated LSI circuit and very
large Scale integrated VLSI circuit represent thousand of transistors
on a chip Advancements in chip making technology have allowed
this miniaturization.
What is Digital Data?
Data which is represented in digits
Pertaining the data in the form of digits
Digital signals convey their information in the form of arrangement
of discrete digits
System output upon the inputs

Digital Signals:
 Variable signals are used by almost computer system.
 Represented by series of numeric values (i.e Binary)
 Discrete or discontinuous signal in nature.

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Logic:

 Sense of arguments.
 Logical arguments can have true of false only.
 May be represented in binary circuits.
 Can be evaluated through digital circuits.

Logic Gates:

 Two- states circuits (low or high).
 More puts but have only one output.
 Often called logic circuits.
 Gates are constructed through.

 Diodes

 Transistors
 Input and output relationship can be represented through truth

tables.

What is Gate?
The most common use of logic elements is to act as switches, although they
have no moving Parts. They open to pass on a pulse if electricity or close to
shut it off. This is why are known as gates.

There are three basic types of gates, i.e. AND,OR and NOT

6.2 BOOLEAN ALGEBRA TERMS:
1. Boolean Constants.
2. Boolean Variables.
3. Complement.
4. Truth Table.
5. Logical Operators (a) OR (b) AND (c) NOT
6. Operator prudence order.
7. Boolean Expressions.
8. Boolean Function.
1. Boolean Constants:

In Boolean algebra, a set of constants has only two elements 0 or 1.
They are stored in Boolean variables. Thus a Boolean constant is either 0
if not 1 of is 1 if not 0.

2. Boolean Variables:

The variables used in Boolean Algebra can be represented by the letters
o the alpha ets su h as A, B, C,……….X,Y,) et . Ea h a ia le ust take o e of
the two constants 1 or 0 at a time. These two values may be given different

names such as TRUE or FALSE YES or NO, HIGH or LOW, UP or DAWN, ON or

OFF etc.

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3. Complement
The complement is the inverse of a variable and is indicated by a bar

over the variable, For example, complement of the variable A is written as A .
EXAMPLES:

1. IF A = 1Then A = 0
2. IF A = 0 Then A=1
4. Truth Table:
A Truth Table is a table that shows the result of a Boolean
Expression for all the possible combination of the values given to the variable
used.

5. Logical Operators:

In Boolean algebra , there are three basic operators, one unary

operator NOT and two binary operators AND and OR as shown in the following

Table.

Operation Symbol used Comments

NOT P i e o a --) Negation of the value

(Unary operation)

OR Plus (+) or union ( ) Logical addition

AND Dot (.) or intersection ( ) Logical Multiplication

(a) NOT Operator:

Not operation is unary operation . It is the negation of a quantity

on which it operates. For example A (read as A Bar) means

complement of A or NOT A.

Truth Table for logical NOT ( --)operator

Input Output

AA

01

10

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Not Gate:
(b) AND Operator :
In Boolean algebra the operation AND means logical
multiplication and it is represented with or without a dot
between the variables. For Example A AND B gives the resulting
value of C in the equation
A.B= C o AB= C ead as A AND B is e ual to C

Truth Table for logical AND (.)operator

INPUT OUTPUT

A B A.B =C

000

010

100

111

Express as AB,A.B OR A B

Two Input AND Gate:

(c) OR Operator:
In Boolean Algebra OR operation means logical addition and it is
represented by Plus sign between two variables. For example observe that
C = 1 if A = 1 or B = 1 or if both are equal to 1. If both are equal to 0 then C =0
Truth Table for logical OR ( + ) operator

INPUT OUTPUT

AB A+B=C

00 0

01 1

10 1

11 1

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Two Input OR Gate:

6. Operator Precedence:

Boolean expression can be evaluated using the following precedence
that Boolean operators must follow:
1. Expression must be scanned from left to right.
2. Parentheses or Brackets are evaluated first.
3. NOT(Complement) operations are performed after parentheses.
4. AND(Multiplication) operations are performed next to NOT operation.
5. OR(Addition) operations are performed at the end.
7. Boolean Expressions:

A Boolean expression. is an arrangement of variables and logical operators
used to express the operation of a logic circuit.

OR
An expression is a logical statement which. is either True or False. These
statements are represented by variables, operated by logical operations.
For examples A + B = C
A + B + C D , A(B+CD)

8. Boolean Functions:
A Boolean function is an expression formed with variables, binary operators
(OR, AND unary operator NOT, 'Parentheses and equal sign). For a given
value of the variable, the value of the function can be either 0 or 1.
For example, the equations

W = X+Y.Z
The variable W is a function of X, Y and Z. This can be written as:

W = F (X, Y, Z)
The Boolean functions are represented as an algebraic expression. A
Boolean function may also be represented in the form of truth table.

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6.3 LAWS OF BOOLEAN ALGEBRA:

Laws of Boolean Algebra are used to define the structure of Boolean Algebra.

There are three basic laws of Boolean algebra. These are the same as in

ordinary algebra,

1. Commutative laws. . 2. Associative laws. 3. Distributive laws.

(1) COMMUTATIVE LAWS:
There are two commutative laws used in Boolean algebra (ordinary data
can be changed).
(a) Commutative Law of Addition:
This commutative law uses binary operator (+) sign as Boolean sum or
Boolean addition
For example:
A+B=B+A
(b) Commutative Law of Multiplication:
This commutative law uses (.) sign as Boolean multiplication.
For example:
A·B=B·A

(2) ASSOCIATIVE LAWS:
These are two Associative laws used in Boolean algebra (Parenthesis can be
changed),

(a) Associative Law of Addition:
The associative laws of addition is A + (B + C) = (A + B) + C
If there are three Elements numbers a, b c the sum if (a + b) + c will, be
equal to a + (b + c)

(b) Associative Law of Multiplication:

If there are three Elements (a, b, c)then the product (a. b).c will be same as

a·(b . c)

(3) DISTRIBUTIVE LAW OF MULTIPICATION OVER ADDITION:.

According to the distributive law: (Similar to multiplying bracket .in algebra)
+= +
(a) A. (B + C) = A . B + A . C { += +

And

(B). A + (B . C) = (A + B) . (A + C)

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