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## Tamilnadu Samacheer Kalvi 11th Computer Science Solutions Chapter 2 Number Systems

### Samacheer Kalvi 11th Computer Science Number Systems Text Book Back Questions and Answers

PART – 1

I. Choose The Correct Answer

Question 1.

Which refers to the number of bits processed by a computer’s CPU?

(a) Byte

(b) Nibble

(c) Word length

(d) Bit

Answer:

(c) Word length

Question 2.

How many bytes does 1 KiloByte contain?

(a) 1000

(b) 8

(c) 4

(d) 1024

Answer:

(d) 1024

Question 3.

Expansion for ASCII ………………..

(а) American School Code for Information Interchange

(b) American Standard Code for Information Interchange

(c) All Standard Code for Information Interchange

(d) American Society Code for Information Interchange

Answer:

(b) American Standard Code for Information Interchange

Question 4.

2^50 is referred as

(a) Kilo

(b) Tera

(c) Peta

(d) Zetta

Answer:

(c) Peta

Question 5.

How many characters can be handled in Binary Coded Decimal System?

(a) 64

(b) 255

(c) 256

(d) 128

Answer:

(a) 64

Question 6.

For 1101_{2} what is the Hexadecimal equivalent?

(a) F

(b) E

(c) D

(d) B

Answer:

(c) D

The Hex Calculator is used to perform addition, subtraction, multiplication and division on two hexadecimal numbers.

Question 7.

What is the 1’s complement of 00100110?

(a) 00100110

(b) 11011001

(c) 11010001

(d) 00101001

Answer:

(b) 11011001

Question 8.

Which amongst this is not an Octal number?

(a) 645

(b) 234

(c) 876

(d) 123

Answer:

(c) 876

II. Short Answers

Question 1.

What is data?

Answer:

The term data comes from the word datum which means a raw fact. The data is a fact about people, places or some objects.

Example: Rajesh, 16, XI

Question 2.

Write the 1’s complement procedure.

Answer:

Step 1 : convert given decimal number into Binary

Step 2 : if the binary bit contains 8 bits if less add 0 at the left most bit, to make it as 8 bits.

Step 3 : Invert all the bits, (i.e.) change 1 as 0 and 0 as 1.

Question 3.

Convert (46)_{10} into Binary number

Answer:

(46)_{10} into binary = 101110_{2}

Question 4.

We cannot find 1’s complement for (28)_{10}. State reason.

Answer:

Since it is a positive number. 1 ’s complement will come only for negative number.

Question 5.

List the encoding systems for characters in memory.

Answer:

- BCD – Binary Coded Decimal
- EBCDIC – Extended Binary Coded Decimal Interchange Code
- ASCII – American Standard Code for Information Interchange
- Unicode
- ISCII – Indian standard code for Information interchange

III. Explain in Brief

Question 1.

What is radix of a number system? Give example.

Answer:

The radix refers to the base of a number system: the total number of possible digits. The decimal number system that we all use is base ten, as it has ten distinct digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).

Example : Binary system – Radix 2

Question 2.

Write note on binary number system.

Answer:

In Binary Number System, there are only two digits namely 0 and 1. The numbers in the binary system are represented to the base 2. In the Binary Number, the left most bit is the Most Significant Bit (MSB) and the right most bit is the Least Significant Bit (LSB). MSB has largest positional weight and LSB has smallest positional weight.

Example : 110010_{2}

Question 3.

Convert (150)_{10} into Binary, then convert that Binary number to Octal.

Step 1:

Change it to Binary.

(150)_{10} = 10010110

Step – 2:

Change it to Octal.

Answer:

Binary 10010110_{2}

Octal 226_{g}

Question 4.

Write short note on ISCII.

Answer:

ISCII – Indian Standard Code for Information Interchange (ISCII) is the system of handling the character of Indian local languages. This is a 8 – bit coding system. Therefore it can handle 256 (2^{8}) characters. It is recognized by Bureau of Indian Standards (BIS). It is integrated with Unicode.

The supported scripts are:

Assamese, Bengali (Bangla), Devanagari, Gujarati, Gurmukhi, Kannada, Malayalam, Oriya, Tamil, and Telugu. ISCII does not encode the writing systems of India based on Arabic, but its writing system switching codes none the less provide for Kashmiri, Sindhi, Urdu, Persian, Pashto and Arabic. The Arabic – based writing systems were subsequently encoded in the PASCII encoding.

ISCII is an 8 – bit encoding. The lower 128 code points are plain ASCII, the upper 128 code points are ISCII – specific. In addition to the code points representing characters, ISCII makes use of a code point with mnemonic ATR that indicates that the following byte contains one of two kinds of information. One set of values changes the writing system until the next writing system indicator or end – of – line.

Question 5.

Add (a) – 22_{10} + 15_{10}

(b) 20_{10} + 25_{10
Answer:
(a) – 2210 = 101102
Binary Equivalent = 101102
Answer:
111110012}

(b) 20_{10} + 25_{10}

Answer:

Answer:

00101101_{2}

IV. Detail Answers

Question 1.

(a) Write the procedure to convert fractional Decimal to Binary.

(b) Convert (98.46)_{10} to Binary

Answer:

(а) Procedure to convert fractional Decimal to Binary.

Step 1 : Multiply the decimal fraction by 2 and note the integer part. The integer part is either Oor 1.

Step 2 : Discard the integer part of the previous product. Multiply the fractional part of the previous product by 2. Repeat step 1 until the same fraction repeats or terminates (0).

Step 3 : The resulting integer part forms a sequence of 0’s and 1 ’s that becomes the binary equivalent of decimal fraction.

Step 4 : The final answer is to be written from first integer part obtained till the last integer part obtained.

(b) Convert (98.46)_{10} to Binary

Answer:

1100010.01110 … _{2}

Question 2.

Find 1’s Complement and 2’s Complement for the following Decimal number (a) – 98 (b) – 135

Answer:

98_{10} = 01100010

8 bit format = 01100010

1’s complement = 10011101

Add 1 bit = + 1

= 10011110

135_{10} = 10000111

1’st complement = 01111000

Add 1 bit = + 1

= 01111001

Question 3.

(a) Add 1101010_{2} + 101101_{2}

(b) Subtract 1101011_{2} – 111010_{2}

Answer:

PART – 1

I. Choose the correct answer

Question 1.

Which is a basic electronic circuit which operates on one or more signals?

(a) Boolean algebra

(b) Gate

(c) Fundamental gates

(d) Derived gates

Answer:

(b) Gate

Question 2.

Which gate is called as the logical inverter?

(a) AND

(b) OR

(c) NOT

(d) XNOR

Answer:

(c) NOT

Question 3.

A + A = ?

(a) A

(b) O

(c) 1

(d) A

Answer:

(a) A

Question 4.

NOR is a combination of?

(a) NOT(OR)

(b) NOT(AND)

(c) NOT(NOT)

(d) NOT(NOR)

Answer:

(a) NOT(OR)

Question 5.

NAND is called as ……………… Gate.

(a) Fundamental Gate

(b) Derived Gate

(c) Logical Gate

(d) Electronic gate

Answer:

(b) Derived Gate

PART – 2

II. Short Answers

Question 1.

What is Boolean Algebra?

Answer:

Boolean algebra is a mathematical discipline that is used for designing digital circuits in a digital computer. It describes the relation between inputs and outputs of a digital circuit.

Question 2.

Write a short note on NAND Gate.

Answer:

The NAND gate is the combination of NOT and AND gates. The NAND gate is generated by inverting the output of a AND gate. The algebraic expression of the NAND is Y = \(\overline{\mathrm{A} . \mathrm{B}}\)

Question 3.

Draw the truth table for XOR gate.

Answer:

The truth table for XOR gate is

Question 4.

Write the associative laws?

Answer:

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

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

Question 5.

What are derived gates?

Answer:

Derived Gates are the gates which are derived from the fundamental gates.

Example : NAND, NOR, XOR, XNOR

PART – 3

III. Explain in Brief

Question 1.

Write the truth table of fundamental gates.

Answer:

(a) AND gate – Truth Table

(b) OR gate – Truth Table

(c) NOT Gate – Truth Table

Question 2.

Write a short note on XNOR gate.

The XNOR (exclusive – NOR) gate is a combination of XOR gate followed by an inverter. Its output is “true” if the inputs are same and false if the inputs are different.

The output of XNOR is C = AB + \(\overline{\mathrm{A}}\) \(\overline{\mathrm{B}}\)

= A \(\odot\) B

Where \(\odot\) indicates included dot.

Truth Table:

Logic symbol:

Logic circuit:

Question 3.

Reason out why the NAND an NOR are called universal gates?

Answer:

NAND and NOR gates are called as Universal gates because the fundamental logic gates can be realized through them.

Question 4.

Give the truth table of XOR gate.

Answer:

XOR – Truth Table

Question 5.

Write the De Morgan’s law.

Answer:

De Morgan’s \(\overline{\mathrm{A}+\mathrm{B}}\) = \(\overline{\mathrm{A}}\) . \(\overline{\mathrm{B}}\)

(\(\overline{\mathrm{A}+\mathrm{B}}\)) = \(\overline{\mathrm{A}}\) + \(\overline{\mathrm{B}}\)

PART – 4

IV. Explain in Detail

Question 1.

Explain the fundamental gates with expression and truth table.

Answer:

A gate is a basic electronic circuit which operates on one or more signals to produce an output signal.

The three fundamental gates are AND, OR and NOT gates.

AND Gate:

The AND gate can have two or more input signals and produce an output signal. The output is “true” only when both inputs are “true”, otherwise, the output is “false”. In other words the output will be 1 if and only if both inputs are 1; otherwise the output is 0. The output of the AND gate is represented by a variable say C, where A and B are two and if input boolean variables. In boolean algebra, a variable can take either of the values ‘0’ or ‘1’. The logical symbol of the AND gate is

The truth table for AND gate is

OR Gate:

The OR gate gets its name from its behaviour like the logical inclusive “OR”. The output is “true” if either or both of the inputs are “true”. If both inputs are “false” then the output is “false”. In other words the output will be 1 if and only if one or both inputs are 1; otherwise, the output is 0. The logical symbol of the OR gate is

The truth table for OR gate is

NOT Gate:

The NOT gate, called a logical inverter, has only one input. It reverses the logical state. In other words the output C is always the complement of the input. The logical symbol of the NOT gate is

The truth table for NOT gate is

Question 2.

How AND and OR can be realized using NAND and NOR gate.

Bubbled AND Gate

(i) Realized of and using only AND gate:

The Boolean function for AND is C = AB. The same can be realized using only NAND gates.

(ii) Realization of or using only NAND’s:

The Boolean function of OR is C = A + B. The same can be realized using only NAND gates.

(iii) Realization of AND using NOR:

By using only the NOR gates, we can get the output equivalent to the output of AND gate.

C = A.B

(iv) Realization of OR using NOR’s:

By using only NOR gates we are getting the output equivalent to OR gate.

Question 3.

Explain the Derived gates with expression and truth table.

Answer:

The logic gates like NAND, NOR, XOR and XNOR are derived gates which are derived from the fundamental gates AND, OR NOT.

(i) NAND gate:

The NAND is the combination of NOT and AND. The NAND is generated by inverting the output of an AND operation.

Output: y = \(\overline{\mathrm{A}\mathrm{B}}\)

Logic circuit

Logic symbol:

Truth table:

Inputs Outputs

(ii) NOR gate:

The NOR is the combination of NOT and OR. The NOR is generated by inverting the output of an OR operation.

Logic function: y = \(\overline{\mathrm{A}\mathrm{B}}\)

Logic circuit:

logic symbol:

Truth table:

(iii) XOR gate:

The XOR (exclusive – OR) gate acts in the same way as the logical either /or.

Logic symbol:

Logic circuit:

Truth table:

(iv) XNOR gate:

XNOR gate (exclusive – NOR) gate is a combination of XOR gate followed by an inverter.

Logic function: y = \(\overline{\mathrm{A}\mathrm{B}}\)

logic symbol:

Truth table:

Boolean function:

### Samacheer kalvi 11th Computer Science Number Systems Additional Questions and Answers

PART – 1

I. Choose the correct answer

Question 1.

The simplest method to represent negative binary number is called ………………..

(a) signed magnitude

(b) sign bit or parity bit

(c) binary

(d) decimal

Answer:

(a) signed magnitude

Question 2.

The term data comes from the word ………………..?

(a) number

(b) datum

(c) nibble

(d) bit

Answer:

(b) datum

Question 3.

Expansion for BCD ………………..

(a) Binary coded decimal

(b) binary complement decimal

(c) binary computer decimal

(d) binary convert decimal

Answer:

(a) Binary coded decimal

Question 4.

……………….. scheme is denoted by hexadecimal numbers.

(a) binary

(b) Unicode

(c) word length

(d) data

Answer:

(b) Unicode

Question 5.

The ……………….. operator is defined in boolean algebra by the use of the dot (.) operator.

(a) AND

(b) OR

(c) NOT

(d) NAND

Answer:

(a) AND

Question 6.

A ……………….. number is represented using base 16.

(a) Hexadecimal

(b) octal

(c) binary

(d) decimal

Answer:

(a) Hexadecimal

Question 7.

The convert (65)_{10} into its equivalent octal number ………………..

(a) (101)_{8}

(b) (101)_{10}

(c) (101)_{12}

(d) (101)_{4}

Answer:

(a) (101)_{8}

Question 8.

Octal number system uses digits ………………..

(a) 7

(b) 5

(c) 8

(d) 10

Answer:

(c) 8

Question 9.

……………….. is the general idea behind positional numbering system.

(a) Radix

(b) Computer memory

(c) Binary number

(d) Decimal number

Answer:

(a) Radix

Question 10.

The NAND gate operates an AND gate followed by a ……………….. gate.

(a) AND

(b) OR

(c) NOT

(d) XOR

Answer:

(c) NOT

Question 11.

Bit means ………………..

(a) nibble

(b) byte

(c) word length

(d) binary digit

Answer:

(d) binary digit

Question 12.

Expand BIT.

(a) Basic Input Term

(b) Binary Input Term

(c) Binary Digit

(d) Binary Inverse Digit

Answer:

(c) Binary Digit

Question 13.

The computer can understand ……………….. languages.

(a) computer

(b) machine

(c) post

(d) pre

Answer:

(b) machine

Question 14.

Identify the wrong pair.

1. 1 YB = 2^80

2. 1 BM = 2^70

3. 1 MB = 2^20

4. 1 TM = 2^40

(a) 2

(b) 3

(c) 4

(d) 1

Answer:

(a) 2

Question 15.

How many bytes does 1 zetta byte contains?

(a) 2^{90}

(b) 2^{80}

(c) 2^{70}

(d) 2^{60}

Answer:

(c) 2^{70}

Question 16.

The collection of 4 bits is ………………..

(a) bit

(b) byte

(c) nibble

(d) KB

Answer:

(c) nibble

Question 17.

Match the following.

(a) 4 12 3

(b) 12 3 4

(c) 4 3 2 1

(d) 4 2 1 3

Answer:

(a) 4 12 3

Question 18.

1 kilo byte represents ……………….. bytes.

(a) 512

(b) 256

(c) 1024

(d) 64

Answer:

(c) 1024

Question 19.

How many mega bytes does 1 GB contains?

(a) 2^{20}

(b) 2^{10}

(c) 2^{30}

(d) 2^{40}

Answer:

(b) 2^{10}

Question 20.

What is the decimal value of 1111_{2}?

(a) 10

(b) 11

(c) 14

(d) 15

Answer:

(d) 15

Question 21.

What is the 1’ s complement of 11001.

(a) 11100110

(b) 01010101

(c) 11110000

(d) 100100111

Answer:

(a) 11100110

Question 22.

The decimal value of Binary number 10 is ………………..

(a) 101010

(b) 2

(c) 100

(d) A

Answer:

(b) 2

Question 23.

The hexadecimal equivalent of 15 is ………………..

(a) A

(b) B

(c) E

(d) F

Answer:

(d) F

Question 24.

Which of the following are data?

(a) Alphabet

(b) Special character

(c) Number

(d) All of these

Answer:

(d) All of these

Question 25.

The radix of hexadecimal number is ………………..

(a) 2

(b) 8

(c) 16

(d) 10

Answer:

(c) 16

Question 26.

Pick the odd one.

(a) BCD

(b) ENIAC

(c) ASCII

(d) EBCDIC

Answer:

(b) ENIAC

Question 27.

The most commonly used number system is ………………..

(a) binary

(b) decimal

(c) octal

(d) hexadecimal

Answer:

(b) decimal

Question 28.

Unicode can handles how many characters?

(a) 64

(b) 128

(c) 256

(d) 65536

Answer:

(d) 65536

Question 29.

What does MSB means?

(a) Major sign bit

(b) Most sign bit

(c) Minor sign bit

(d) Most significant bit

Answer:

(d) Most significant bit

Question 30.

Which one is the right most bit?

(a) MSB

(b) LSB

(c) USB

(d) USRB

Answer:

(b) LSB

Question 31.

The binary equivalent of hexadecimal number B is ………………..

(a) 1011

(b) 1100

(c) 1001

(d) 1010

Answer:

(a) 1011

Question 32.

The left most bit of a positive binary number in signed notation is ………………..

(a) 0

(b) 1

(c) 2

(d) A

Answer:

(a) 0

Question 33.

What is the range of ASCII values for lower case alphabets?

(a) 65 to 90

(b) 65 to 122

(c) 97 to 122

(d) 98 to 122

Answer:

(c) 97 to 122

Question 34.

The radix for octal number system is ………………..

(a) 2

(b) 8

(c) 1

(d) 16

Answer:

(b) 8

Question 35.

What is the ASCII value for blank space?

(a) 8

(b) 2

(c) 18

(d) 32

Answer:

(d) 32

Question 36.

Which one of the following company have formulated EBCDIC?

(a) Microsoft

(b) 1 BM

(c) Sun

(d) Apple

Answer:

(b) 1 BM

Question 37.

Which one of the following bit has smallest positional weight?

(a) MSB

(b) LSB

(c) UPS

(d) USB

Answer:

(b) LSB

Question 38.

The base value of hexadecimal number is ………………..

(a) 2

(b) 8

(c) 16

(d) 18

Answer:

(c) 16

Question 39.

Name the person who proposed the basic principles of Boolean Algebra?

(a) Wiliam Boole

(b) George Boole

(c) James Boole

(d) Boolean George

Answer:

(b) George Boole

Question 40.

How many truth values are there?

(a) 2

(b) 3

(c) 4

(d) 5

Answer:

(a) 2

Question 41.

What is the other name for logical statement?

(a) Truth values

(b) Truth functions

(c) Truth table

(d) Truth variables

Answer:

(b) Truth functions

Question 42.

The variables which can store the truth values are called as ………………..

(a) logical variable

(b) binary valued variable

(c) boolean variables

(d) all of these

Answer:

(d) all of these

Question 43.

The NOT operator is represented by the symbol.

(a) over bar

(b) single apostrophe

(c) a and b

(d) plus

Answer:

(c) a and b

Question 44.

Which is not a logical operator?

(a) dot

(b) plus

(c) over bar

(d) command

Answer:

(d) command

Question 45.

The output for the AND operator is ………………..

(a) A + B

(b) –

(c) A.B

(d) AB + C

Answer:

(c) A.B

Question 46.

Which symbol is used to in OR operator?

(a) –

(b) •

(c) *

(d) +

Answer:

(d) +

Question 47.

Which gate takes only one input?

(a) OR

(b) AND

(c) NOT

(d) XOR

Answer:

(c) NOT

Question 48.

Which among the following can be replaced by a bubbled AND gate?

(a) AND

(b) NAND

(c) OR

(d) not

Answer:

(b) NAND

Question 49.

Which is not a derived date?

(a) AND

(b) NAND

(c) NOR

(d) XOR

Answer:

(a) AND

Question 50.

Find the universal gates from the following.

(a) XOR

(b) XNOR

(c) a and b

(d) NOR

Answer:

(d) NOR

Question 51.

The statement “C equal the complement of A or B” means

(a) C = A + B

(b) C = \(\overline{\mathrm{A}}+\overline{\mathrm{B}}\)

(c) C = \(\overline{\mathrm{A}}\) + \(\overline{\mathrm{B}}\)

(d) C = \(\overline{\mathrm{A}\mathrm{B}}\)

Answer:

(a) C = A + B

Question 52.

Which symbol is used in XOR gate?

(a) \(\odot\)

(b) \(\otimes\)

(c) \(\oplus\)

(d) –

Answer:

(c) \(\oplus\)

Question 53.

Included dot means ………………..

(a) \(\odot\)

(b) [•]

Answer:

(a) \(\odot\)

Question 54.

What is the output of XOR gate?

(a) C = A% B

(b) C = A \(\otimes\) A

(c) C = A \(\odot\) B

(d) C = A \(\oplus\) B

Answer:

(d) C = A \(\oplus\) B

Question 55.

Identify the statement which is wrong.

(a) A . 1 = A

(b) A . A = A

(c) A + O = A

(d) A . 1 = 0

Answer:

(b) A . A = A

Question 56.

Find A + \(\overline{\mathrm{A}}\) .B = ………………..

(a) A + B

(b) A.B

(c) \(\overline{\mathrm{A}}\).B

(d) A.\(\overline{\mathrm{B}}\)

Answer:

(d) A.\(\overline{\mathrm{B}}\)

Question 57.

Identify the statements which are true.

(i) A + 0 = A

(ii) A.A = A

(iii) A + \(\overline{\mathrm{A}}\) = 1

(iv) A. O = O

(a) (iii) (iv) are true

(b) (i) (ii) are true

(c) (i) (ii) (iii) are true.

(d) all are true

Answer:

(d) all are true

Question 58.

Find the wrong pair from the following:

(a) Null element : A + 1 = 1

(b) Involution : \(\overset { = }{ A }\) = A

(c) Demorgan’s : \(\overline{\mathrm{A+B}}\) =\(\overline{\mathrm{A}}\) . \(\overline{\mathrm{A}}\)

(d) Commutative : A + B = B . A

Answer:

(d) Commutative : A + B = B . A

Question 59.

Match the following

(a) 3 4 2 1

(b) 1 2 3 4

(c) 4 3 2 1

(d) 3 4 1 2

Answer:

(a) 3 4 2 1

Question 60.

With 2 inputs in the truth table, how many set of values will be obtained.

(a) 4

(b) 8

(c) 2

(d) 1

Answer:

(a) 4

PART – 2

II. Short Answers

Question 1.

What is nibble?

Answer:

Nibble is a collection of 4 bits. A nibble is a half a byte.

Question 2.

Expand ASCII.

Answer:

American Standard Code for Information Interchange

Question 3.

What is radix?

Answer:

The base value of a number is also known as the radix.

Question 4.

What is MSB and LSB?

Answer:

MSB means Most Significant Bit; LSB-Least Significant Bit

Question 5.

Expand: BCD, EBCDIC, ASCII

Answer:

BCD – Binary Coded Decimal; EBCDIC – Extended Binary Coded Decimal Interchange Code; ASCII – American Standard Code for Information Interchange.

Question 6.

Define word length?

Answer:

Word length refers to the number of bits processed by a computers CPU.

Question 7.

What are the methods of converting a number from decimal to binary.

Answer:

- Repeated division by two.
- Sum of powers of 2.

Question 8.

How will you convert a number from octal to Binary.

Answer:

For each octal digit in the given number write its 3 digits binary equivalent using positional notation.

Question 9.

What are the various ways for Binary representation of signed numbers?

Answer:

- Signed magnitude representation
- 1’s complement
- 2’s complement

Question 10.

Write a short note on BCD.

Answer:

- BCD stands for Binary Coded Decimal system.
- BCD is 26 bit encoding system.
- IT can handle 64 characters.

PART – 3

III. Explain in Brief

Question 1.

What is binary number system?

Answer:

There are only two digits in the Binary system, namely, 0 and 1. The numbers in the binary system are represented to the base 2 and the positional multipliers are the powers of 2. The left most bit in the binary number is called as the Most Significant Bit (MSB) and it has the largest positional weight. The right most bit is the Least Significant Bit (LSB) and has the smallest positional weight.

Example:

The binary sequence (11101)_{2} has the decimal equivalent:

(1101)_{2} = 1 x 2^{3} + 1 x 2^{2} + 0 x 2^{1} + 1 x 2^{0}

= 8 + 4 + 0 + 1

= (13)_{10}

Question 2.

What is octal number system?

Answer:

Octal number system uses digits 0, 1, 2, 3, 4, 5, 6 and 7 (8 digits): Each octal digit has its own positional value or weight as a power of 8.

Example:

The Octal sequence (547)_{8} has the decimal equivalent:

Question 3.

What is decimal number system?

Answer:

It consists of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9(10 digits). It is the oldest and most popular number system used in our day to day life. In the positional number system, each decimal digit is weighted relative to its position in the number. This means that each digit in the number is multiplied by 10 raised to a power corresponding to that digit’s position.

Example:

(123)_{10} = (1 x 10^{2}) + (2 x 10^{1}) + (3 x 10^{0})

= 100 + 20 + 3

= (123)_{10}

Question 4.

Write the distributive law.

Answer:

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

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

Question 5.

What is truth table?

Answer:

A truth table represents all the possible values of logical variable or statements along with ail the possible results of given combination of truth values.

PART – 4

IV. Explain in Detail

Question 1.

Explain the different types of number systems?

Answer:

Different Types of Number Systems

A numbering system is a way of representing numbers. The most commonly used numbering system in real life is Decimal number system. Other number systems are Binary, Octal, Hexadecimal number system. Each number system is uniquely identified by its base value or radix. Radix or base is the count of number of digits in each number system. Radix or base is the general idea behind positional numbering system.

Decimal Number System:

It consists of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9(10 digits). It is the oldest and most popular number system used in our day to day life. In the positional number system, each decimal digit is weighted relative to its position in the number. This means that each digit in the number is multiplied by 10 raised to a power corresponding to that digit’s position.

Example:

(123)_{10} = 1 x 10^{2} + 2 x 10^{1} + 3 x 10^{0}

= 100 + 20 + 3

= (123)_{10}

Binary Number System:

There are only two digits in the Binary system, namely, 0 and 1. The numbers in the binary system are represented to the base 2 and the positional multipliers are the powers of 2. The left most bit in the binary number is called as the Most Significant Bit (MSB) and it has the largest positional weight. The right most bit is the Least Significant Bit (LSB) and has the smallest positional weight.

Example:

The binary sequence (1101)_{2} has the decimal equivalent:

(1101)_{2} = 1 x 2^{3} + 1 x 2^{2} + 0 x 2^{1} + 1 x 2^{0}

= 8 + 4 + 0 + 1

= (13)_{10}

Octal Number System:

Octal number system uses digits 0,1,2,3,4,5,6 and 7 (8 digits). Each octal digit has its own positional value or weight as a power of 8.

Example:

The Octal sequence (547)_{8} has the decimal equivalent:

(547)_{8} = 5 x 8^{2} + 4 x 8^{2} + 7 x 8^{0}

= 5 x 64 + 4 x 8 + 7 x 1

= 320 + 32 + 7

= (359)_{10}

Hexadecimal Number System:

A hexadecimal number is represented using base 16. Hexadecimal or Hex numbers are used as a shorthand form of binary sequence. This system is used to represent data in a more compact manner. Since 16 symbols are used, 0 to F, the notation is called hexadecimal. The first 10 symbols are the same as in the decimal system, 0 to 9 and the remaining 6 symbols are taken from the first 6 letters of the alphabet sequence, A to F, where A represents 10, B is 11, C is 12, D is 13, E is 14 and F is 15.

Question 2.

Explain the octal to decimal conversion and hexadecimal to decimal conversion.

Answer:

Octal to Decimal Conversion:

To convert Octal to Decimal, we can use positional notation method.

- Write down the Octal digits and list the powers of 8 from right to left(Positional Notation).
- For each positional notation of the digit write the equivalent weight.
- Multiply each digit with its corresponding weight.
- Add all the values.

Example:

Convert (1265)_{8} to equivalent Decimal number

(1265)_{8} = 512 x 1 + 64 x 2 + 8 x 6 + 1 x 5

= 512 + 128 + 48 + 5

(1265)_{8} = (693)_{10}

Hexadecimal to Decimal Conversion:

To convert Hexadecimal to Decimal we can use positional notation method.

- Write down the Hexadecimal digits and list the powers of 16 from right to left (Positional Notation).
- For each positional notation written for the digit, now write the equivalent weight.
- Multiply each digit with its corresponding weight.
- Add all the values to get one final value.

Example:

Convert (25F)_{16} into its equivalent Decimal number.

Question 3.

Explain the binary addition and binary subtraction

Answer:

Binary Addition:

The following table is useful when adding two binary numbers.

Example Add: 1011_{2} + 1001_{2}

Example : Perform Binary addition for the

following : 23_{10} + 12_{10}

Step 1 : Convert 23 and 12 into binary form

Step 2 : Binary addition of 23 and 12:

Binary Subtraction:

The table for Binary Subtraction is as follows:

When subtracting 1 from 0, borrow 1 from the next Most Significant Bit, when borrowing from the next Most Significant Bit, if it is 1, replace it with 0. If the next Most Significant Bit is 0, you must borrow from a more significant bit that contains 1 and replace it with 0 and Os upto that point become Is.

Example : Subtract 1001010_{2} – 10100_{2}

Example Perform binary addition for the following: (-21)_{10} + (5)_{10}

Step 1 : Change – 21 and 5 into binary form

Step 2:

Step 3.

Binary Addition of – 21 and 5:

Question 4.

Explain the theorems of boolean algebra.

Answer:

Theorems of Boolean Algebra:

Identity:

A + 0 = A

A. 1 = A

Complement:

A + \(\overline{\mathrm{A}}\) = 1

A. \(\overline{\mathrm{A}}\) = 0

Commutative:

A + B = B + A

A. B = B .A

Associative:

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

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

Distributive

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

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

Null Element:

A + 1 = 1

A. 0 = 0

Involution

(\(\overset { = }{ A }\)) = A

Indempotence:

A +A = A

A.A = A

Absorption:

A + (A . B) = A

A . (A + B) = A

3rd Distributive:

A + \(\overline{\mathrm{A}}\). B = A + B

De Morgan’s:

\(\overline{\mathrm{A+B}}\) = \(\overline{\mathrm{A}}\).\(\overline{\mathrm{B}}\)

(\(\overline{\mathrm{A.B}}\)) = \(\overline{\mathrm{A}}\) + \(\overline{\mathrm{B}}\)