Number Representations and Computer Arithmetic MCQ Quiz - Objective Question with Answer for Number Representations and Computer Arithmetic - Download Free PDF

Answer: Option 3

Explanation: 

Step 1: first we convert 147 to the Binary number system.

(147)10 = (10010011)2

Step 2: Now we group 4 bits to convert the Hexadecimal number system.

1001 0011

≡ (93)16

The correct answer is Gray code.

Key Points

• The Gray code, also referred to as the reflected binary code, is a binary numeral system in which two consecutive numbers differ in only one bit. The unique property of Gray code is that each transition from one value to the next value involves changing only one bit.
• This system was invented by Frank Gray at Bell Labs to prevent spurious output from electromechanical switches. For instance, while switching from one position to another in standard binary code, there is a risk that switches will change at different times leading to invalid numbers, but with Gray code, since only one bit changes at a time, such misinterpretations are avoided.
• So, Gray code is also known as reflected binary code because the sequence of binary values reflects it about its midpoint. For example:
• Binary: 000, 001, 010, 011, 100, 101, 110, 111
• If we reflect this sequence, i.e., reverse it, we get:111, 110, 101, 100, 011, 010, 001, 000

• Now let's take the first half of the original sequence and the first half of the reflected sequence.

• By inverting the bits in the second half, we get the sequence of Gray codes:  000, 001, 011, 010, 110, 111, 101, 100

So, every binary number has a unique Gray code, and vice versa, attributing to why the term "reflected binary code" is used to describe the Gray code.

Convert binary to decimal :-

• For binary number with n digits:
• dn-1 ... d3 d2 d1 d0
• The decimal number is equal to the sum of binary digits (dn) times their power of 2 (2n):
• decimal = d0×20 + d1×21 + d2×22 + ...

Calculation:

Decimal equivalent of binary number 1001110 :-
=  1 × 2⁶ + 0 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 1 x 21 + 0 x 20

​= 64 + 0 + 0 + 8 + 4 + 2 + 0 = 78​

Binary to decimal conversion →

(129)₁₀ = (10000001)₂

0.25 → 0.25 × 2 = 0.5

0.5 × 2 = 1

(0.25)₁₀ = (.01)₂

(129.25)₁₀ = (10000001.01)₂

2's Complement - It is a type of mathematical and logical (binary) representation that helps in representing signed numbers and performing arithmetic operations such as subtraction, addition, etc.

To perform 2's complement of (1000)₂ we will perform the following steps -

1. We will perform 1's complement on (1000)₂ by flipping 1s to 0s and 0s to 1s.
   (1000)₂ ===> (0111)₂
2. Now we will add 1 to the resultant value, that is, (0111)2.
   (0111)₂ + (1)₂ ===> (1000)₂
3. Hence, we get (1000)2 back after 2's complement.

• The correct answer is option 3, i.e., 374.
• Binary number 101110110 is equal to decimal number 374.
• Following method can be used to convert Binary number to Decimal number:

1. (101110110)₂ = (1 x 2⁸) + (0 x 2⁷) + (1 x 2⁶) + (1 x 2⁵) + (1 x 2⁴) + (0 x 2³) + (1 x 2²) + (1 x 2¹) + (0 x 2⁰)
2. (101110110)2 = 256 + 0 + 64 + 32 + 16 + 0 + 4 + 2 + 0
3. (101110110)2 = 374

The correct answer is 220 bytes.

Key Points

• 1 Megabyte is equal to 1000000 bytes (decimal).
• 1 MB = 106 B in base 10 (SI).
• 1 Megabyte is equal to 1048576 bytes (binary).
• 1 MB = 220 B in base 2.
• Byte is the basic unit of digital information transmission and storage, used extensively in information technology, digital technology, and other related fields. It is one of the smallest units of memory in computer technology, as well as one of the most basic data measurement units in programming.
• The earliest computers were made with the processor supporting 1 byte commands, because in 1 byte you can send 256 commands. 1 byte consists of 8 bits,
• Megabyte (MB) is a unit of transferred or stored digital information, which is extensively used in information and computer technology.
• In SI, one megabyte is equal to 1,000,000 bytes. At the same time, practically 1 megabyte is used as 220 B, which means 1,048,576 bytes.

The correct answer is 11000110

Key Points

• To convert the hexadecimal number C6 to a binary number, you can convert each hexadecimal digit to its 4-bit binary representation.
• C in hexadecimal is 12 in decimal, which is 1100 in binary.
• 6 in hexadecimal is 6 in decimal, which is 0110 in binary.
• So, the binary representation of C6 is 11000110.

Additional InformationHere are the decimal numbers 1 to 15 represented in both hexadecimal and binary forms:

•  Decimal 1: Hexadecimal 1, Binary 0001
• Decimal 2: Hexadecimal 2, Binary 0010
• Decimal 3: Hexadecimal 3, Binary 0011
• Decimal 4: Hexadecimal 4, Binary 0100
• Decimal 5: Hexadecimal 5, Binary 0101
• Decimal 6: Hexadecimal 6, Binary 0110
• Decimal 7: Hexadecimal 7, Binary 0111
• Decimal 8: Hexadecimal 8, Binary 1000
• Decimal 9: Hexadecimal 9, Binary 1001
• Decimal 10: Hexadecimal A, Binary 1010
• Decimal 11: Hexadecimal B, Binary 1011
• Decimal 12: Hexadecimal C, Binary 1100
• Decimal 13: Hexadecimal D, Binary 1101
• Decimal 14: Hexadecimal E, Binary 1110
• Decimal 15: Hexadecimal F, Binary 1111 

Answer: Option 2

Explanation:

An octal Equivalent of a binary number is obtained by grouping 3 bits from right to left.

So Octal Equivalent: 1353

Important Points

Binary to Octal code

Calculation:

14 in binary form is represented as:

1410 = (00001110)2

Taking the 1's complement of the above, we get 11110001

Adding 1 to the 1's complement, we get the 2's complement representation of the number, i.e. 11110010

Since there is a 1 in the MSB, the number is a negative number with value -14.

∴ The 2's complement of -6410 contains 7 bits.

Application:

Decimal value = (3 ⋆ 4096 + 15 ⋆ 256 + 5 ⋆ 16 + 3)

It can be written as:

(2 + 1) × 212 + (8 + 4 + 2 + 2⁰) × 28 + (4 + 1) × 24  + (2 + 1) × 20

2¹ × 2¹² + 2⁰ × 212 + (2³ + 2² + 2¹ + 2⁰) × 2⁸ + (2² + 2⁰) × 2⁴ + (2¹ + 2⁰) × 2⁰

This can be written as:

213 + 212 + 211 + 210 + 29 × 28 + 26 + 24 + 21 + 20

The binary representation will be:

(11111101010011)₂

The correct answer is (11010)2 = (62)8

Key Points

Binary numbers and octal numbers are both used in computing. They are different ways of representing the same value - just like how "10" and "ten" are different ways of expressing the same quantity in decimal.

• Each digit of an octal number represents three binary digits because 2³ = 8. Here's the mapping:
  • "000" => "0"
  • "001" => "1"
  • "010" => "2"
  • "011" => "3"
  • "100" => "4"
  • "101" => "5"
  • "110" => "6"
  • "111" => "7"
• Now let's convert the binary numbers to their equivalent octal numbers.
  • (111 110 111)₂ = (7 6 7)₈
  • (110 110 101)₂ = (6 6 5)₈
  • (10 101 . 110)₂ = (2 5 . 6)₈
  • (11 010)₂ = (3 2)₈ - Corrupted as the corresponding octal number should be (32)₈ instead of (62)₈.

Therefore, the 4th pair, (11010)₂ = (62)₈, is not equal.

28₁₀ = (11100)₂ = (0000 0000 0001 1100)₂

-28₁₀ = 2’s complement of 0000 0000 0001 1100

2’s complement of 0000 0000 0001 1100 = 1111 1111 1110 0100

Note:

Tricks to find: 2’s complement

Mistake PointsThe question is asking for the 12^th Digit in Hexadecimal representation, i.e. 0 will be the first digit, 1 will be the second, and so on.

The correct answer is (option 2) i.e. B

Explanation:

Digits in hexadecimal number systems are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Therefore, the total number of different digits in hexadecimal number systems is 16.

Hence 12th digit in the hexadecimal system is B. And it is equivalent to 11 for Decimal and 1011 for the Binary number system,

Important Points

• Digits in decimal number systems are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Therefore, number of different digits in decimal number systems is 10.
• Digits in Octal number systems are 0, 1, 2, 3, 4, 5, 6, 7. Therefore, number of different digits in octal number systems is 8.
• Digits in binary number systems are 0, 1. Therefore, number of different digits in binary number systems is 2.
• In the standard hexadecimal system, each digit can have 16 possible values, ranging from 0 to 9 and then A to F, representing the values 10 to 15.
• To determine the 12th digit in the standard hexadecimal system, we need to convert the number 12 from decimal to hexadecimal.
• 12 in decimal is equal to B in hexadecimal. Therefore, the 12th digit in the standard hexadecimal system is option 4) B.

The hexadecimal system is a number system with a base of 16. It is commonly used in computing and digital systems because it provides a convenient way to represent binary numbers. In hexadecimal, the digits range from 0 to 9, and then use the letters A to F to represent values 10 to 15.

Here's a breakdown of the digits leading up to the 12th position:

• 1st digit: 0
• 2nd digit: 1
• 3rd digit: 2
• 4th digit: 3
• 5th digit: 4
• 6th digit: 5
• 7th digit: 6
• 8th digit: 7
• 9th digit: 8
• 10th digit: 9
• 11th digit: A
• 12th digit: B

Therefore, the 12th digit in the standard hexadecimal system is 'B'.