Number System MCQ Quiz - Objective Question with Answer for Number System - Download Free PDF

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

Explanation:

Hexadecimal to Octal Conversion

Definition: Hexadecimal and octal are both positional numeral systems, widely used in computing and digital electronics. Hexadecimal (base-16) uses sixteen symbols, 0-9 and A-F, to represent values, while octal (base-8) uses eight symbols, 0-7.

To convert a hexadecimal number to its octal equivalent, it is often easiest to first convert it to binary (base-2), and then from binary to octal. This method works because both octal and hexadecimal are powers of 2 (octal is 2³ and hexadecimal is 2⁴).

Step-by-Step Solution:

Given hexadecimal number: 1A3

1. Convert Hexadecimal to Binary:

Each hexadecimal digit can be represented by a 4-bit binary number:

• 1 (Hex) = 0001 (Binary)
• A (Hex) = 1010 (Binary)
• 3 (Hex) = 0011 (Binary)

So, the hexadecimal number 1A3 can be written in binary as: 0001 1010 0011

2. Group Binary Digits into Sets of Three:

Since octal is base-8 and each octal digit corresponds to 3 binary digits, we group the binary digits in sets of three, starting from the right:

• 0001 1010 0011 (Binary)
• 000 110 100 011 (Binary, grouped in sets of three)

3. Convert Binary Sets to Octal:

Each group of three binary digits can be converted directly to its octal equivalent:

• 000 (Binary) = 0 (Octal)
• 110 (Binary) = 6 (Octal)
• 100 (Binary) = 4 (Octal)
• 011 (Binary) = 3 (Octal)

So, the binary number 000 110 100 011 can be written in octal as: 0643

Hence, the correct octal representation of the hexadecimal number 1A3 is 643.

Important Information:

To analyze other options, let's convert the hexadecimal number 1A3 using the same method:

• Option 1: 346
  • 346 (Octal) in binary: 011 100 110
  • Groups: 011 100 110 (Binary) = 3 4 6 (Octal)
  • Binary to Hexadecimal: 011100110 (Binary) = 1C6 (Hex)
  • 1C6 ≠ 1A3
• Option 2: 124
  • 124 (Octal) in binary: 001 010 100
  • Groups: 001 010 100 (Binary) = 1 2 4 (Octal)
  • Binary to Hexadecimal: 001010100 (Binary) = 54 (Hex)
  • 54 ≠ 1A3
• Option 3: 634
  • 634 (Octal) in binary: 110 011 100
  • Groups: 110 011 100 (Binary) = 6 3 4 (Octal)
  • Binary to Hexadecimal: 110011100 (Binary) = 19C (Hex)
  • 19C ≠ 1A3

Therefore, the correct option is confirmed as option 4: 643.

The correct answer is 010010

Key Points

• The two's complement of a binary number is found by inverting all of the bits, known as the one's complement, and then adding 1.
• The binary number 101110's inverted binary number (one's complement) would be 010001.
• Next, you add 1 to the inverted binary number:

         010001

                 +1

         010010

Therefore, the 2's complement of the binary number 101110 is 010010.

Explanation:

Hexadecimal to Octal Conversion

Definition: Hexadecimal and octal are both positional numeral systems, widely used in computing and digital electronics. Hexadecimal (base-16) uses sixteen symbols, 0-9 and A-F, to represent values, while octal (base-8) uses eight symbols, 0-7.

To convert a hexadecimal number to its octal equivalent, it is often easiest to first convert it to binary (base-2), and then from binary to octal. This method works because both octal and hexadecimal are powers of 2 (octal is 2³ and hexadecimal is 2⁴).

Step-by-Step Solution:

Given hexadecimal number: 1A3

1. Convert Hexadecimal to Binary:

Each hexadecimal digit can be represented by a 4-bit binary number:

• 1 (Hex) = 0001 (Binary)
• A (Hex) = 1010 (Binary)
• 3 (Hex) = 0011 (Binary)

So, the hexadecimal number 1A3 can be written in binary as: 0001 1010 0011

2. Group Binary Digits into Sets of Three:

Since octal is base-8 and each octal digit corresponds to 3 binary digits, we group the binary digits in sets of three, starting from the right:

• 0001 1010 0011 (Binary)
• 000 110 100 011 (Binary, grouped in sets of three)

3. Convert Binary Sets to Octal:

Each group of three binary digits can be converted directly to its octal equivalent:

• 000 (Binary) = 0 (Octal)
• 110 (Binary) = 6 (Octal)
• 100 (Binary) = 4 (Octal)
• 011 (Binary) = 3 (Octal)

So, the binary number 000 110 100 011 can be written in octal as: 0643

Hence, the correct octal representation of the hexadecimal number 1A3 is 643.

Important Information:

To analyze other options, let's convert the hexadecimal number 1A3 using the same method:

• Option 1: 346
  • 346 (Octal) in binary: 011 100 110
  • Groups: 011 100 110 (Binary) = 3 4 6 (Octal)
  • Binary to Hexadecimal: 011100110 (Binary) = 1C6 (Hex)
  • 1C6 ≠ 1A3
• Option 2: 124
  • 124 (Octal) in binary: 001 010 100
  • Groups: 001 010 100 (Binary) = 1 2 4 (Octal)
  • Binary to Hexadecimal: 001010100 (Binary) = 54 (Hex)
  • 54 ≠ 1A3
• Option 3: 634
  • 634 (Octal) in binary: 110 011 100
  • Groups: 110 011 100 (Binary) = 6 3 4 (Octal)
  • Binary to Hexadecimal: 110011100 (Binary) = 19C (Hex)
  • 19C ≠ 1A3

Therefore, the correct option is confirmed as option 4: 643.

Explanation:

Correct Option - The carry-out bit is ignored

When subtracting two n-bit numbers using 2's complement and the result is positive, the carry-out bit does not have a significant role in determining the outcome of the subtraction. The carry-out bit is essentially ignored in this scenario. This is due to the way 2's complement arithmetic handles overflow and underflow conditions.

Understanding 2's Complement Representation:

2's complement representation is a method used to encode signed integers in binary. In this representation, positive numbers are represented as usual in binary, while negative numbers are represented by taking the binary representation of the absolute value, inverting all the bits (creating the 1's complement), and then adding one to the least significant bit.

For example:

• +5 in 4-bit binary: 0101
• -5 in 4-bit 2's complement: 1011 (1's complement of 0101 is 1010, add 1 to get 1011)

Subtraction Using 2's Complement:

When performing subtraction using 2's complement, we add the 2's complement of the number to be subtracted. For instance, to compute A - B, we actually calculate A + (-B). The process involves the following steps:

• Compute the 2's complement of B (which gives -B).
• Add this value to A.
• Check the result for any overflow or underflow conditions.

Carry-out Bit:

The carry-out bit is a bit that is carried out of the most significant bit position when performing binary addition or subtraction. In the context of 2's complement subtraction, the carry-out bit can indicate whether an overflow has occurred, but it does not affect the correctness of the result itself. If the result of the subtraction is positive, it means that the result fits within the range of representable values for the given bit-width, and thus, the carry-out bit is ignored.

Analysis of Other Options:

Option 1: The carry-out bit indicates an overflow

This option is incorrect because the carry-out bit alone does not indicate an overflow in the context of 2's complement arithmetic. Overflow is determined by examining the sign bits of the operands and the result, not just the carry-out bit. For example, if adding two positive numbers results in a negative number, or adding two negative numbers results in a positive number, an overflow has occurred.

Option 2: The carry-out bit is added to the result

This option is incorrect because the carry-out bit is not added to the result in 2's complement arithmetic. The carry-out bit is discarded, and the result is taken as-is unless overflow has occurred, which needs to be handled separately.

Option 3: The carry-out bit is inverted

This option is incorrect because inverting the carry-out bit has no meaningful impact on the result of the 2's complement subtraction. The carry-out bit does not affect the final result and is simply ignored if the result is within the valid range.

Option 4: The carry-out bit is ignored

This is the correct option, as explained above. The carry-out bit does not affect the result of the subtraction if the result is positive and within the representable range of the bit-width.

Important Information:

Understanding the carry-out bit and its role in binary arithmetic is crucial for anyone working with digital systems and computer architecture. The carry-out bit can indicate different conditions depending on the context, such as overflow or borrow in subtraction, but in the case of 2's complement arithmetic, it is ignored if the result is positive and valid.

Additionally, understanding the principles of 2's complement representation and arithmetic is foundational for designing and analyzing algorithms that involve signed integer calculations. This knowledge is essential for fields like embedded systems, computer engineering, and digital signal processing, where precise manipulation of binary data is required.

• 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.

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

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

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'.

Excess - 3 - code is also known as self-complementing code which means 1's complement of an excess - 3 number is the excess - 3 code for the 9's complement of the corresponding decimal number.

Example:

1  in binary is 0001

excess 3 code is 0001 + 0011 = 0100

1's complement of the above code is 1011 which is 11 

11 is excess 3 code for 8

and 9's complement of 1 is 8

Concept:

• To convert a decimal number system to hexadecimal, we follow the successive division approach i.e. we divide the decimal number by 16 and note down the remainder.
• Each remainder is then expressed in hexadecimal.

Calculation:

So, The hexadecimal equivalent of decimal number 4096 is 1000. 

The correct answer is option 4.

Concept:

Hexadecimal number system:

The hexadecimal number system is a kind of number system with a base value of 16 characters. It is also spelled 'hex' at times. Only 16 symbols are used to represent hexadecimal values. These are the values or symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Each number corresponds to a decimal value.

Octal Number System:

The Octal Number System has an eight-digit basis and employs numbers ranging from 0 to 7. When paired in pairs of three, octal numbers are commonly represented as binary numbers in the number system.

Explanation:

The given data,

If (356)8 = (x)16

Convert the octal number to binary and then convert it into Hexa decimal.

3= 011

5= 101

6= 110

(011 101 110)₂= (0 1110 1110)₂

1110= E

(0 E E)16

Hence the correct answer is EE.

Alternate Method The given data,

If (356)8 = (x)16

Convert the octal number to decimal and then convert it into Hexa decimal.

(356)8 =(3 x 8²+ 5 x 8¹+6 x 8⁰)₁₀

(356)8 =(238)10

(356)8 =(238)10 =(EE)₁₆

Number System

The octal number system is represented by base 8.

The number system is a way to represent or express numbers. You have heard of various types of number systems such as the whole numbers and the real numbers. But in the context of computers, we define other types of number systems. They are: