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

Explanation:

Decimal Equivalent of a Hexadecimal Number

Definition: A hexadecimal number is a base-16 number system that uses digits from 0 to 9 and letters from A to F (where A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15). To convert a hexadecimal number to its decimal equivalent, we use the positional value of each digit, multiplying it by powers of 16, starting from 0 for the rightmost digit. The sum of these products gives the decimal equivalent.

Given Problem: Convert the hexadecimal number (8A6) to its decimal equivalent.

Step-by-Step Conversion:

We are tasked with converting (8A6) from hexadecimal to decimal. Let's calculate:

Step 1: Write down the given hexadecimal number:

8A6

Step 2: Expand the hexadecimal number into its positional components:

Each digit in a hexadecimal number is multiplied by 16 raised to a power. The power starts at 0 from the rightmost digit and increases by 1 for each digit to the left.

For 8A6, the positional values are:

• 6: Rightmost digit, position 0 → \( 6 × 16^0 \)
• A: Middle digit, position 1 → \( A × 16^1 \) (where A = 10 in decimal)
• 8: Leftmost digit, position 2 → \( 8 × 16^2 \)

Step 3: Convert each positional value and calculate:

• \( 6 × 16^0 = 6 × 1 = 6 \)
• \( A × 16^1 = 10 × 16 = 160 \)
• \( 8 × 16^2 = 8 × 256 = 2048 \)

Step 4: Add the results of all positional values:

\( 2048 + 160 + 6 = 2214 \)

Step 5: Verify the calculation:

Rechecking the calculation confirms that the decimal equivalent of the hexadecimal number (8A6) is 2214.

Final Answer: The decimal equivalent of (8A6) is 2214. Hence, the correct option is Option 4.

Additional Information

To further analyze the other options:

Option 1: 1422

If we calculate incorrectly or misplace the powers of 16, we might arrive at this value. For example, if we mistakenly calculate \( 8 × 16^1 + A × 16^0 + 6 × 16^2 \), we would get a wrong result. However, this is not the correct approach for hexadecimal to decimal conversion.

Option 2: 1242

This value could result from a misinterpretation of the hexadecimal digits or incorrect multiplication of the powers of 16. Double-checking the procedure rules out this option.

Option 3: 2122

While close to the correct answer, this value might arise from an addition error or a slight miscalculation of one of the positional terms. For instance, miscalculating \( 8 × 16^2 \) or \( A × 16^1 \) could lead to this value.

Option 5: No value provided

This option is invalid as it does not represent a potential decimal equivalent of the given hexadecimal number.

Conclusion:

Understanding the conversion of hexadecimal numbers to decimal requires careful application of positional values and powers of 16. The correct decimal equivalent of the hexadecimal number (8A6) is 2214. This value matches Option 4, confirming its correctness.

Explanation:

The Radix of a Binary Number

Definition: In numerical systems, the term "radix" refers to the base of a number system. It is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, the decimal system has a radix of 10 because it uses 10 unique digits (0 through 9).

Radix of Binary Numbers: The binary number system is a numeral system that uses only two unique digits, 0 and 1. Hence, the radix of the binary number system is 2. Binary is the foundation of all modern computing systems and digital electronics because it aligns well with the two-state nature of electronic components (on and off).

Understanding Binary Numbers:

Binary numbers represent values using two symbols: 0 and 1. Each digit in a binary number is referred to as a "bit" (binary digit). The value of each position in a binary number is a power of 2, starting from the rightmost digit (least significant bit, or LSB) and increasing to the leftmost digit (most significant bit, or MSB). For example:

• The binary number 101 represents the decimal value: (1 × 2²) + (0 × 2¹) + (1 × 2⁰) = 4 + 0 + 1 = 5.

Practical Applications:

• The binary system is the basis of computer programming and hardware design. Computers process data in binary because it is easier to implement in digital circuits, where only two states are needed: on (1) and off (0).
• Binary is also used in data transmission, coding systems, and digital communication protocols.

Correct Option Analysis:

The correct option is:

Option 3: 2

This option correctly identifies the radix of a binary number system, which is 2. Binary numbers use only two unique symbols (0 and 1), making the radix or base of the system equal to 2.

Important Information

To further understand the analysis, let’s evaluate the other options:

Option 1: 1

This option is incorrect because a radix of 1 would mean the numeral system uses only one unique digit. Such a system is not practical for representing numbers, as it would fail to distinguish between different values.

Option 2: 0

This option is incorrect because a radix of 0 is not defined or meaningful in numeral systems. A base of 0 would imply no symbols are available for representing numbers, which is impossible.

Option 4: 8

This option is incorrect because a radix of 8 corresponds to the octal numeral system, not the binary system. The octal system uses eight unique digits (0 through 7) to represent numbers.

Option 5: No answer provided

While this option is left blank, the correct answer remains option 3, as explained above.

Conclusion:

The radix of the binary number system is 2, reflecting the two unique digits (0 and 1) used to represent numbers. Understanding the concept of radix is crucial for working with various numeral systems, including binary, decimal, octal, and hexadecimal. This knowledge is fundamental in computing, electronics, and digital communication, where binary plays a pivotal role.

Explanation:

Two's Complement Representation

Definition: Two's complement is a mathematical operation used in computer science to represent signed integers in binary. It is widely used because of its simplicity in handling arithmetic operations and representing negative numbers. In this system, positive numbers are represented in standard binary form, while negative numbers are obtained by inverting all the bits of the absolute value (in binary) and adding 1 to the least significant bit (LSB).

Steps to Find Two's Complement:

1. Start with the binary representation of the absolute value: Represent the number in binary. For example, the absolute value of -8 is 8, which in binary (for a 4-bit system) is 1000.
2. Invert all the bits: Flip all the bits of the binary representation. For 1000, the inverted bits become 0111.
3. Add 1 to the result: Adding 1 to the inverted binary number gives 0111 + 1 = 1000.

Thus, the two's complement representation of -8 (in a 4-bit system) is 1000.

Correct Option Analysis:

The correct option is:

Option 1: 1000

This is the correct two's complement representation of -8 in a 4-bit system. Following the steps outlined above, the binary value 1000 is derived by inverting the bits of the binary representation of 8 and adding 1.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: 1010

This option is incorrect because 1010 is not derived using the two's complement method for -8 in a 4-bit binary system. The inverted bits of 1000 are 0111, and adding 1 results in 1000, not 1010.

Option 3: 110

This option is incorrect because 110 does not represent -8 in two's complement form within a 4-bit binary system. It neither matches the expected binary format nor adheres to the two's complement operation rules.

Option 4: 1110

This option is incorrect as it does not represent the two's complement of -8 in a 4-bit binary system. The correct result, as derived earlier, is 1000. 1110 represents a different value altogether in two's complement.

Conclusion:

Understanding two's complement representation is essential for dealing with signed integers in binary arithmetic. The process involves inverting the bits of the absolute value and adding 1 to the least significant bit. This method simplifies arithmetic operations and provides a consistent way to handle negative numbers in binary systems.

In the context of the given problem, -8 is correctly represented as 1000 in two's complement form for a 4-bit binary system. This representation is derived systematically by following the steps outlined above, ensuring accuracy in binary arithmetic operations and signed integer representation.

The correct answer is option 4: 0101

Key Points

• Gray Code is a binary numeral system where two successive values differ in only one bit (also known as reflected binary code).
• To convert binary to Gray Code:
  • Keep the most significant bit (MSB) the same.
  • For the remaining bits: each bit = previous binary bit XOR current binary bit.

Step-by-Step Conversion:

• Decimal 7 in binary = 0111
• Now convert 0111 to Gray Code:
  • First bit (MSB) = 0
  • Second bit = 1 XOR 0 = 1
  • Third bit = 1 XOR 1 = 0
  • Fourth bit = 1 XOR 1 = 0
• Gray Code = 0101

Additional Information

• Gray Code is used in error correction and minimizing transitions in digital circuits.
• Binary to Gray Code formula: Gray[i] = Binary[i] XOR Binary[i-1]

Hence, the correct answer is: option 4: 0101

The correct answer is Four.

Key Points

• Each hexadecimal digit can be represented by 4 binary digits (bits).
• The hexadecimal system is base-16, which includes the digits 0-9 and the letters A-F.
• Binary system is base-2, which includes only the digits 0 and 1.
• To convert a hexadecimal digit to binary, each hex digit is mapped to a 4-bit binary equivalent.

Important Points

• The hexadecimal numeral system is widely used in computing and digital electronics because it is more compact than binary.
• One hexadecimal digit represents four binary digits (bits), making it easier to read and interpret large binary numbers.

1 0 0 1 0 0 0 0
-  1 1 1 1 0 0 1
0 0 0 1 0 1 1 1

Step 1: 0 – 1 = Borrow to make 10 – 1 = 1.
Step 2: 1 – 0 = 1.
Step 3: 1 – 0 = 1.
Step 4: 1 – 1 = 0.
Step 5: 0 – 1 = Borrow to make 10 – 1 = 1.
Step 6: 1 – 0 = 1.
Step 7: 1 – 0 = 1.

Remember: When zero takes 1 as carry from its left side number, '0' will become '10' which is equal to '2' (2-1=1) and if that '10' further gives carry then it will become '1' not '0'.

Solution:

The binary representation of 257 is 100000001

∴ The total number of the binary digit in 257 is 9

The correct answer is 'option 2'

Concept

Divide 127 by 2. Use the integer quotient obtained in this step as the dividend for the next step. Repeat the process until the quotient becomes 0.

Solution:

Write the remainder from bottom to top i.e. in the reverse chronological order.

This will give the binary equivalent of 127. 

Therefore, the binary equivalent of decimal number 127 is 1111111.

The correct answer is option 1): 45

Concept:

To convert the binary number 101101 to decimal, follow these two steps:

• Start from one's place in 101101: multiply one place with 2^0, tens place with 2^1, hundreds place with 2^2 and so on from right to left
• Add all the products we got from step 1 to get the decimal equivalent of 101101. Using the above steps, here is the work involved in the solution for converting 101101 to a decimal number (Don't forget that we start from one place to so on...)
  • Decimal equivalent of "1" = 1 × 2^0 = 1
  • Decimal equivalent of "0" = 0 × 2^1 = 0
  • Decimal equivalent of "1" = 1 × 2^2 = 4
  • Decimal equivalent of "1" = 1 × 2^3 = 8
  • Decimal equivalent of "0" = 0 × 2^4 = 0
  • Decimal equivalent of "1" = 1 × 2^5 = 32
  • The decimal equivalent of "101101" =   45
  • Here is the final answer, The binary number 101101₂ converted to decimal is therefore equal 45₁₀

Step 1: Divide (21)10 successively by 2 until the quotient is 0.

21/2 = 10, remainder is 1

10/2 = 5, remainder is 0

5/2 = 2, remainder is 1

2/2 = 1, remainder is 0

1/2 = 0, remainder is 1

Step 2: Read from bottom (MS2) to top (LS2) as 10101

This is the binary equivalent of decimal number 21

Step 3:Binary equivalent of 0.125 is, multiplying by 2 until we get 1 and writing down the integer after each multiplication,

⇒ 0.125 × 2 = 0.25

⇒ 0.25 × 2 = 0.5

⇒ 0.5 × 2 = 1

⇒ Binary equivalent of 0.125 = 001

∴ The binary code of (21.125)₁₀ is,

\({\left( {21.125} \right)_{10}} = 10101.001\)

Tips and Tricks:

Steps to writing 2’s complement to any binary number:

• Start from right to left and search for the first ‘1’
• Write down the bits until that first ‘1’ as it is.
• Write down the remaining left bits with their respective complement.

Example: Let the given number is: 100100

The 2’s complement is as shown.

Given number is 1010101

Binary Coded Decimal (BCD) code:

• BCD is a way to express each of the decimal digits with a binary code.
• In this code, each decimal digit is represented by its 4-bit binary equivalent.
• Also, with four bits we can represent sixteen numbers (0000 to 1111)
• But as there are 10 decimal digits from 0 to 9, BCD code uses only the first ten of these (0000 to 1001). The remaining six code combinations i.e. 1010 to 1111 are invalid in BCD.

Mistake Points In the question it is asking that " not an invalid BCD code" here, not invalid means valid only. so among the given options, 1001 is a valid BCD Code

The given hexadecimal number is: 236

To convert the given Hexadecimal number, first, we need to convert it into binary and then into octal.

Hexadecimal to binary: 0010 0011 0110

Binary to octal: To convert a binary number into octal, we need to make a group of three digits from right to left before the decimal pointer and left to right after the decimal pointer.

= 001 000 110 110

= 1066 

Concept:

1's complement of Binary: 1's complement of a Binary number is defined by the value obtained by inverting all the bit, i.e, 0 as 1 and 1 as 0.

2's complement of Binary: It is the sum of 1's complement of Binary number and 1 to the least significant bit (LSB).

∴ 2's complement = 1's complement + 1 (LSB)

Analysis:

The 1's complement of the given binary digit will be:

11011111 → 00100000

Adding 1 to the above, we get the 2's complement as:

00100000 → 00100001

Shortcut Trick

Steps to writing 2’s complement to any binary number:

• Start from right to left and search for the first ‘1’
• Write down the bits until that first ‘1’ as it is.
• Write down the remaining left bits with their respective complement.

The correct answer is 10000001

Additional Information

• In BCD each decimal digit is represented by a 4-bit binary number.
• The binary representation of 8 → 1000
• The binary representation of 1 → 0001
• (81)₁₀ = 10000001