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

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.

The correct answer is 0 - 9, A - F.

Key Points

• The hexadecimal number system is a base-16 numeral system that uses 16 distinct symbols.
• These symbols include digits 0 to 9 and letters A to F, where A represents 10, B represents 11, and so on, up to F which represents 15.
• Hexadecimal is commonly used in computing and digital electronics as a more human-friendly representation of binary-coded values.
• This system is particularly useful for representing memory addresses, color codes in web design (e.g., #FFFFFF for white), and machine-level instructions.
• Hexadecimal is widely applied in programming languages such as HTML, CSS, and assembly language.

Additional Information

• Binary Number System:
  • The binary system is a base-2 numeral system that uses only two symbols: 0 and 1.
  • It is the foundation for all modern computer systems and digital circuits.
  • Every hexadecimal digit corresponds to exactly four binary digits, making conversions between the two systems straightforward.
• Decimal Number System:
  • The decimal system is a base-10 numeral system and uses digits 0 to 9.
  • This is the most commonly used number system in everyday life.
  • Decimal numbers can be converted into hexadecimal using division and remainder calculations.
• Octal Number System:
  • The octal system is a base-8 numeral system that uses digits 0 to 7.
  • It is sometimes used in computing as a shorthand for binary numbers.
  • Each octal digit corresponds to three binary digits.
• Applications of Hexadecimal:
  • Hexadecimal is widely used in memory addressing and debugging.
  • It is also employed in defining color codes for web design (e.g., #FF5733).
  • Microcontrollers and embedded systems often use hexadecimal to represent machine-level instructions.

Calculation:

To convert the binary number \((1100.1011)_2\) to:

The binary number \(1100_2 \) is equal to:

\(1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 8 + 4 + 0 + 0 = 12\)

Fractional part: \(.1011_2\)

The binary fractional part \(.1011_2\) can be converted by adding the powers of 2, starting from \(2^{-1}\):

\(1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} + 1 \times 2^{-4} = 0.5 + 0 + 0.125 + 0.0625 = 0.6875\)

Combine the integer and fractional parts:

\(12 + 0.6875 = 12.6875\)

Thus, the decimal equivalent of \((1100.1011)_2\)  is \(12.6875 \).

Thus, option '1' is correct.

Explanation:

To convert the decimal number 26.85 into its binary equivalent, we need to handle both the integer part (26) and the fractional part (0.85) separately.

Step 1: Convert the integer part (26) to binary

We start by dividing the integer part by 2 and recording the remainder for each division until the quotient becomes zero. The binary equivalent is then obtained by reading the remainders from bottom to top.

• 26 ÷ 2 = 13, remainder = 0
• 13 ÷ 2 = 6, remainder = 1
• 6 ÷ 2 = 3, remainder = 0
• 3 ÷ 2 = 1, remainder = 1
• 1 ÷ 2 = 0, remainder = 1

Reading the remainders from bottom to top, we get the binary equivalent of 26 as 11010.

Step 2: Convert the fractional part (0.85) to binary

To convert the fractional part, we multiply it by 2 and record the integer part of the result. We then repeat this process with the new fractional part until we achieve the desired precision or the fractional part becomes zero.

• 0.85 × 2 = 1.70, integer part = 1, fractional part = 0.70
• 0.70 × 2 = 1.40, integer part = 1, fractional part = 0.40
• 0.40 × 2 = 0.80, integer part = 0, fractional part = 0.80
• 0.80 × 2 = 1.60, integer part = 1, fractional part = 0.60
• 0.60 × 2 = 1.20, integer part = 1, fractional part = 0.20
• 0.20 × 2 = 0.40, integer part = 0, fractional part = 0.40
• 0.40 × 2 = 0.80, integer part = 0, fractional part = 0.80
• 0.80 × 2 = 1.60, integer part = 1, fractional part = 0.60

Repeating this process, we notice that the fractional part starts repeating after some iterations. Reading the integer parts from top to bottom, we get the binary equivalent of 0.85 as approximately 0.110110...

Combining both parts:

The binary equivalent of the integer part 26 is 11010 and the binary equivalent of the fractional part 0.85 is approximately 0.110110. Therefore, the binary equivalent of 26.85 is approximately 11010.110110.

The correct option is: 11010.110110

Explanation:

To convert the binary number 1101.1101 into its decimal equivalent, we need to understand the binary numbering system and how to convert binary fractions to decimal.

Step-by-Step Conversion:

1. **Separate the integer and fractional parts**: The given binary number is 1101.1101, where 1101 is the integer part and 1101 is the fractional part.

2. **Convert the integer part**: The integer part of the binary number is 1101. We convert this to decimal by multiplying each digit by 2 raised to the power of its position, starting from 0 on the right.

Integer Part Conversion:

• 1 × 2³ = 1 × 8 = 8
• 1 × 2² = 1 × 4 = 4
• 0 × 2¹ = 0 × 2 = 0
• 1 × 2⁰ = 1 × 1 = 1

Adding these values together, we get:

8 + 4 + 0 + 1 = 13

So, the integer part 1101 in binary is 13 in decimal.

3. **Convert the fractional part**: The fractional part of the binary number is 1101. We convert this to decimal by multiplying each digit by 2 raised to the negative power of its position, starting from -1 on the left.

Fractional Part Conversion:

• 1 × 2^-1 = 1 × 0.5 = 0.5
• 1 × 2^-2 = 1 × 0.25 = 0.25
• 0 × 2^-3 = 0 × 0.125 = 0
• 1 × 2^-4 = 1 × 0.0625 = 0.0625

Adding these values together, we get:

0.5 + 0.25 + 0 + 0.0625 = 0.8125

So, the fractional part 1101 in binary is 0.8125 in decimal.

4. **Combine the integer and fractional parts**: Now that we have both parts converted, we combine them to get the final decimal number.

13 (integer part) + 0.8125 (fractional part) = 13.8125

Therefore, the decimal equivalent of the binary number 1101.1101 is 13.8125.

Conclusion:

The correct option is:

Option 2: 13.8125

Additional Information

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

Option 1: 12.8125

This option is incorrect. The value 12.8125 would imply an error in either the integer or fractional conversion steps. In this case, the decimal equivalent of the given binary number (1101.1101) does not match 12.8125.

Option 3: 12.625

This option is also incorrect. The fractional part conversion to 0.8125 means that a value of 12.625 is not the correct representation of the binary number 1101.1101.

Option 4: 13.625

This option is close but still incorrect. The correct fractional part conversion of 1101 to 0.8125 means that 13.625 is not the accurate decimal equivalent of 1101.1101.

Conclusion:

Understanding the process of converting binary numbers to their decimal equivalents involves careful handling of both integer and fractional parts separately. The correct decimal equivalent of the binary number 1101.1101 is 13.8125, as accurately detailed in the steps above.

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