Hexadecimal Number System MCQ Quiz - Objective Question with Answer for Hexadecimal Number System - 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.

The correct answer is Hexadecimal number system.

Additional Information

• Hexadecimal numbers are used extensively in microprocessor work. 
• The hexadecimal number system has a base of 16.
• After reaching 9 in the hexadecimal system, we continue as A, B, C, D, E, F. 
• For converting a decimal number to a hexadecimal number, the number is successively divided by 16 with remainders occupying the successive positions from the right.

The procedure is exactly similar to the procedure for converting a decimal number to binary. 

For example: N = A_n Bⁿ + A_{n − 1} B^{n − 1} + . . . + A₁ B¹ + A₀ B⁰ . . .  

where, N = number, B = base, A_n = (n + 1)^th digit in that base. 

Converting hexadecimal to the decimal.

Let hexadecimal number =11

So, N = 1*161 + 1*160 = 1*16 + 1*1 =16 +1 = 17

The decimal number 11 is smaller than the hexadecimal number 11.

The correct answer is option 2

Concept:

For conversion of hexadecimal to binary, each hexadecimal digit takes 4 binary bits.

(FA)16 = (1111 1010)2

Additional Information

Concept:

Hexadecimal

• The hexadecimal number system has a base of 16.
• After reaching 9 in the hexadecimal system, we continue as A, B, C, D, E, F.

Additional Information

Octal to Binary Conversion:

• Every octal number is represented by a 3-bit binary equivalent.
• The table of binary representation for octal numbers (0 to 7) is given below:

• Since one hex digit can represent 4 binary bits, it will take 5 hex digits to represent the 20-bit address of a memory location.

• To find the 5-digit hexadecimal address of the 500th memory location, we need to convert the decimal value of 500 to hexadecimal.
• Given that the computer system uses a 20-bit address code to identify over 1 million memory locations, we can determine the range of addresses using the following formula:
• Number of memory locations = 2^number of bits
• In this case, the number of memory locations is over 1 million, which is 1,000,000. Therefore: 2^20 = 1,048,576
• Since the 500th memory location falls within this range, we can proceed with the conversion.
• Converting decimal 500 to hexadecimal, we have: 500 = 1F4
• Therefore, the 5-digit hexadecimal address of the 500th memory location is: 001F4
• The hexadecimal equivalent is hence, (001F4)₁₆
• The 5-digit hexadecimal address of the 500th memory location is 001F4 -1 = 001F3.

Concept:

Hexadecimal number: In this, value of the base is 16. Each digit is represented by 4-bit binary no.

Octal number: For octal number, value of base is 8. Each digit of an octal number is represented by 3-bit binary no.

Explanation:

Octal number = 657

Binary representation for this number (each digit of a octal number is converted into 3 binary bits) 

So, 657 in binary is equivalent to 110 101 111

Now group this binary number into 4 bits starting from right to left. 

i.e. 0001 1010 1111

Hexadecimal representation for this number is : 1AF

The correct answer is The system uses 10 numerical digits and 6 alphabets –0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, A, B, C, D, E, F.

 Key Points

• The Hexadecimal numbering system uses the Base of 16 system and is a popular choice for representing long binary values because their format is quite compact and much easier to understand.
• Being a Base-16 system, the hexadecimal numbering system, therefore, uses 16 (sixteen) different digits with a combination of numbers from 0 through to 15. In other words, there are 16 possible digit symbols.
• However, there is a potential problem with using this method of digit notation caused by the fact that the decimal numerals of 10, 11, 12, 13, 14 and 15 are normally written using two adjacent symbols.
• To get around this tricky problem hexadecimal numbers that identify the values of ten, eleven, twelve, thirteen, fourteen, and fifteen are replaced with capital letters of A, B, C, D, E and F respectively.

Additional Information

• A number system that corresponds only to two numerical digits i.e., 0 and 1 is known as a binary number system.
  • As it is a 2-digit system thus known as a base 2 system.
  • The operation of a computer system solely relies on the binary number system.
  • As it only understands binary values i.e., 0 and 1.
  • The two binary digits 0 and 1 are known as bits.
  • A bit is considered the primary unit of information.
• In the Decimal number system, the digits 0 to 9 represents numbers.
  • As it uses 10 digits to represent a number, it is also called the base 10 number system.
  • Each digit has a value based on its position called place value.

The correct answer is Hexadecimal number system.

Additional Information

• Hexadecimal numbers are used extensively in microprocessor work. 
• The hexadecimal number system has a base of 16.
• After reaching 9 in the hexadecimal system, we continue as A, B, C, D, E, F. 
• For converting a decimal number to a hexadecimal number, the number is successively divided by 16 with remainders occupying the successive positions from the right.

The procedure is exactly similar to the procedure for converting a decimal number to binary. 

For example: N = A_n Bⁿ + A_{n − 1} B^{n − 1} + . . . + A₁ B¹ + A₀ B⁰ . . .  

where, N = number, B = base, A_n = (n + 1)^th digit in that base. 

Converting hexadecimal to the decimal.

Let hexadecimal number =11

So, N = 1*161 + 1*160 = 1*16 + 1*1 =16 +1 = 17

The decimal number 11 is smaller than the hexadecimal number 11.

Conversion: 

Hexadecimal number to decimal number

(C3B1)16 = (C × 16³) + (3 × 16²) + (B × 16) + (1 × 160)

(C3B1)16 = (12 × 163) + (3 × 162) + (11 × 16) + (1 × 160)

(C3B1)16 = (50097)10

Binary to Hexadecimal conversion table:

Convert Decimal to Hex:

• Decimal to hexadecimal conversion can be achieved by applying the repeated division and remainder algorithm.
• Decimal to Hex step by step Method:
• If the given decimal number is less than 16, the hex equivalent is the same. Remember that the letters A, B, C, D, E, and F are used for the values 10, 11, 12, 13, 14, and 15, convert accordingly
• For example, the decimal number 15 will be F in Hex.
• If the given decimal number is 16 or greater, divide the number by 16.
• Write down the remainder.
• Divide the part before the decimal point of your quotient by 16 again. Write down the remainder.
• Continue this process of dividing by 16 and noting the remainders until the last decimal digit you are left with is less than 16.
• When the last decimal digit is less than 16, the quotient will be less than 0 and the remainder will be the digit itself.
• The last remainder you get will be the most significant digit of Hex value while the first remainder from Step 3 is the least significant digit.
   

Solution:

So, (82)10 = (52)₁₆

(0.25)10  

⇒ 0.25 × 16 = 0.4

(0.25) 10  = (0.4)16

The correct answer is FFFF

Explanation:

• Here we have to perform an XOR operation on two Hexadecimal numbers.
• The boolean expression for XOR is A \(\bigoplus\) B = \(A\bar B\) + \(\bar AB\)
• The truth table for XOR is:

• Symbol	Truth Table
  2-input Ex-OR Gate	B	A	Q
  	0	0	0
  	0	1	1
  	1	0	1
  	1	1	0
  Boolean Expression Q = A ⊕ B	A OR B but NOT BOTH gives Q

• Means A and B are the expressions given to us in Hex code. So, we need to find the binary equivalent of the Hexadecimal numbers. This is because these codes will be executed in 1 and 0 in the CPU.
• The following table is to be memorized in order to convert Hexadecimal to Binary:

• Hex	Binary
  0	0000
  1	0001
  2	0010
  3	0011
  4	0100
  5	0101
  6	0110
  7	0111
  8	1000
  9	1001
  A	1010
  B	1011
  C	1100
  D	1101
  E	1110
  F	1111

• Therefore, (4AC0)₁₆ = 0100101011000000
• And (B53F)16 = 1011010100111111
• Now follow the Truth Table to perform the operation on every bit in XOR gate:
  • (4AC0)16   \(\bigoplus\)  (B53F)16  = 0100101011000000   \(\bigoplus\)   1011010100111111
  • 0100101011000000   \(\bigoplus\)   1011010100111111   =  1111111111111111
• So the result obtained is (1111111111111111)₂ which is the Binary Equivalent of (FFFF)₁₆ (using the table given above).
• Hence the correct answer is FFFF.

Concept: 

Hexa decimal division process:

Step 1: Convert the given hexadecimal numbers to decimal.

Step 2: perform division operation to the decimal numbers.

Step 3: Convert the result to requires number system.

Calculation:

Convert the given hexadecimal numbers to decimal and perform division operation and for the remainder obtained, convert that remainder into hexadecimal.

⇒ 37H = 3 × 16¹ + 7 × 16⁰ = (55)₁₀

⇒ 17H =  1 × 16¹ + 7 × 16⁰ = (23)₁₀

⇒ remainder of (55 / 23) = (9)₁₀

(9)10 in hexa decimal form is represented as shown

⇒ (9)10 = 0 × 161 + 9 × 160 = 09H

∴ The remainder is 09H

Note: From 0 - 9 vales of the both decimal and hexadecimal number systems are same.

Hexadecimal:

• Hexadecimal numbers are used extensively in microprocessor work. 
• The hexadecimal number system has a base of 16.
• After reaching 9 in the hexadecimal system, we continue as A, B, C, D, E, F.

Concept:-

Hexadecimal number system: Hexadecimal is a radix-16 or base-16 number system, while in our day-to-day life in computers and digital logic we use a base-2 (binary) number system, but it is visually more efficient to use base-16 (Hexadecimal) number system.

Decimal number system: The most common base-10 number system that we use frequently.

In general
(an × rn) +… (a1 × r1) + (a0 × r0) + (a-1 × r-1) +…….. (a-m × r-m)  -------(1)

Where an = Hexadecimal number at the nth place.

The above expansion of a number in any radix r, always gives back the decimal equivalent of the number.

Calculation:-

Given,

Hexadecimal number is (0101)16              

Expanding according to equation 1,

(0 × 163) + (1 × 162) + (0 × 161) + (1 × 160)

= (0 × 4096) + (1 × 256) + (0 × 16) + (1 × 1)

= (257)10

So, the decimal equivalent of the given hexadecimal number is (257)10

Conversion: 

Hexadecimal number to decimal number

(2F)₁₆ = (2 × 16) + (F × 160)

(2F)₁₆ = 2 × 16 + 15 × 160

(2F)₁₆ = (47)₁₀

Binary to Hexadecimal conversion table: