Types of Number System MCQ Quiz - Objective Question with Answer for Types of 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.

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.

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.

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

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'.

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

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:

Another number system to Decimal

In each and every representation of numbers with different bases, the maximum value in a number system with the base ‘r’ is r – 1. Since numbers vary from 0 to r – 1.

To convert any number which is in the different base to decimal number system we use binary-weighted representation.

Eg: let the number be ( abc⋯ ⋯ yz)r

Now to convert the above number into the decimal number system

a × rn-1 + b × rn-2 + ⋯ ⋯ + y × r1 + z × r0

If we convert all numbers into decimal then we can perform normal addition and subtraction etc.

Application:

Given:

(110)_x = (132)₄,

The decimal equivalent of this number will be:

1 x x² + 1 x x¹ + 0 x x⁰ = 1 x 4² + 3 x 4 + 2 x 4⁰

x² + x + 0 = 16 + 12 + 2

x² + x - 30 = 0 

On solving this quadratic equation we'll get:

x = 5, -6

Base can't be a negative so;

x = 5

The correct answer is 13.

• Step by step solution:
  • Step 1: Write down the binary number: 1101
  • Step 2: Multiply each digit of the binary number by the corresponding power of two: 1x2³ + 1x2² + 0x2¹ + 1x2⁰
  • Step 3: Solve the powers: 1x8 + 1x4 + 0x2 + 1x1 = 8 + 4 + 0 + 1
  • Step 4: Add up the numbers written above: 8 + 4 + 0 + 1 = 13
    So, (1101)₂ = (13)₁₀

Key Points

• (1101)₂ = (?)₈  (Binary to octal conversation)
  • Step 1: Write down the binary number (001101)₂, Group all the digits in sets of three starting from the LSB (far right). Add zeros to the left of the last digit if there aren't enough digits to make a set of three. (001 101)
  • Step 2: Use the table below to convert each set of three into an octal digit. In this case, 001=1, 101=5.
    So, the number 1101 in binary is equivalent to 15 in octal.
  • Table for binary to the octal conversation:

    BINARY	OCTAL
    000	0
    001	1
    010	2
    011	3
    100	4
    101	5
    110	6
    111	7

Decimal to Octal Conversion: If we want to change any octal number into decimal number we have to start multiplying the digits of the number from right hand side with increasing powers of 8 staring from 0 and finally summing up all the products.

Application:

Concept:

A number system includes the number of independent digits used in the number system (the base), the place values of the different digits constituting the number, and the maximum numbers that can be written with the given number of digits.

Octal numbers: These numbers use digits from 0 to 7, total of 8 digits, and hence they are called octal number system. Octal numbers have base 8.

Hexadecimal numbers: The numbers which have base 16. It uses 16 different digits to represent the numbers. It is denoted as h16, where h is a hexadecimal number. It may be a combination of alphabets and numbers. Thus, it includes numbers from 0 to 9 and alphabets A to F.

Calculation:

Hexadecimal to Octal Conversion:

Given, B3416 is a hexadecimal number.

B → 1011, 3 → 0011, 4 → 0100

1011 0011 0100

Now group them from right to left, each having 3 digits.

101, 100, 110, 100

101 → 5, 100 → 4, 110 → 6, 100 → 4

Hence, B3416 = 54648

Key Points

There is grouping of 3 from right to left since octal has base of 8, and 8 = 2³, i.e. power 3.

Concept:

The radix or base of a number system is defined as the number of different digits which can occur in each position in the number system

Application:

For octal number system base is 8 hence there are 8 different digits from (0 - 7)

Any number containing digit greater than 7 is invalid octal number.

Concept:

Decimal to binary:

• Take decimal number as dividend.

• Divide the number by 2.
• Get the integer quotient for the next iteration.

• Get the remainder (it will be either 0 or 1 because of divisor 2).
• Repeat the steps until the quotient is equal to 0
• Write the remainders in reverse order (which will be equivalent binary number of given decimal number).

Decimal to binary: (fractional part)

• Take decimal number as multiplicand.
• Multiple this number by 2 (2 is base of binary so multiplier here).
• Store the value of integer part of result in an array (it will be either 0 or 1 because of multiplier 2).
• Repeat the above two steps until the number became zero.
• Write these resultant integer part

Calculation:

Binary of  57:

Now, write remainder from bottom to up (in reverse order), this will be 111001 which is equivalent binary number of decimal integer 57.

Convert decimal fractional number 0.375 into binary number.

Here, decimal fraction: 0.375

Now, write these resultant integer part, this will be 0.0110 which is equivalent binary fractional number of decimal fractional 0.375.

∴ 57.375 can be written as 111001.011 in binary

Hence, option (1) is correct.