Binary Codes Conversion MCQ Quiz - Objective Question with Answer for Binary Codes Conversion - Download Free PDF

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.

Given: Binary code 1101

The procedure to convert a gray code to a binary one is as shown:

A3 → Copy MSB → 1

A2 → 1 ⊕ 1 → 0

A1 → 1 ⊕ 0 → 1

A0 → 0 ⊕ 1 → 1

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.

The Binary Number System: The binary number system with only two independent digits, 0 and 1, is a base-2 number system. All larger binary numbers are represented in terms of ‘0’ and ‘1’.

Decimal to Binary Conversion:

Step 1: Divide 10 by 2 to get the Quotient. Keep the Whole part for the next step and set the Remainder aside.

10 / 2 = 5 with 0 remainder

Step 2: Divide the Whole part of the Quotient from Step 1 by 2. Again, keep the Whole part and set the Remainder aside.

5/ 2 = 2 with 1 remainder

Step 3: Repeat Step 2 above until the Whole part is 0.

2 / 2 = 1 with 0 remainder

1 / 2 = 1 with 1 remainder

Step 4: Write down the Remainders in reverse order to get the answer to 10 as a binary.

When we put the remainders together in reverse order, we get the answer 1010

Hence, the binary equivalent of decimal number 10is 1010.

NOTE:

The decimal number system has a base of 10 as it has 10 independent digits, i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.  The octal and hexadecimal number systems have a radix (or base) of 8 and 16, respectively.

Concept:

A decoder is a multiple input and multiple output logic circuit which coverts coded inputs into coded outputs. There is one to one mapping from input code to output code words. A decoder which has n bit binary input code and a one activated output out of 2n output code is called binary decoder.

Explanation:

A device which converts BCD to seven segment display is called decoder. Seven segment displays are used to give a visual indication of output states of digital ICs. These outputs are in BCD form and not suitable for directly driving seven segment displays. For this reason, decoder are used. 

The correct answer is option 3): (0.1100)2

Concept:

Conversion of Fractional Decimal Numbers into Binary Numbers:

Fractional numbers can be converted to binary form by successive multiplication by 2.

In each step, the digit before the decimal point is transferred binary record and the process is repeated with the remaining fraction.

The last step is reached if the fraction part is zero or it is terminated when the desired accuracy is attained.

Calculation:

(0.7854)10

0.7854 × 2 = 1.5708  it gives 1

0.5708  × 2  = 1.1416 it gives 1

0.1416 × 2   = 0.2832 it gives 0

0.2832 × 2 = 0.5664 it gives 0

 (0.7854)10 is equal to  (0.1100)2

Solution:

The binary representation of 257 is 100000001

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

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,

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

A binary coded decimal code in which the combination for the complement of a digit is the complement of the combination for that digit.

Binary codes only consists of '0' and '1'. 

Hence option 2 is correct. 

The correct answer is option 3): (0.1100)2

Concept:

Conversion of Fractional Decimal Numbers into Binary Numbers:

Fractional numbers can be converted to binary form by successive multiplication by 2.

In each step, the digit before the decimal point is transferred binary record and the process is repeated with the remaining fraction.

The last step is reached if the fraction part is zero or it is terminated when the desired accuracy is attained.

Calculation:

(0.7854)10

0.7854 × 2 = 1.5708  it gives 1

0.5708  × 2  = 1.1416 it gives 1

0.1416 × 2   = 0.2832 it gives 0

0.2832 × 2 = 0.5664 it gives 0

 (0.7854)10 is equal to  (0.1100)2

CONCEPT:

• Following are the steps to convert decimal to octal:

1. If the provided decimal number is less than the number 8 then the octal number is the same.
2. If the provided decimal number is larger than 7 then divide it by number 8.
3. take a note of the remainder, received after division
4. Repeat step 2 and 3 with the quotient until it is less than 8
5. Now, note the remainders in the reverse order (bottom to top)
6. The resultant thus received is the equivalent octal number to the provided decimal number.
    

Explanation:

Hence, (177)10 = (261)8

• 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.

So, (10101)₁₀ = (2775)₁₆

Alternate Method:

The conversion from decimal to hexadecimal can also take place by first converting the decimal to binary, i.e.

Taking a pair of 4 and converting it into its hexadecimal equivalent, we get:

0010 0111 0111 0101 = (2775)₁₆

Given BCD number = 0001 0010 0110

BCD is converted to a decimal by taking a pair of 4 and representing it in its equivalent decimal, i.e.

The decimal equivalent of the given BCD will be:

Decimal equivalent = 126

Converting this to an equivalent binary through successive division, we get:

Decimal to Hexadecimal →

11 is represented as B in hexadecimal.

(43)₁₀ = (2B)₁₆

Concept:

A decoder is a multiple input and multiple output logic circuit which coverts coded inputs into coded outputs. There is one to one mapping from input code to output code words. A decoder which has n bit binary input code and a one activated output out of 2ⁿ output code is called binary decoder.

Explanation:

A device which converts BCD to seven segment display is called decoder. Seven segment displays are used to give a visual indication of output states of digital ICs. These outputs are in BCD form and not suitable for directly driving seven segment displays. For this reason, decoder are used. 

The correct answer is option '2'.

Concept: 

1. To convert the binary number into an octal number, we must first understand the relationship between binary and octal numbers.
2. To convert binary into octal, we need to use a 3-bit set or grouping.

Octal defines the number 8, so we can say that in the octal number system there is a total of 8 digits ranging from 0 to 7.

  From the table below we can convert binary into octal or octal into binary and vice-versa.

In an integral part of a binary number, we take a grouping of 3 from LSB (Least significant bits) to MSB (Most significant bits) and on the decimal side, we take a grouping of 3 from left to right.

i.e., (1101.1101)₂ can be written as (001101.110100)₂ and grouped as

(Here zeroes are added to complete the grouping of three)

From the above table, we can write the grouping as follows

• 001→1
• 101→5
• 110→6
• 100→4​

​So, the binary number (1101.1101)₂ converted into an octal number as (15.64)₈

Tips:

It can also be solved by converting binary into its decimal equivalent and then converting that decimal number into its Octal equivalent i.e.,

Binary→Decimal→Octal

Octal to Hexadecimal conversion:

For converting octal to hexadecimal, we need to first convert octal into binary equivalent and then by taking a group to 4 instead of taking 3 and follow the same as above.

For example, for converting the above binary into hexadecimal i.e., (1101.1101)₂   we need to take grouping as 

And its equivalent would be (1101)₂ → (13)₁₆

So hexadecimal equivalent of (1101.1101)₂ would be (13.13)₁₆