All questions of Number System & Representation for Electrical Engineering (EE) Exam

Understanding 1's Complement
1's complement is a method for representing negative binary numbers. To find the 1's complement of a binary number, you simply invert all the bits: change 0s to 1s and 1s to 0s.
Given Number
The number provided is -011012. Here, the 'minus' sign indicates that we need to find the 1's complement of the positive binary equivalent of the number.
Identify the Binary Equivalent
- The number is given in binary as 01101.
- The first step is to identify the bits:
- 0 → 0
- 1 → 1
- 1 → 1
- 0 → 0
- 1 → 1
- Thus, 01101 is the binary representation of the number 13 in decimal.
Calculating 1's Complement
To find the 1's complement, we invert each bit of 01101:
- 0 → 1
- 1 → 0
- 1 → 0
- 0 → 1
- 1 → 0
Thus, the 1's complement of 01101 is 10010.
Final Representation
Since the original number was negative (-01101), its 1's complement is represented in the binary system as 10010.
Options Analysis
Now, let's examine the options provided:
- a) 01010
- b) 10011
- c) 10010
- d) 01011
From the analysis, the correct answer is option C: 10010.
This confirms that the 1's complement of -01101 is indeed 10010.

Conversion to Octal

To convert a decimal number to octal, we need to divide the decimal number by 8 successively and note down the remainders at each step. The octal equivalent will be the remainders read in reverse order.

Calculation

Starting with the decimal number (432267)10:

Step 1: 432267 ÷ 8 = 54033 with a remainder of 3
Step 2: 54033 ÷ 8 = 6754 with a remainder of 1
Step 3: 6754 ÷ 8 = 844 with a remainder of 2
Step 4: 844 ÷ 8 = 105 with a remainder of 4
Step 5: 105 ÷ 8 = 13 with a remainder of 1
Step 6: 13 ÷ 8 = 1 with a remainder of 5
Step 7: 1 ÷ 8 = 0 with a remainder of 1

Octal Equivalent

Reading the remainders in reverse order, we get the octal equivalent as (1514213)8.

Therefore, the octal equivalent of the decimal number (432267)10 is not provided in the options given.

Octal Equivalent of Hexadecimal Number (FB2)16

To find the octal equivalent of a hexadecimal number, we need to convert each hexadecimal digit into its equivalent binary representation and then group the binary digits into groups of three from right to left. Finally, we convert each group of three binary digits into its equivalent octal digit.

Step 1: Convert Hexadecimal to Binary
To convert the hexadecimal number (FB2)16 to binary, we can use the following table:

Hexadecimal	Binary
F	1111
B	1011
2	0010

So, (FB2)16 in binary is 111110110010.

Step 2: Group Binary Digits
Starting from the right, group the binary digits into groups of three:
1 111 101 100 10

Step 3: Convert Binary to Octal
Now, we convert each group of three binary digits into its equivalent octal digit:
1 7 5 4 2

Therefore, the octal equivalent of (FB2)16 is (17542)8.

Conclusion
The correct answer is option 'B' (7662)8.

Conversion of Octal to Decimal
To convert an octal number to a decimal number, each digit of the octal number is multiplied by 8 raised to the power of its position from right to left, starting from 0.

Given Octal Number: (145)₈
- The octal number (145)₈ can be expanded as (1 * 8²) + (4 * 8¹) + (5 * 8⁰).
- Calculating the values, we get (1 * 64) + (4 * 8) + (5 * 1) = 64 + 32 + 5 = 101.

Therefore, the decimal equivalent of (145)₈ is (101)₁₀.

To find the sum of X and Y represented in 2's complement format, we need to follow these steps:

1. Convert the binary representation of X and Y to their decimal equivalent:
- X = 00110 = 2^1 + 2^2 = 6
- Y = 10011 = -2^0 + 2^1 + 2^4 = -1 + 2 + 16 = 17

2. Add the decimal values of X and Y:
- 6 + 17 = 23

3. Convert the decimal sum back to binary representation:

- If the sum is positive (greater than or equal to 0), convert it to binary normally:
- 23 = 2^4 + 2^3 + 2^2 + 2^1 + 2^0 = 16 + 8 + 4 + 2 + 1 = 10111

- If the sum is negative (less than 0), convert the absolute value to binary and take the 2's complement:
- Absolute value of -23 = 23 = 10111
- Take the 2's complement by inverting all the bits and adding 1:
- Inverting: 10111 -> 01000
- Adding 1: 01000 + 1 = 01001

4. If the binary representation of the sum has more bits than the desired number of bits, truncate the most significant bits until the desired number of bits is reached. In this case, the desired number of bits is 5, so we can keep only the last 5 bits.

Therefore, the sum of X and Y represented in 2's complement format using 5 bits is 11001.

The number of 1 can refer to different things depending on the context. It could refer to:

- The quantity of the digit 1: In this case, the number of 1 would simply mean the count or total number of occurrences of the digit 1.

- The value of 1: This could refer to a specific number or quantity represented by the digit 1. For example, if we're talking about a specific number like 111, then the number of 1 would be 3.

- A placeholder or position in a number system: In a positional number system like decimal or binary, the number of 1 in a specific position represents its value. For example, in the decimal number 123, the number of 1 in the hundreds place is 1.

Without further information or context, it is not possible to determine the specific meaning of "the number of 1."

BCD stands for Binary Coded Decimal. It is a binary representation of decimal numbers where each decimal digit is represented by a four-bit binary code.

To convert the decimal number 10 into its BCD form, we need to represent each digit (1 and 0) in binary using four bits each.

- Converting the first digit 1 into binary:
- The binary representation of 1 is 0001.

- Converting the second digit 0 into binary:
- The binary representation of 0 is 0000.

Therefore, the BCD representation of the decimal number 10 is obtained by concatenating the binary representations of each digit:

BCD representation = 0001 0000

So, the correct answer is option C: 00010000.

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When adding -33 and -40 using 2's complement, we follow these steps:

1. Convert -33 and -40 to their 2's complement representation.
-33 in 2's complement: 00100001
-40 in 2's complement: 00101000

2. Add the two numbers together, including any carry from the previous bit.
00100001
+ 00101000
__________
01001001

3. Check if the result is negative by looking at the leftmost bit. If it is 1, the result is negative.

In this case, the result is 01001001, which is positive. Therefore, the addition of -33 and -40 using 2's complement is 73.

A three-digit decimal number can range from 100 to 999. In the conventional BCD (Binary Coded Decimal) format, each decimal digit is represented by a 4-bit binary code.

Let's break down the representation of a three-digit decimal number in the BCD format:

- The first digit can have values from 1 to 9. In the BCD format, it requires 4 bits to represent each value. So, for the first digit, we need 4 bits.

- The second and third digits can have values from 0 to 9. Again, each digit requires 4 bits to represent each value. So, for the second and third digits, we need 4 bits each.

To find the total number of bits required for the representation of a three-digit decimal number, we add up the bits required for each digit.

- Bits required for the first digit: 4 bits
- Bits required for the second digit: 4 bits
- Bits required for the third digit: 4 bits

Adding them up, we get a total of 4 + 4 + 4 = 12 bits.

Therefore, a three-digit decimal number requires 12 bits for representation in the conventional BCD format.

Understanding the Conversion of Decimal to Binary
To convert a decimal number into binary, we need to understand the binary number system, which is base 2, using only the digits 0 and 1. Each binary digit represents a power of 2, starting from the rightmost digit (which represents 2^0).
Step-by-Step Conversion of 127 to Binary
1. Identify the Largest Power of 2
The largest power of 2 less than or equal to 127 is 2^6, which is 64.
2. Subtract and Find Remaining Value
- 127 - 64 = 63
- The next largest power of 2 less than or equal to 63 is 2^5 (32).
- 63 - 32 = 31
- Next, 2^4 (16) gives 31 - 16 = 15.
- Next, 2^3 (8) gives 15 - 8 = 7.
- Next, 2^2 (4) gives 7 - 4 = 3.
- Next, 2^1 (2) gives 3 - 2 = 1.
- Finally, 2^0 (1) gives 1 - 1 = 0.
3. Write the Binary Representation
Since we used powers of 2 from 2^0 to 2^6, we can represent 127 in binary as:
- 2^6 (1), 2^5 (1), 2^4 (1), 2^3 (1), 2^2 (1), 2^1 (1), 2^0 (1) → 1111111
Final Answer Evaluation
Among the options provided:
- a) 1100111: This is 103 in decimal.
- b) 1111111: This is 127 in decimal (Correct Answer).
- c) 1111011: This is 123 in decimal.
- d) 111111: This is 63 in decimal.
Thus, option 'b' (1111111) is the accurate binary representation of the decimal number 127.

Octal and Hexadecimal Number Systems

In the field of computer science and digital electronics, different number systems are used to represent numbers. Two commonly used number systems are octal and hexadecimal.

Octal Number System:
The octal number system is a base-8 system, meaning it uses 8 digits (0-7) to represent numbers. Each digit in the octal system represents a power of 8.

Hexadecimal Number System:
The hexadecimal number system is a base-16 system, meaning it uses 16 digits (0-9 and A-F) to represent numbers. Each digit in the hexadecimal system represents a power of 16.

Conversion from Hexadecimal to Octal:

To convert a hexadecimal number to its octal equivalent, we need to follow these steps:

1. Group the hexadecimal number into sets of 3 digits from right to left.
- If the number of digits is not a multiple of 3, add leading zeros to form complete sets of 3 digits.
2. Write down the octal equivalent of each group of 3 digits.
- Use the following table to find the octal equivalent of each hexadecimal digit:

Hexadecimal | Octal
------------|------
0 | 0
1 | 1
2 | 2
3 | 3
4 | 4
5 | 5
6 | 6
7 | 7
8 | 10
9 | 11
A | 12
B | 13
C | 14
D | 15
E | 16
F | 17

3. Combine the octal equivalents of each group to get the final octal equivalent.

Conversion of Hexadecimal 100 to Octal:

In this case, the hexadecimal number is 100. Let's convert it to octal:

1. Group the digits into sets of 3: 1|00.
2. Find the octal equivalent of each group: 1 -> 1, 00 -> 0.
3. Combine the octal equivalents: 10.
4. The octal equivalent of hexadecimal 100 is 10.

So, the correct answer is option A: 400.

2's complement of 11001011
To find the 2's complement of a binary number, we have to first find the 1's complement (flipping all the bits) and then add 1 to the result.
- 1's complement of 11001011
- The 1's complement of 11001011 is obtained by flipping all the bits: 00110100
- Adding 1 to the 1's complement
- To get the 2's complement, we add 1 to the 1's complement:
00110100
+ 00000001
00110101
Therefore, the 2's complement of 11001011 is 00110101, which corresponds to option C.

Understanding Binary to Octal Conversion
To convert the binary number 10101110 to octal, follow these steps:
Step 1: Group the Binary Digits
- Start from the right and group the binary digits into sets of three.
- If necessary, add leading zeros to the leftmost group to make it a full set of three.
For 10101110, we group it as follows:
- 010 101 110
Step 2: Convert Each Group to Decimal
- Now, convert each group of three binary digits to its octal equivalent.
- 010 in binary = 2 in decimal
- 101 in binary = 5 in decimal
- 110 in binary = 6 in decimal
Step 3: Combine the Octal Digits
- Combine the decimal values obtained from each group to form the final octal number.
- Therefore, the octal representation is 256.
Final Result
- The octal equivalent of the binary number 10101110 is 256.
Conclusion
- The correct answer for the conversion of the binary number 10101110 to octal is option 'C' - 256.

Understanding Octal Conversion
To convert the decimal number 149 into octal, we follow a systematic process of dividing the number by 8 (the base of octal) and keeping track of the remainders.
Step-by-Step Conversion
1. Divide 149 by 8:
- 149 ÷ 8 = 18 with a remainder of 5.
- Record the remainder (5).
2. Divide the quotient (18) by 8:
- 18 ÷ 8 = 2 with a remainder of 2.
- Record the remainder (2).
3. Divide the next quotient (2) by 8:
- 2 ÷ 8 = 0 with a remainder of 2.
- Record the remainder (2).
Collecting Remainders
- Now we have the remainders in reverse order:
- From the last division to the first: 2, 2, 5.
Final Octal Representation
- Reading the remainders from the last division to the first gives us:
- 225 in octal.
Conclusion
Thus, the decimal number 149 is represented as 225 in octal code. Therefore, the correct answer is option 'D'.

Logic XOR of hexadecimal numbers

XOR or exclusive OR is a logical operation that outputs true only when the two binary inputs are different. XOR operation is represented by the symbol ⊕. When used with hexadecimal numbers, the XOR operation is performed on each pair of corresponding bits.

Given hexadecimal numbers (4AC0)16 and (B53F)16, the XOR operation can be performed as follows:

4AC0 = 0100 1010 1100 0000
B53F = 1011 0101 0011 1111
XOR = 1111 1111 1111 1111

The result of the XOR operation is (FFFF)16, which is option C.

Explanation

- XOR operation: The XOR operation is a binary operation that takes two binary inputs and outputs true only if the inputs are different. The XOR operation is represented by the symbol ⊕.
- Conversion to binary: To perform the XOR operation on hexadecimal numbers, we need to convert them to binary. Each hexadecimal digit represents four binary digits or bits. For example, (4AC0)16 is equivalent to (0100 1010 1100 0000)2 in binary.
- Performing XOR: Once the numbers are in binary form, we can perform the XOR operation on each pair of corresponding bits. If both bits are the same, the result is 0. If the bits are different, the result is 1.
- Result: After performing the XOR operation on each pair of bits, we get the binary result of (1111 1111 1111 1111)2, which is equivalent to (FFFF)16 in hexadecimal form.

Conclusion

The logic XOR operation of (4AC0)16 and (B53F)16 results in (FFFF)16. This result is obtained by converting the hexadecimal numbers to binary, performing the XOR operation on each pair of corresponding bits, and converting the binary result back to hexadecimal form.

To determine the number of bits required to encode decimal numbers from 0 to 9999 in straight binary codes, we need to consider the range of values and calculate the number of bits needed to represent the largest value in that range.

Range of Values:
The range of decimal numbers is from 0 to 9999.

Calculation:
To calculate the number of bits required to represent a decimal number in binary, we use the formula log2(n), where n is the decimal number.

For the largest number in the range, which is 9999, we calculate the number of bits required as follows:

Number of bits = log2(9999)

Using the logarithmic identity log2(x) = log10(x) / log10(2), we can rewrite the calculation as:

Number of bits = log10(9999) / log10(2)

Using a scientific calculator or logarithmic tables, we find:

Number of bits ≈ 13.29

Answer:
The number of bits required to encode decimal numbers 0 to 9999 in straight binary codes is approximately 13.29. Since we cannot have fractional bits, we round up to the nearest whole number, which is 14.

Therefore, the correct answer is option 'B' - 14.

To subtract (001100)2 from (101001)2 using base 2, we can set up the subtraction as follows:

101001 (top number)
- 001100 (bottom number)
____________

Starting from the rightmost digit, we subtract 0 from 1, which gives us 1. We then subtract 0 from 0, which gives us 0. Continuing with the subtraction, we subtract 1 from 1, which gives us 0. Next, we subtract 1 from 0, but we need to borrow from the next digit.

To borrow from the next digit, we change it from 0 to 1. Therefore, the next digit becomes 1, and we subtract 1 from 2, which gives us 1. Finally, we subtract 0 from 1, which gives us 1.

Thus, the result of subtracting (001100)2 from (101001)2 is (11101)2.

To subtract 28 from 29 using 2, you would start with 29 and subtract 2 repeatedly until you reach 28.

29 - 2 = 27
27 - 2 = 25
25 - 2 = 23
23 - 2 = 21
21 - 2 = 19
19 - 2 = 17
17 - 2 = 15
15 - 2 = 13
13 - 2 = 11
11 - 2 = 9
9 - 2 = 7
7 - 2 = 5
5 - 2 = 3
3 - 2 = 1

So, subtracting 28 from 29 using 2 results in 1.