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

To find the remainder when the value 37H is divided by 17H, we can perform long division.

Step 1: Convert the hexadecimal values to decimal.
37H = 3 * 16^1 + 7 * 16^0 = 48
17H = 1 * 16^1 + 7 * 16^0 = 23

Step 2: Perform long division.
_______
23 | 48
- 46
_______
2

Step 3: The remainder is 2 in decimal. We need to convert it back to hexadecimal.
2 = 2 * 16^0 = 02H

Therefore, the remainder when 37H is divided by 17H is 02H, which corresponds to option D.

Radix of Octal Number System

Definition: Radix of a number system is the number of digits or symbols used to represent the values of the numbers in that system. For example, in the decimal number system, the radix is 10 because it uses 10 digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) to represent the values of the numbers.

Octal Number System: The octal number system is a base-8 number system that uses 8 digits (0, 1, 2, 3, 4, 5, 6, 7) to represent the values of the numbers.

Calculation of Radix: To calculate the radix of a number system, we need to find the base of the number system. The base of the number system is the smallest number that can be represented with a single digit in that system.

In the octal number system, the smallest number that can be represented with a single digit is 0. The next highest number that can be represented with a single digit is 1. Therefore, the base of the octal number system is 8.

Conclusion: Hence, the radix of the octal number system is 8.

Converting Binary to Decimal:

To convert a binary number to a decimal number, you can use the positional notation system. Each digit in a binary number represents a power of 2, starting from 2^0 on the rightmost digit.

Binary Number (1101)2:

In the binary number (1101)2, from right to left:
- The rightmost digit is 1 * 2^0 = 1
- The next digit is 0 * 2^1 = 0
- The next digit is 1 * 2^2 = 4
- The leftmost digit is 1 * 2^3 = 8

Calculating Decimal Equivalent:

To find the decimal equivalent, add up the values of each digit:
1 + 0 + 4 + 8 = 13

Therefore, the decimal equivalent of the binary number (1101)2 is 13.

So, the correct answer is option 'C'.

Code is a symbolic representation of Discrete information.

Code is a method of representing information using symbols or signals. It is used in various fields, including computer programming, communication systems, and electrical engineering. The purpose of coding is to convert information into a format that can be easily processed, transmitted, or stored.

Continuous vs. Discrete Information:

To understand why code is a symbolic representation of discrete information, let's first define continuous and discrete information.

Continuous information refers to data that can take on any value within a certain range. It is represented by analog signals, which are continuous and can have infinite possible values. For example, the temperature in a room can vary continuously between 20 to 25 degrees Celsius.

Discrete information, on the other hand, can only take on specific values within a certain set. It is represented by digital signals, which are discrete and can only have a finite number of possible values. For example, the number of people in a room can only be a whole number, such as 10 or 15.

The Nature of Code:

Code is a symbolic representation of discrete information because it uses discrete symbols to represent data. In computer programming, for example, code consists of a series of instructions written using a specific set of symbols, such as letters, numbers, and punctuation marks. Each symbol represents a specific command, operation, or value.

When a computer program is executed, the code is interpreted or compiled into machine language, which consists of binary digits (0s and 1s). These binary digits, also known as bits, represent the fundamental units of discrete information in computing. Each bit can have a value of either 0 or 1, making it a discrete representation.

Advantages of Discrete Representation:

Using discrete representation in coding has several advantages:

1. Precision: Discrete representation allows for precise and accurate representation of information. Each symbol or bit carries a specific meaning, making it easier to interpret and process the data.

2. Storage Efficiency: Discrete representation requires less storage space compared to continuous representation. In digital systems, data can be stored using binary digits, which are compact and efficient for storage and transmission.

3. Error Detection and Correction: Discrete representation allows for error detection and correction techniques. By using error detection codes, it is possible to identify and correct errors that may occur during transmission or storage of data.

In conclusion, code is a symbolic representation of discrete information because it uses discrete symbols or bits to represent data. This allows for precise representation, efficient storage, and error detection and correction in various fields such as computer programming and communication systems.

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.

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.

To convert a decimal number to a binary number, we can use the method of successive division by 2. Here's how we can convert the decimal number (57.375)10 to its binary equivalent:

1. Convert the integer part:
- Divide 57 by 2: 57 ÷ 2 = 28, remainder 1
- Divide 28 by 2: 28 ÷ 2 = 14, remainder 0
- Divide 14 by 2: 14 ÷ 2 = 7, remainder 0
- Divide 7 by 2: 7 ÷ 2 = 3, remainder 1
- Divide 3 by 2: 3 ÷ 2 = 1, remainder 1
- Divide 1 by 2: 1 ÷ 2 = 0, remainder 1

Reading the remainders from bottom to top, the binary equivalent of the integer part is 111001.

2. Convert the fractional part:
- Multiply the fractional part (0.375) by 2: 0.375 × 2 = 0.75
- Multiply the fractional part (0.75) by 2: 0.75 × 2 = 1.5
- Multiply the fractional part (0.5) by 2: 0.5 × 2 = 1.0

Reading the whole numbers from left to right, the binary equivalent of the fractional part is 011.

3. Combine the integer and fractional parts:
Putting the integer part and the fractional part together, we get the binary representation of (57.375)10 as 111001.011.

Therefore, the correct answer is option A, (111001.011)2.

Binary Code of (21.125)10

To convert a decimal number to binary, we need to divide the decimal number by 2 repeatedly until the quotient becomes zero. The binary representation is obtained by writing the remainders in reverse order. In this case, we need to convert the decimal number 21.125 to binary.

Step 1: Integer Part Conversion

To convert the integer part of the decimal number (21), we divide it by 2 repeatedly until the quotient becomes zero.

21 ÷ 2 = 10 with a remainder of 1
10 ÷ 2 = 5 with a remainder of 0
5 ÷ 2 = 2 with a remainder of 1
2 ÷ 2 = 1 with a remainder of 0
1 ÷ 2 = 0 with a remainder of 1

The remainders in reverse order are 10101, which is the binary representation of the integer part of the decimal number.

Step 2: Fractional Part Conversion

To convert the fractional part of the decimal number (0.125), we multiply it by 2 repeatedly until it becomes zero or until the desired number of binary places are obtained.

0.125 × 2 = 0.25 (0)
0.25 × 2 = 0.5 (0)
0.5 × 2 = 1.0 (1)

The binary representation of the fractional part is 001. Therefore, the binary representation of the decimal number 21.125 is 10101.001.

Answer: Option A - 10101.001

Explanation:

Gray code is a binary numeral system where two successive values differ in only one bit. This is also known as the reflected binary code. The 4-bit Gray code can be represented as:

0000
0001
0011
0010
0110
0111
0101
0100
1100
1101
1111
1110
1010
1011
1001
1000

To find the Gray code for decimal number 5, we need to convert 5 into its binary equivalent which is 0101.

Steps to convert binary to Gray code:

1. Write the most significant bit of the binary number to the Gray code.
2. Write the next bit of the binary number XOR with the previous bit of the binary number to the Gray code.
3. Repeat step 2 for all remaining bits of the binary number.

Conversion of 0101 to Gray code:

Step 1: Write the most significant bit of the binary number to the Gray code.

0

Step 2: Write the next bit of the binary number XOR with the previous bit of the binary number to the Gray code.

01

Step 3: Repeat step 2 for all remaining bits of the binary number.

011

0111

Hence, the 4-bit Gray code for decimal number 5 is 0111 which is option A.

Decimal Equivalent of Excess-3 Number 110010100011.01110101

To find the decimal equivalent of the given excess-3 number, we need to understand the concept of excess-3 code and convert it to decimal representation.

1. Excess-3 Code
Excess-3 code is a binary-coded decimal (BCD) representation of decimal digits. In this code, each decimal digit is represented by a 4-bit binary code. The excess-3 code is obtained by adding 3 to the corresponding BCD code.

For example, the BCD representation of decimal 0 is 0000. In excess-3 code, it becomes 0011 (0000 + 0011 = 0011). Similarly, the BCD representation of decimal 1 is 0001, and in excess-3 code, it becomes 0100 (0001 + 0011 = 0100).

2. Conversion Steps
To convert the given excess-3 number to decimal, we can follow these steps:

Step 1: Split the number into integer and fraction parts.
The given number is 110010100011.01110101. Splitting it into integer and fraction parts, we get:
Integer part: 110010100011
Fraction part: 01110101

Step 2: Convert the integer part to decimal.
To convert the integer part of the excess-3 number to decimal, we need to convert each group of 4 bits to its corresponding decimal digit. We can use the following table for reference:

Excess-3 Code | Decimal
0000 | 3
0001 | 4
0010 | 5
0011 | 6
0100 | 7
0101 | 8
0110 | 9
0111 | 10
1000 | 11
1001 | 12
1010 | 13
1011 | 14
1100 | 15
1101 | 16
1110 | 17
1111 | 18

Converting the integer part of the given number:
1100 1010 0011 -> 15 8 6 -> 1586

Step 3: Convert the fraction part to decimal.
To convert the fraction part of the excess-3 number to decimal, we can use the place value method. Each bit in the fraction part represents a negative power of 2.

0111 0101 -> (0*1/2) + (1*1/4) + (1*1/8) + (1*1/16) + (0*1/32) + (1*1/64) + (0*1/128) + (1*1/256)
-> 0 + 0.25 + 0.125 + 0.0625 + 0 + 0.015625 + 0 + 0.00390625
-> 0.45703125

Step 4: Combine the integer and fraction parts.
Combining the integer and fraction parts, we get:
1586.45703125

3. Final Answer
The decimal

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.

Octal is a numeral system that uses a base of 8, meaning it only includes the digits 0-7. To convert a binary number to octal, we need to group the binary digits into sets of three from right to left and assign an octal digit to each group.

Given binary number: 111010

To convert this binary number to octal, we will group the digits into sets of three from right to left:

1 1 1 0 1 0

Now let's assign an octal digit to each group:

Group 1: 1 1 1 -> The octal digit equivalent of binary 111 is 7.

Group 2: 0 1 0 -> The octal digit equivalent of binary 010 is 2.

So the octal equivalent of 111010 is 72.

Therefore, the correct answer is option C) 72.

The range of numbers represented by an 8-bit two's complement system is -128 to 127. This means that the system can represent both positive and negative integers within this range. The most significant bit (MSB) is used as the sign bit, with 0 indicating a positive number and 1 indicating a negative number. The remaining 7 bits are used to represent the magnitude of the number.

Conversion of Decimal to Binary
To convert the decimal number (98.75)10 into binary:
1. Whole Number Part (98)10
- Divide by 2 and keep track of the remainders:
- 98 ÷ 2 = 49 remainder 0
- 49 ÷ 2 = 24 remainder 1
- 24 ÷ 2 = 12 remainder 0
- 12 ÷ 2 = 6 remainder 0
- 6 ÷ 2 = 3 remainder 0
- 3 ÷ 2 = 1 remainder 1
- 1 ÷ 2 = 0 remainder 1
- Reading from bottom to top gives (1100010)2.
2. Fractional Part (0.75)10
- Multiply by 2 and take the integer part:
- 0.75 × 2 = 1.5 (integer part 1)
- 0.5 × 2 = 1.0 (integer part 1)
- Thus, (0.75)10 = (0.11)2.
- Combined: (1100010.11)2.
Conversion of Decimal to Octal
To convert (98.75)10 to octal:
1. Whole Number Part (98)10
- Divide by 8:
- 98 ÷ 8 = 12 remainder 2
- 12 ÷ 8 = 1 remainder 4
- 1 ÷ 8 = 0 remainder 1
- Reading from bottom to top gives (142)8.
2. Fractional Part (0.75)10
- Multiply by 8:
- 0.75 × 8 = 6.0 (integer part 6)
- Thus, (0.75)10 = (0.6)8.
- Combined: (142.6)8.
Conversion of Decimal to Hexadecimal
To convert (98.75)10 to hexadecimal:
1. Whole Number Part (98)10
- Divide by 16:
- 98 ÷ 16 = 6 remainder 2
- Thus, (62)16.
2. Fractional Part (0.75)10
- Multiply by 16:
- 0.75 × 16 = 12.0 (integer part 12, which is C in hex)
- Thus, (0.75)10 = (0.C)16.
Final Results
- Binary: (1100010.11)2
- Octal: (142.6)8
- Hexadecimal: (62.C)16
Thus, the correct answer is option 'D'.

One possible way to answer this question is by providing a personal example of a time when you faced a difficult situation and how you overcame it. Here is an example response:

One difficult situation I faced was during my sophomore year of college when I was struggling with a particularly challenging math course. I had always considered myself to be strong in math, but this class was proving to be a real struggle for me. I was spending hours studying and trying to understand the material, but it just wasn't clicking.

Instead of giving up, I decided to seek help. I visited my professor during office hours and explained my difficulties. He was very understanding and offered to set up weekly one-on-one tutoring sessions with a graduate student who specialized in the subject matter. I also formed a study group with some classmates who were also struggling.

Through these tutoring sessions and study group meetings, I was able to gain a better understanding of the material. I learned different study techniques and approaches that helped me grasp the concepts. I also found that teaching others helped solidify my own understanding of the material.

By the end of the semester, I was able to improve my grades and pass the course. This experience taught me the importance of seeking help and not being afraid to ask for assistance when facing a difficult situation. It also showed me the value of collaboration and working with others to overcome challenges.

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.

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.

's complement method, we get:

-46 = 1101 0010 (in 8-bit binary)
To get its 2's complement, we need to invert all bits and add 1:
0010 1110 + 1 = 0010 1111

28 = 0001 1100 (in 8-bit binary)

Now we can add these two numbers using binary addition:

0010 1111 (2's complement of -46)
+ 0001 1100 (28)
-----------
0100 1011

The result in binary is 0100 1011, which is equal to 75 in decimal. However, since we used the 2's complement method to represent negative numbers, we need to check if the result is negative by looking at the leftmost bit (the sign bit). If it is 1, the result is negative. In this case, the leftmost bit is 0, so the result is positive. Therefore, the final answer is 75.

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."

To subtract (010110)2 from (1011001)2, we can use the binary subtraction method.

```
1 0 1 1 0 0 1 (carry)
- 0 1 0 1 1 0 (010110)
--------------
1 0 0 0 1 1 (1011001 - 010110 = 100011)
```

Therefore, (1011001)2 - (010110)2 = (100011)2.

Understanding BCD Code
Binary-Coded Decimal (BCD) is a class of binary encodings for decimal numbers where each digit is represented by its own binary sequence. In standard BCD, each digit from 0 to 9 is represented using four bits.
Valid and Invalid BCD Codes
In BCD, only the binary representations of decimal digits 0 to 9 are valid. Therefore:
- Valid BCD Codes (0-9):
- 0000 (0)
- 0001 (1)
- 0010 (2)
- 0011 (3)
- 0100 (4)
- 0101 (5)
- 0110 (6)
- 0111 (7)
- 1000 (8)
- 1001 (9)
- Invalid BCD Codes:
- 1010 (A in hex)
- 1011 (B in hex)
- 1100 (C in hex)
- 1101 (D in hex)
- 1110 (E in hex)
- 1111 (F in hex)
Analysis of the Options
Given the options:
- a) 1011: Invalid BCD (represents 11)
- b) 1010: Invalid BCD (represents 10)
- c) 1001: Valid BCD (represents 9)
- d) 1100: Invalid BCD (represents 12)
Conclusion
The correct answer is option C (1001), as it is the only valid BCD representation among the given options. It represents the decimal digit 9 and adheres to the rules of BCD encoding. The other options (1010, 1011, and 1100) represent values outside the valid range for BCD.

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.

To add 28 and 18 using base 2 (binary), we can convert the numbers to their binary representations and then perform the addition.

28 in binary is 11100.
18 in binary is 10010.

Now let's add them:

11100
+10010
_______
101010

Therefore, 28 + 18 in base 2 is equal to 101010.

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.

Conversion of BCD to Binary

To convert BCD (Binary-Coded Decimal) to binary, we need to understand the BCD representation and the corresponding binary values.

BCD Representation:
BCD is a binary encoding of decimal numbers, where each decimal digit is represented by a 4-bit binary code. The BCD representation of decimal digits from 0 to 9 is as follows:

Decimal | BCD
--------|-----
0 | 0000
1 | 0001
2 | 0010
3 | 0011
4 | 0100
5 | 0101
6 | 0110
7 | 0111
8 | 1000
9 | 1001

Given BCD:
BCD 0001 0010 0110

Conversion Process:
To convert BCD to binary, we need to convert each BCD digit to its corresponding binary value and concatenate them.

Step 1:
Convert the first BCD digit (0001) to binary. Looking at the BCD table, we find that 0001 corresponds to the binary value 0001.

Step 2:
Convert the second BCD digit (0010) to binary. According to the BCD table, 0010 corresponds to the binary value 0010.

Step 3:
Convert the third BCD digit (0110) to binary. Referring to the BCD table, 0110 corresponds to the binary value 0110.

Concatenation:
Concatenate the binary values obtained from each BCD digit.

Binary value after concatenation: 0001 0010 0110

Therefore, the binary representation of the given BCD (0001 0010 0110) is 0001 0010 0110.

Conclusion:
The correct answer is option 'A' (1111110).