Here is a quiz on binary representation and arithmetic, drawing on the provided sources.

Quiz on Binary Representation and Arithmetic

Multiple Choice

  1. What base do human beings typically use for counting? a) Binary (base 2) b) Octal (base 8) c) Decimal (base 10) d) Hexadecimal (base 16) Answer: c) Decimal (base 10)

  2. What base do computers primarily use? a) Decimal (base 10) b) Binary (base 2) c) Hexadecimal (base 16) d) Duodecimal (base 12) Answer: b) Binary (base 2)

  3. What bases are used in computing as compact forms for representing binary numbers? a) Decimal and Duodecimal b) Octal and Decimal c) Hexadecimal and Decimal d) Hexadecimal and Octal Answer: d) Hexadecimal and Octal

  4. Which type of integer representation can represent zero and positive integers but not negative integers? a) Signed integer b) Unsigned integer c) Two's complement d) Sign-magnitude Answer: b) Unsigned integer

  5. Which bit is typically used as the sign bit in binary number representations like sign-magnitude, one's complement, and two's complement? a) The least significant bit (LSB) b) Any bit in the middle c) The most significant bit (MSB), or leftmost bit d) There is no sign bit in binary Answer: c) The most significant bit (MSB), or leftmost bit

  6. Which of the following is NOT listed as a common method for representing signed numbers in binary in the sources? a) Sign–magnitude b) One's complement c) Two's complement d) Ten's complement e) Offset binary f) Base −2 Answer: d) Ten's complement

  7. Which signed number representation is the most common method used in computers for signed integers? a) Sign-magnitude b) One's complement c) Two's complement d) Offset binary Answer: c) Two's complement

Fill in the Blank

  1. The two symbols used in the binary number system are ___ and ___. Answer: 0 and 1

  2. Binary digits are called ___. Answer: bits

  3. Eight bits is called a ___. Answer: byte

  4. The binary number system is a ___ notation. Answer: positional

  5. A n-bit storage location can represent up to ___ distinct entities. Answer: 2^n

  6. The value of an unsigned integer is interpreted as "the ___ of its underlying binary pattern". Answer: magnitude

  7. In ___ representation, the most significant bit, the leftmost bit, holds the sign. Answer: sign-magnitude

  8. Two's complement is an alternative way of representing negative binary numbers that has the unique property that subtraction can be performed using ___ hardware. Answer: addition

  9. Most digital systems use ___ for representing negative binary numbers for arithmetic operations. Answer: 2's Complement

True/False

  1. True or False: The binary number system uses only two digits, 0 and 1. Answer: True

  2. True or False: A 3-bit memory location can represent up to 8 distinct entities. Answer: True

  3. True or False: In computing, negative numbers are represented by directly using a minus sign ("−") symbol in memory. Answer: False

  4. True or False: A zero (0) in the leftmost bit of a sign-magnitude number means the number is positive. Answer: True

  5. True or False: In sign-magnitude representation, arithmetic operations are straightforward because the standard rules of arithmetic apply directly to the binary patterns. Answer: False

  6. True or False: In 2's complement addition, the carry from the most significant bit (MSB) is typically discarded as long as the result is within the accepted range. Answer: True

  7. True or False: The value of each symbol in a positional number system depends on its position within the number. Answer: True

  8. True or False: Overflow occurs in 2's complement addition when two numbers are added and the result is too large to fit in the allotted number of bits. Answer: True

  9. True or False: In 1's complement representation, positive numbers are the bit complement of the corresponding negative value. Answer: False (Negative values are the bit complement of the corresponding positive value.)

Short Answer

  1. Describe how the value of the binary number 10110B is calculated using positional notation, as shown in the sources. Answer: The value is calculated as the sum of each digit multiplied by its positional weight (power of 2). For 10110B: 1×2^4 + 0×2^3 + 1×2^2 + 1×2^1 + 0×2^0.

  2. Convert the binary number (110)2 to decimal. Answer: (110)2 = 1 * 2^2 + 1 * 2^1 + 0 * 2^0 = 1 * 4 + 1 * 2 + 0 * 1 = 4 + 2 + 0 = (6)10.

  3. Convert the binary number (10110.1)2 to decimal. Answer: (10110.1)2 = 1 * 2^4 + 0 * 2^3 + 1 * 2^2 + 1 * 2^1 + 0 * 2^0 + 1 * 2^-1 = 1 * 16 + 0 * 8 + 1 * 4 + 1 * 2 + 0 * 1 + 1 * 0.5 = 16 + 0 + 4 + 2 + 0 + 0.5 = (22.5)10.

  4. Convert the decimal number (11)10 to binary using the Repeated Division by 2 method. Answer: 11 / 2 = 5 R 1 (LSB) 5 / 2 = 2 R 1 2 / 2 = 1 R 0 1 / 2 = 0 R 1 (MSB) Reading remainders from bottom to top gives (1011)2.

  5. Convert the decimal number (65)10 to binary using the Repeated Division by 2 method. Answer: 65 / 2 = 32 R 1 (LSB) 32 / 2 = 16 R 0 16 / 2 = 8 R 0 8 / 2 = 4 R 0 4 / 2 = 2 R 0 2 / 2 = 1 R 0 1 / 2 = 0 R 1 (MSB) Reading remainders from bottom to top gives (1000001)2.

  6. Convert the decimal fraction (0.625)10 to binary. Answer: 0.625 * 2 = 1.25 (Carry 1, MSB) 0.25 * 2 = 0.5 (Carry 0) 0.5 * 2 = 1.0 (Carry 1, LSB) Reading carries from top to bottom gives (0.101)2.

  7. How is the positional value of a symbol (digit) in a number system computed? Answer: The positional value is computed using the symbol's position value and the base value (radix) of the number system. For example, in base 10, the symbol at position number 3 has a positional value of 10^3.

  8. What is the maximum decimal number that can be represented with n bits, starting with zero? Answer: The highest decimal number is 2^n – 1.

  9. What is the range of an 8-bit unsigned integer? (Provide the minimum and maximum decimal values) Answer: An 8-bit unsigned integer can represent values from 0 to 2^8 - 1 = 256 - 1 = 255. So, the range is 0 to 255.

  10. Describe the sign bit convention for positive and negative numbers in sign-magnitude representation using an 8-bit number as an example. Answer: The leftmost bit is the sign bit. A 0 in this bit indicates a positive number, and a 1 indicates a negative number. The remaining 7 bits represent the magnitude of the number. For example, (00101101)2 is +45 in decimal, and (11000011)2 is -67 in decimal in sign-magnitude (Note: source gives these examples but they appear to be in 2's complement, the sign-magnitude description matches and). Using the rule from: 01011001 (89) would be +89, 11011001 would be -89 in sign-magnitude. Let's use the rule. For an 8-bit number, the sign bit is the first bit. 0 for positive, 1 for negative..

  11. How is the 2's complement of a binary number found by using its 1's complement? Answer: The 2's complement of a binary number is found by adding 1 to the 1's complement of that number.

  12. Convert the binary number 101 to its 2's complement (assuming it is the magnitude part). Answer: The 1's complement of 101 is 010. Adding 1 to 010 gives 011. So the 2's complement of 101 is 011.

  13. Describe one method mentioned in the sources to decode a 2's complement binary number if the sign bit is 1 (indicating negative). Answer: One method is to "invert the n-1 bits and plus 1" to get the absolute value of the negative number. Alternatively, scan the remaining n-1 bits from the right (least-significant bit). Look for the first occurrence of 1. Flip all the bits to the left of that first occurrence of 1. The flipped pattern gives the absolute value.

  14. Using 8 bits, what is the range of a signed integer in 2's complement representation? (Provide the minimum and maximum decimal values) Answer: The range is from -(2^7) to +(2^7)-1. For 8 bits, this is -128 to +127.

  15. Using 16 bits, what is the range of a signed integer in 2's complement representation? (Provide the minimum and maximum decimal values) Answer: The range is from -(2^15) to +(2^15)-1. For 16 bits, this is -32,768 to +32,767.

  16. Perform the binary addition: 111 + 11. Answer: 111

    • 011

    1010 [43 (Example a)]

  17. Perform the binary subtraction: 1011 – 0110. Answer: 1011

    • 0110

    0101 [Matches Answer b in 46, based on rules in 43]

  18. Perform the binary multiplication: 1010 * 1001. Answer: 1010 x 1001

    1010 (1010 * 1) 00000 (1010 * 0, shifted) 000000 (1010 * 0, shifted) 1010000 (1010 * 1, shifted)

1011010 [Matches Answer c in 46, based on rules in 44]

  1. Perform the binary division: 1100 ÷ 0100. Answer: 0011

    0100 | 1100 -0100 ----- 0100 -0100 ----- 0000 The result is 0011. [Matches Answer d in 46, based on procedure in 45]

  2. Perform the subtraction 100 – 011 using the 1's complement method. Answer: Direct Method: 100 - 011 = 001 1's Complement Method: 100 (minuend)

    • 100 (1's complement of 011)

    000 (result with carry 1) Add end-around carry: 000 + 1 = 001. Final answer: 001. [Matches Answer a in 48, based on method in 47]

  3. Perform the subtraction 100 – 011 using the 2's complement method. Answer: Direct Method: 100 - 011 = 001 2's Complement Method: 100 (minuend)

    • 101 (2's complement of 011)

    001 (result with carry ignored) Final answer: 001. [Matches Answer a in 49, based on method in 49]

  4. In computer systems, data can be represented using a fixed number of bits. What does a binary code represent using a two-symbol system (like 0 and 1)? Answer: A binary code represents text, computer processor instructions, or any other data.

  5. What is the American Standard Code for Information Interchange (ASCII) used for? Answer: ASCII uses a 7-bit binary code to represent text and other characters within computers, communications equipment, and other devices. Each letter or symbol is assigned a number from 0 to 127.

  6. What is Binary-Coded Decimal (BCD)? Answer: BCD is a binary encoded representation of integer values that uses a 4-bit group (nibble) to encode each decimal digit. Only ten values (0-9) in each nibble are legal, while the remaining six are illegal.

  7. How is the character 'A' typically represented internally by a computer after being typed on a keyboard? Answer: When the key 'A' is pressed, it is internally mapped to a unique code (like ASCII code 65 in decimal), which is then converted to its equivalent binary value (like 1000001) for the computer to understand.

  8. Using 8 bits, what is the sign-magnitude representation of +88? (Provide the 8-bit binary pattern) Answer: First, find the binary representation of 88: 1011000. Using 8 bits, this is 01011000. For +88, the sign bit is 0. So, the representation is 01011000. [Based on 14, 50]

  9. Using 8 bits, what is the sign-magnitude representation of -88? (Provide the 8-bit binary pattern) Answer: First, find the binary representation of 88: 1011000. Using 8 bits, this is 01011000. For -88 in sign-magnitude, the sign bit is 1, and the magnitude is the same. So, the representation is 11011000. [Based on 14, 50]

  10. Using 8 bits, what is the 1's complement representation of -88? (Provide the 8-bit binary pattern) Answer: First, find the 8-bit binary representation of +88, which is 01011000. The 1's complement representation of -88 is the bit complement of +88. Flipping all the bits gives 10100111. [Based on 70]

  11. Using 8 bits, what is the 2's complement representation of -88? (Provide the 8-bit binary pattern) Answer: First, find the 8-bit binary representation of +88, which is 01011000. The 2's complement of -88 is found by flipping all bits (1's complement) and adding 1. The 1's complement is 10100111. Adding 1 gives 10100111 + 1 = 10101000. [Based on 48, 87]