Digital Engineering - Fall 2023, Lecture 01
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Questions and Answers

Which numeral system has the least number of digits?

  • Octal
  • Hexadecimal
  • Binary (correct)
  • Decimal

    • What is the primary reason for using binary in computer design?

  • It is easier to use than decimal.
  • It simplifies the design with on/off switches. (correct)
  • It supports more digits than other systems.
  • It is based on human counting habits.

    • How many digits are used in the hexadecimal numeral system?

  • 2 digits
  • 8 digits
  • 16 digits (correct)
  • 10 digits

    • What is the equivalent of the binary number 1010 in hexadecimal?

    <p>A</p> Signup and view all the answers

    What is the size of a standard byte in terms of bits?

    <p>8 bits</p> Signup and view all the answers

    In positional notation for the decimal number 43, what digit corresponds to the 10’s place?

    <p>4</p> Signup and view all the answers

    Which base number system includes the digits 0 through 7?

    <p>Octal</p> Signup and view all the answers

    What is a common unit of data that consists of 4 bytes?

    <p>Word</p> Signup and view all the answers

    What is the decimal equivalent of the binary number 1010?

    <p>10</p> Signup and view all the answers

    Which binary number represents the decimal number 7?

    <p>111</p> Signup and view all the answers

    In binary addition, what is the sum of the binary numbers 110 and 101?

    <p>1011</p> Signup and view all the answers

    What is the binary representation for the decimal number 9?

    <p>1001</p> Signup and view all the answers

    Which addition statement is correct for base 10?

    <p>310 + 610 = 910</p> Signup and view all the answers

    What is the decimal equivalent of the binary number 1101 0110?

    <p>214</p> Signup and view all the answers

    How many different numbers can be represented with a 16-digit binary number?

    <p>65,536</p> Signup and view all the answers

    Using the range formula R = B^K, what is the range for a base 2 system with 20 bits?

    <p>4,194,304</p> Signup and view all the answers

    Which base system requires the fewest symbols to represent the same numbers?

    <p>Base 2</p> Signup and view all the answers

    What is the number of digits required to represent the number 256 in decimal using base 2?

    <p>8</p> Signup and view all the answers

    In a base 16 system, which of the following digits is not used?

    <p>G</p> Signup and view all the answers

    If a number in base 8 is represented as 147, what is its decimal equivalent?

    <p>112</p> Signup and view all the answers

    How many bits are needed to represent 1,024 different numbers?

    <p>10</p> Signup and view all the answers

    What is the output of the AND operation when both inputs are 0?

    <p>0</p> Signup and view all the answers

    What happens when a binary number is shifted left by one position?

    <p>Its value is multiplied by the base</p> Signup and view all the answers

    In binary multiplication, how are the bits arranged based on their place values?

    <p>Bits are aligned based on the multiplier's place value</p> Signup and view all the answers

    What is the result of 1010 shifted right by one position?

    <p>110</p> Signup and view all the answers

    What does a carry bit represent in binary multiplication?

    <p>An overflow from an AND operation</p> Signup and view all the answers

    Which of the following correctly describes the AND operation?

    <p>It outputs 1 only if both inputs are 1</p> Signup and view all the answers

    If you perform a left shift on the binary number 102, what will the result be?

    <p>1002</p> Signup and view all the answers

    What is indicated when the last bit in the multiplication result is a 0?

    <p>The 1's place was not lowered</p> Signup and view all the answers

    What is the base 16 equivalent of the base 10 number 5,735?

    <p>1667</p> Signup and view all the answers

    How do you express the base 8 number 72638 in base 10?

    <p>3,763</p> Signup and view all the answers

    What is the most significant bit when converting 8,039 to base 16?

    <p>7</p> Signup and view all the answers

    In binary subtraction, what would the difference be for the minuend 10110 and the subtrahand 10010?

    <p>00100</p> Signup and view all the answers

    Which of the following is a reason why hexadecimal is often used in computing?

    <p>It is easier to read and write than binary.</p> Signup and view all the answers

    What is the binary equivalent of the base 16 digit F?

    <p>1111</p> Signup and view all the answers

    In the conversion process from base 2 to base 16, what is the binary representation of the hexadecimal digit 7?

    <p>0111</p> Signup and view all the answers

    When converting the number 3,763 from base 10 to base 8, which power of 8 is not utilized?

    <p>8^1</p> Signup and view all the answers

    What occurs when there is a borrow into the most significant bit (msb) position during subtraction?

    <p>The subtrahend is larger than the minuend.</p> Signup and view all the answers

    How is the 1’s complement of a binary number defined?

    <p>By inverting each bit of the number.</p> Signup and view all the answers

    What does the 2’s complement of a binary number entail?

    <p>Subtracting the number from 2n and adding 1.</p> Signup and view all the answers

    What is the 2’s complement representation of the number -6 in a 4-bit system?

    <p>1010</p> Signup and view all the answers

    If you add 4 (0100) and -6 (1010) using 2’s complement, what is the final result?

    <p>1110</p> Signup and view all the answers

    In which way can the 1's complement of the number 1011001 be derived?

    <p>By inverting each bit.</p> Signup and view all the answers

    Which of the following correctly describes the relationship between the 1's complement and the 2's complement?

    <p>The 2's complement is obtained by adding 1 to the 1's complement.</p> Signup and view all the answers

    What defines a borrow situation during subtraction in binary?

    <p>The subtrahend is greater than the minuend.</p> Signup and view all the answers

    Study Notes

    Digital Engineering - Fall 2023, Lecture 01 - Data Representation

    • Course title: Digital Engineering
    • Class year: Fall 2023
    • Lecture topic: Data Representation
    • Instructor: Dr. Tarek Abdul Hamid

    Why Binary?

    • Early computer design was decimal
    • Examples: Mark I and ENIAC
    • John von Neumann proposed binary data processing (1945)
    • Simplified computer design
    • Used for both instructions and data
    • Natural relationship between on/off switches and calculation using Boolean logic

    Counting and Arithmetic

    • Decimal or base 10 number system
    • Origin: counting on the fingers
    • "Digit" from Latin word digitus meaning "finger"
    • Base: the number of different digits including zero in the number system
    • Example: Base 10 has 10 digits, 0 through 9
    • Binary or base 2
    • Bit (binary digit): 2 digits, 0 and 1
    • Octal or base 8: 8 digits, 0 through 7
    • Hexadecimal or base 16: 16 digits, 0 through F
    • Examples: 1010₂ = A₁₆, 1110₂ = B₁₆

    Keeping Track of the Bits

    • Bits commonly stored and manipulated in groups
    • 8 bits = 1 byte
    • 4 bytes = 1 word (in many systems)
    • Number of bits used in calculations affects accuracy of results
    • Limits size of numbers manipulated by the computer

    Positional Notation: Base 10

    • Example: 43₁₀ = (4 x 10¹) + (3 x 10⁰)
    • Example 2: 527₁₀ = (5 x 10²) + (2 x 10¹) + (7 x 10⁰)

    Positional Notation: Octal

    • Example: 624₈ = (6 x 64) + (2 x 8) + (4 x 1) = 384 + 16 + 4 = 404₁₀

    Positional Notation: Hexadecimal

    • Example: 6,704₁₆ = (6 x 4096) + (7 x 256) + (0 x 16) + (4 x 1) = 24,576 + 1,792 + 0 + 4 = 26,372₁₀

    Positional Notation: Binary

    • Example: 1101 0110₂ = (1 x 128) + (1 x 64) + (0 x 32) + (1 x 16) + (0 x 8) + (1 x 4) + (1 x 2) + (0 x 1) = 128 + 64 + 0 + 16 + 0 + 4 + 2 + 0 = 214₁₀

    Estimating Magnitude: Binary

    • Example: 1101 0110₂ > 192₁₀ (128 + 64 + additional bits to the right)

    Range of Possible Numbers

    • R = Bᵏ where R = range, B = base, K = number of digits
    • Example 1: Base 10, 2 digits, R = 10² = 100 different numbers (0...99)
    • Example 2: Base 2, 16 digits, R = 2¹⁶ = 65,536 or 64K
    • 16-bit PC can store 65,536 different number values

    Decimal Range for Bit Widths

    • Shows the decimal range for different bit widths.

    Base or Radix

    • Base: the number of different symbols required to represent any given number
    • Larger the base, the more numerals are required
    • Examples:
      • Base 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
      • Base 2: 0, 1
      • Base 8: 0, 1, 2, 3, 4, 5, 6, 7
      • Base 16: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

    Number of Symbols vs. Number of Digits

    • For a given number, the larger the base, the more symbols required, but the fewer digits needed.

    Counting in Base 2

    • Shows table for converting binary to decimal notation.

    Base 10, 8, and 2 Addition Tables

    • Shows addition tables for various bases

    Base 10 and 8 Multiplication Tables

    • Shows multiplication tables for various bases

    Addition

    • Examples of addition in different bases

    Addition - Carry

    • Examples of addition in different bases, including carry values

    Binary Arithmetic

    • Discusses binary addition, multiplication, shift and division using XOR and AND operations

    Binary Multiplication

    • Boolean logic without performing arithmetic
    • AND (carry bit): output is "1" if both are "1," shift method
    • Examples provided for shifting

    Binary Multiplication - Examples (2-25, 2-26)

    • Further examples of binary multiplication

    Converting from Base 10

    • Powers Table
    • Examples converting from base 10 to base 2

    From Base 10 to Base 2

    • Examples converting from base 10 to base 2

    From Base 10 to Base 16

    • Examples converting from base 10 to base 16

    From Base 8 to Base 10

    • Examples converting from base 8 to base 10

    From Base 16 to Base 2

    • The nibble method is used
    • Hexadecimal is used for troubleshooting in modern computer operating systems and networks

    Binary Subtraction

    • Subtraction operation in binary
    • Shows example using borrow values

    Binary Subtraction - Example

    • A full example of Binary substraction is provided

    In General

    • Conditions of binary subtraction: borrow/no borrow and positive/negative result

    Two's Complement

    • Representing negative numbers
    • Two types of complements: r's complement, (r − 1)'s complement
    • For base 2: 2's complement and 1's complement

    l's Complement

    • Definition of 1's complement (2ⁿ⁻¹ - N)

    2's Complement

    • Definition of 2's complement (2ⁿ⁻N or 1's complement + 1)

    Operations with 2's Complement

    • Addition using 2's complement

    A 2's Complement Table for 4 bits

    • Table demonstrating the 2's complement values

    Binary Division

    • Binary division
    • Method for dividing in binary, including quotient and remainder

    Binary Division - Example

    • A full example of binary division is provided.

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    Digital Engineering Lecture 01 PDF

    Description

    This quiz covers the key concepts of data representation in digital engineering, highlighting the historical context of binary processing and various numbering systems. It includes discussions on the significance of binary in computer design, counting systems, and arithmetic operations. Test your understanding of these foundational principles to pave the way for more advanced topics in digital engineering.

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