Digital Engineering - Fall 2023, Lecture 01

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.