ECE 2020 Digital Design

Last time

• Last time
  • Mixed logic
  • Number systems
• To be effectively prepared for today you should have:
  • Read your notes
• Homework 4 will be posted by Wednesday October 22, 2025
  • Coverage includes gates, logic, and number systems
• Withdrawal date: Saturday October 25, 2025
  • We're working to give you Exam 2 grades before
  • Reach out to Academic Advisors in your units if you need guidance
• Today
  • Number systems
• Be ready!
  • I expect you to take notes
  • Take your quizz at 10:15am

Positional Number Systems

• Key objective: go beyond binary operations and represent others

• Positional number system
  • Number is represented by a string of digits where each digit position has an associated weight
  • Value of the number is the weighted sum of the digits \[ D = d_{p-1}d_{p-2}\cdots d_{1}d_0.d_{-1}\cdots d_{-n} = \sum_{i=-n}^{p-1}d_i r^i \]
    • \(r\) is the radix (basis) of the number system (typically \(r\) is an integer with \(r\geq 2\))
    • a digit in position \(i\) (\(d_i\)) has weight \(r^i\)
• Typical systems
  • \(r=2\): binary (\(0,1\))
  • \(r=8\) octal (\(0,1,2,3,4,5,6,7\))
  • \(r=16\) hexadecimal (\(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F\))
  • \(r=10\) decimal (\(0, 1, 2, 3, 4, 5, 6, 7, 8, 9\))

Conversion between number systems

• The formula \(D = \sum_{i=-n}^{p-1}d_i r^i\) converts any number system to decimal

• We can use binary representations as pivot point between binary, hexadecimal, octal

• We will mainly worry about decimal to binary and vice versa

Unsigned integers and addition

Signed integers and addition/subtraction

• Signed integer representation
  • signed magnitude
  • 1's complement
  • 2's complement
• We will only focus on 2's complement for addition and subtraction

Until next time

• To be effectively prepared for Wednesday October 15, 2025, you should:
  • Review your notes and homework solution for CMOS logic
  • Don't forget what we did before CMOS logic - that is still relevant