Lecture 3 (05-20)

Hierarchal design

• Building high-level parts from lower-level parts (“black box” small units and building up to bigger units)

todo go over the example with 4 bits. Always remember -1, 0, 8. -8 since they don’t abide by the “slide hit invert” rule

• Acc its in the slide right after. Shortcuts for signed numbers

There are two ways to add signed numbers. He says this is a common source of confusion

1. You can either add two signed numbers
2. subtract unsigned numbers
   • We can implicitly negate the number and add them?
   • Uhh i’m lost

Implementation of subtraction circuit

• Invert all the Y bits. Also $C_{in}$ will be 1 (as we add 1 as per the algorithm above)
• If we’re adding, then we don’t want to invert. We could use a multiplexer (so depending on if we’re adding or subtraction, we simply don’t invert and don’t add 1 at the start).
• Alternatively we can use an XOR gate. X XOR 1 = NOT(X)
  • Let Sub be the bit that determines if we’re subtracting or nah. Then you do things mhmmh.

Problem when adding

• Consider two signed 4-bit numbes 0110 + 0011. The result would be 1001 but that’s a negative number! 9 exceeds the range of 4-bit signed intgers (-8 to 7)
• Same happens if we add two negative numbers.
  • 1000 + 1111 = 0111 but that’s positive. Oh no!
• Positive + Negative number will never overflow
• This is called overflow

Multiplication / Division

• We can shift left to multiply by 2
  • Works for signed and unsigned numbers
• Shifting to the right divides by 2.
  • But this only works for unsigned numbers
• Shift right arithmetic
  • Will be what we talk about next week.todo
  • todo there is another things we gotta watch outside of class time.