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[\<- Number systems and adders](9.md)
---
# Signed numbers and subtraction
## Signed numbering systems
### Negative numbers, how?
- All our numbers so far have been positive
- N bits a range of 0 to (2^N)-1
- Referred to as unsigned numbers
- Signed numbers are numbers that can be either positive or negative
- But we still only have 1s and 0s to work with
- Requires a scheme/convention/agreement on how to interpret a given set of 1s and 0s
- That convention applies to both positive and negative numbers
### Signed Numbers
- Dedicate the Most Significant Bit (MSB) to be a "sign" bit, with 1 meaning negative
### Considerations
- Must be clear about how many bits are being used
- Different sizes can be used in different contexts: 4-bit, 8-bit, 16-bit, 32-bit
- With 4-bit numbers, bit 3 is the sign bit
- Leading 0's not always expressed
- Ideally we'd like to use the same adder circuit and it just works, whether the numbers are positive or negative
- N + (-N) should give us 0
### Schemes we will NOT use
- Sign-and-Magnitude
- Easy to understand but not useful in reality
- 0101 -> 5
- 1101 -> -5
- What would we get if we added these two?
- 1's complement
- Flip all the bits for negative numbers
- 0101 -> 5
- 1010 -> -5
- Adding yields all 1s
---
## 2's complement numbers and the process of negation
### 2's Complement, our standard
- To "negate", flip all the bits and add 1
- Find -5 from 5
- X = 0101 (5)
- ~X = 1010
- -X = 1011 (-5)
- Find -2 from 2
- X = 0010 (2)
- ~X = 1101
- -X = 1110 (-2)
- Works both ways
- Find what 1101 is
- X = 1101 (?)
- ~X = 0010
- -X = 0011 (3)
- **X = -3**
- Find what 1001 is
- X = 1001 (?)
- ~X = 0110
- -X = 0111 (7)
- **X = -7**
---
## Adding 2's complement numbers
### Examples of addition
- Final carry-out can be ignored (for now)
- 4-bit addition => a 4-bit result
### A comparison
- Schemes cover both positive and negative
|b3b2b1b0|Sign and magnitude|1's complement|2's complement|
|--------|------------------|--------------|--------------|
|0111 |+7 |+7 |+7 |
|0110 |+6 |+6 |+6 |
|0101 |+5 |+5 |+5 |
|0100 |+4 |+4 |+4 |
|0011 |+3 |+3 |+3 |
|0010 |+2 |+2 |+2 |
|0001 |+1 |+1 |+1 |
|0000 |+0 |+0 |+0 |
|1000 |-0 |-7 |-8 |
|1001 |-1 |-6 |-7 |
|1010 |-2 |-5 |-6 |
|1011 |-3 |-4 |-5 |
|1100 |-4 |-3 |-4 |
|1101 |-5 |-2 |-3 |
|1110 |-6 |-1 |-2 |
|1111 |-7 |-0 |-1 |
### Converting number "size"
- Size is typically a constraint of the circuit
- A 4-bit adder can only take 4-bit inputs and generate a 4-bit result
- Sometimes there might be a mismatch
- You're given a 4-bit number but need to use it in an 8-bit adder
- Sign extension => fill in with the sign bit
- 0xF as a 4-bit number is -1
- 0xF as an 8-bit number is 16 (leading 0's)
- 0xFF is -1 as an 8-bit number
---
## 2's complement subtraction
- We know X-Y is the same as X+(-Y)
- In 2's complement, -Y is (~Y+1)
- `~` means we invert all the bits
- So if we have X and Y in 2's complement form, and we want to subtract
- Use an adder
- Invert all the bits in Y
- Add 1
- This allows us to reuse the adder
- don't need to separate circuit for subtraction
---
## Enhancing an adder circuit to do subtraction
### How to reuse adder
- Want to use same circuit for both addition and subtraction
- Need a control signal to determine which
- Adder takes two operand inputs, A and B
- If calculating X-Y, need to pass -Y to input B
- Otherwise pass Y directly thru to B
- Two choices for B: Y if adding, -Y if subtracting
- We use muxes when there are choices
### Selective inversion of 2nd op
- Control signal "invert" will cause the B port of the adder to get ~Y
- To subtract, assert "invert" and "carryin"
### Special characteristic of XOR
- We've talked about viewing one of the inputs to AND/OR gates as a control signal
- AND: 1 enables, 0 disables
- OR: 1 disables, 0 enables
- Applying the same concept to XOR:
- 0 causes the other input to pass thru
- 1 causes the other input to invert
- i.e. passes true to complement version of input based on control signal
- Just like we need to reuse our adder for subtraction
### Using XOR for selective inv.
- The XOR gates allow for selective inversion
- If subtracting, invert (and add 1)
- If adding, don't
---
## Subtraction is negation and then addition
### About subtraction
- When calculating on paper, it is possible to directly calculate the subtraction operation
- WE WILL NOT DO THIS IN THIS CLASS
- We are learning things in the context of underlying circuit implementations
- We have learned how to build an adder, not a subtractor
- When you need to show subtraction, you \*must\* negate and add
---
[Overflow, Comparison, ALU ->](11.md)
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