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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
+
+![diagram](10.1.png)
+
+### 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
+
+![diagram](10.2.png)
+
+### 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"
+
+![diagram](10.3.png)
+
+### 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
+
+![diagram](10.4.png)
+
+---
+
+## 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