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authorlshprung <lshprung@yahoo.com>2020-10-14 11:20:21 -0700
committerlshprung <lshprung@yahoo.com>2020-10-14 11:20:21 -0700
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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
diff --git a/9.md b/9.md
index 2bec00d..3731fb4 100644
--- a/9.md
+++ b/9.md
@@ -123,3 +123,7 @@ module fulladd(Cin, x, y, s, Cout);
endmodule
```
+
+---
+
+[Signed numbers and subtraction ->](10.md)