2's Complement
Two's complement is a method of representing negative binary numbers. It allows arithmetic operations to work correctly with negative values, in a similar manner as positive numbers. The MSB (most significant bit) of each register is reserved to hold the sign of the number. A 0 denotes a positive value, so positive values remain unchanged. A leading 1 denotes a negative value.
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Normally, a 3 bit binary number ranges from 0 (000) to 7 (111). With 2's complement the range is from plus 3 (011) to minus 3 (101). Note that binary zero (all 0's) has a single representation. Applying the 2's complement function a second time reverts back to the original number. |
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Both these layouts calculate the value of A minus B using the 2's complement method of invert and add 1. The result is held in register B , so the original value of B is 'lost'. Three functions are used:
- Inverter function, to invert B .
- Incrementor function, to increment (add 1) to B .
- Accumulator function, to add A to B .
Operation
Enter the values of A and B and start the train from the station. When the train halts, the result can be read from register B .
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| Click layout to pause/run train | Click points to switch 0/1 | Click start circle to reset train/points |
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Lazy points switch between upper
0
or lower
1
branch lines
Trains arriving on a branch line switch the point to that line |
|
Sprung points allow branch line trains to join the main line
All main line trains go straight ahead and never 'branch off' |
How it works
The lower Inverter and Incrementor functions calculate the 2's complement of B . The Inverter uses compact flip-flops . The Adder then adds the value of A to B .
This second layout is more compact as the Incrementor is not needed. Instead, a 'first return' latch ensures the train returns to the Adder to add the final 1 . The operation is the same as the layout above.
|
| Click layout to pause/run train | Click points to switch 0/1 | Click start circle to reset train/points |
|
|
Lazy points switch between upper
0
or lower
1
branch lines
Trains arriving on a branch line switch the point to that line |
|
|
Sprung points allow branch line trains to join the main line
All main line trains go straight ahead and never 'branch off' |
How it works
From the station, the train enters the Inverter function which inverts the value of
B
. It then passes up to the Adder, resetting the 'first return' latch to
0
on the way. Therefore there is no need to reset this latch before each new calculation.
The Accumulator function adds
A
to
B
. As the train returns, it uses the top set of latches to set
A
to
1
. The 'first return' latch returns the train to the Adder, so that the
1
is added to
B
.
On its second visit, the 'first return' latch lets the train pass through, so the train returns to the 'Halt' siding to end the calculation.
Note
- The 2's complement method only works when all registers are of the same size. Add and Incrementor overflows are simply 'lost',so there are no overflow error lines.
- If B is greater than A , then the result of A - B is negative, and is read as a 2's complement value.