Why not Subtractors

Subtraction and addition are intertwined, so pure subtractors are not common.

How can we perform subtraction with an adder?

Negate the subtrahend... add (12 -7 = 12 + (-7))

Look at ways to represent signed binary numbers:

	Signed magnitude - most significant bit used for "minus sign"
	2's complement
	1's complement

Signed Magnitude

Most significant bit is set to 1 for negative numbers:

	5 is 00101
	and -5 is 10101

2 cases when subtracting:

Positive result	Negative result
1101	0110
-0110	- 1101
-------	-------
0111	1001
	-10000
	=======
	-0111

	No carry from the most significant bit... means positive result and we're done
	Most significant carry... means that the result is negative, subtract 2n

There might be an easier way

2's Complement

In general: R's complement to a radix R number is:

	The most significant bit equals -(2n)
	Remaining bits normal binary value

Examples:

Now, let's subtract using 2's complement representation:

Example: 2's complement, positive result
45 - 19 ======== 26	101101 - 010011 ========	2's comp -->	0101101 + 1101101 ======== 0011010
Check: 0011010 = 16+8+2 = 26... ok

 

Example: 2's complement, negative result
19 - 45 ======== -26	010011 - 101101 ========	2's comp -->	0010011 + 1010011 ======== 1100110
Check: 1100110 = -64+32+4+2 = -26... ok

General algorithm when subtracting with 2's complement:

	If carry into the most significant, then results is positive and you're done
	If no carry, then result is the correct 2's complement negative answer

1's Complement

Generally case is called "Radix diminished complement" for radix R numbers.

	Most significant bit equals (2n -1)
	Remaining bits have normal binary value

Biggest change from 2's complement. Negating a number is simple: take the bit-wise complement.

Example: 00110 = 6 What is -6? Take Bit-wise complement 11001 = -15 + 8 + 1 = -6

Ok, subtract using 1's complement:

Example: 1's complement, positive result
45 - 19 ======== 26	101101 - 010011 ========	1's comp -->     End around carry -->	0101101 + 1101100  ======== 0011001 +1 ======== 0011010
Check: 0011010 = 16+8+2 = 26... ok

 

Example: 1's complement, negative result
19 - 45 ======== -26	010011 - 101101 ========	1's comp -->	0010011 + 1010010 ======== 1100101
Check: 1100110 = -63+32+4+1 = -26... ok

Algorithm:

	A carry into the most significant bit is added to the result... "end-around carry"
	No, end-around carry means you have the 1's complement negative answer

Summary

	Signed-magnitude - requires translation to 2's complement and back again... this means extra hardware
	1's complement - better, but the "end-around carry" is an extra step in the process
	2's complement - best for the subtraction process