1. Binary Arithmetic Stephen Boyd March 14, 2011

2. Two's Complement Most significant bit represents sign. 0 = positive 1 = negative Positive numbers behave exactly like unsigned integers. Negative numbers are added to the negative value of the most significant bit. 1 101 = (-8) + 5 = -3 1 001 1011 = (-128) + 27 = -101 1 111 1111 = (-128) + 127 = -1

3. Two's Complement To convert between negative and positive, invert every bit and add 1. 1 101 (-3) → 0 010 + 1 = 0 011 (3) 0 011 (3) → 1 100 + 1 = 1 101 (-3) For n number of bits, an integer has a possible range of -2 n-1 to 2 n-1 -1. 1 000 to 0 111 for a 4-bit integer, or -8 to 7

4. Binary Addition and Subtraction Similar to decimal addition 3 + 2 → 0 011 + 0 010 00 10 carry 0 011 3 0 010 2 0 101 = 5 Subtraction can be achieved by adding a negative. 3 – 5 = 3 + (-5)

5. Overflow Overflow occurs when the result is beyond the possible range allowed by the number of bits. 4 + 5 → 0 100 + 0 101 01 00 carry 0 100 4 0 101 5 1 001 = -7 (WRONG!) Typically only occurs when the two left-most carries are either 10 or 01. 11 produces no problems. Can be detected with a simple XOR of the last two carries.

6. Half Adders A simple addition of 1-bit operands (A and B) produces two values: The summation S = A XOR B The carry C = A AND B This type of circuit is called a half adder:

7. Full Adders The drawback of half adders is that they do not account for the carry-in from a previous bit addition. For that, there are full adders: S = A XOR B XOR C in C = (A AND B) OR (C in AND (A XOR B))

8. Ripple Adders Multiple full adders can be strung together to create a ripple adder, with the carry-out from each going into the carry-in of the next.

9. Carry Lookahead Adders A problem with the ripple adder is that each 1-bit adder must wait for the previous adder to determine if there will be a carry before completing the summation. Lookahead adders are able to determine simultaneously for each 1-bit adder if there will be an incoming carry, and then allow each adder to perform the summation without waiting.

10. Carry Lookahead Adders In lookahead adders each 1-bit adder first informs the look ahead logic of two values: G – If the bit will generate a carry. This will happen regardless of an incoming carry if both A and B are 1. (G = A AND B) P – If the bit will propagate an incoming carry into the next bit adder. This will happen if either A or B is at least 1. (P = A OR B) Once P and G are known for all bit adders, it is possible to determine incoming carries without waiting for the full summation.

11. Carry Lookahead Adders

12. Subtraction As stated, a two's complement integer can be negated by inverting each bit and adding 1. X XOR Y will return Y if X = 0, and Y' if X = 1. With that in mind, any adder can be adapted to subtract instead with a toggle input XOR'd with each of the input bits and carried in to the first 1-bit adder.

13. Multiplication of Unsigned Integers Again, similar to decimal. Multiplying integers with x and y number of bits will yield a product of bits x+y. (at most) Example w/4-bit integers: 12 * 5 = 60 1 1 0 0 0 1 1 1 0 0 1 1 1 1 0 0 Multiply the numbers as usual for decimal, then add them. Because binary only uses 1s and 0s, a shortcut is to simply repeat the multiplicand for each 1 in the multiplier, shifted left. Multiplicand Multiplier Product

14. Multiplication of Unsigned Integers

15. Fast Multiplication: Booth's Algorithm Multiplying integers M * N of lengths m and n respectively, need the following values: A = M followed by n+1 0's. B = M' followed by n+1 0's. P = m 0's followed by N and a 0. Start with P and examine 2 rightmost bits. 10 → P = P+B, shift right 01 → P = P+A, shift right 00 or 11 → shift right Do this n times, then drop the right most bit. P is now the product in two's complement. Example: -5 * 7 → 1011 * 0111 A = 1011 0000 0 B = 0101 0000 0 P = 0000 0111 0 1) P=P+B 0 0 0 0 0 1 1 1 0 shift 0 1 0 1 0 1 1 1 0 2) shift 0 0 1 0 1 0 1 1 1 3) shift 0 0 0 1 0 1 0 1 1 4) P=P+A 0 0 0 0 1 0 1 0 1 shift 1 0 1 1 1 0 1 0 1 Product = 1 1 0 1 1 1 0 1