1. Building Adders & Sub tractors Dr Ahmed Telba

2. Introducing adder circuits Adder circuits are essential inside microprocessors as part of the ALU, or arithmetic logic unit, where the processing and manipulation of binary numbers takes place. How does binary addition work? The simplest addition you can do is to add together two 1-bit binary numbers. Suppose the numbers are called A and B. Each of these numbers can take the values 0 or 1. There are four possible additions:

3. Half Adder nput Binput A CARRY digit SUM digit 0000 0101 1001 1110

4. Goals By the end of this lecture, you should understand… How to build a Half-Adder How to build a Full-Adder How to use Adders to perform subtraction

5. What We Can Do Last time, we learned how to build a circuit that met functional requirements. Today, let’s build a circuit that performs a specific engineering function and supports two output lines – we’ll build a circuit to add two binary numbers …

6. Problem Definition We will start out simply, adding two single digit binary numbers Let’s make certain we can do this by hand before we construct the circuit There are four possible combinations of single digit, two number addition: – 0 + 0 =0 – 0 + 1 = 1 – 1 + 0 = 1 – 1 + 1 = 10

7. Problem Analysis Notice that one of the combinations of two single digit additions is a double digit number, created by a carry: 1 + 1 = 10

8. Standardized Output We’ll use leading zeros in order to produce a standard output: 0 + 0 = 00 (0 plus 0 equals 0) 0 + 1 = 01 (0 plus 1 equals 1) 1 + 0 = 01 (1 plus 0 equals 1) 1 + 1 = 10 (1 + 1 equals 10) Notice that all of our answers include two bits. By putting the output in standardized form, we can stipulate the behavior of both outputs given any combination of two inputs with the following truth table …

9. Truth Table for Adding Two 1-Bit Numbers INPUTSOUTPUTS A BQR 0000 0101 1001 1110

10. Breaking Down the Truth Table Notice the Q column (carry line) from the truth table? Recognize it? It’s an AND!: Q = A B INPUTSOUTPUTS A BQR 0000 0101 1001 1110

11. Breaking Down the Truth Table What about the R column (sum line)? What sub-expression could we use? What do you notice about the R column? R is high iff one and exactly one input is high … INPUTSOUTPUTS A BQR 0000 0101 1001 1110

12. Breaking down the Truth Table “R will be high iff one and exactly one input is high.” What expression can we use to prove this statement? R = (A’ B) + (A B’)

13. Circuit Diagram Q = (~A B) + (A ~B) QAA BB

14. More on XOR We have already established that any circuit can be created as a combination of 3 basic gates: NOT, AND and OR. Although not one of the fundamental gates, the XOR gate is common enough that it is called a derived gate. We can build the XOR from the primitive gates. However, engineers use the XOR so often they’ve given XOR its own symbol …

15. XOR Symbol The symbol for the XOR Gate: A B Q Q = A B ++

16. Re-Writing the Truth Table The XOR circuit delivers the Sum Line (R) of adding two single digits. The AND circuit produces the Carry Line (Q). We’ve just created a fundamental circuit called a Half Adder, which provides the capability of adding two single bit numbers.INPUTSOUTPUTSA BQR ABABABAB A B 0000 0101 1001 1110 ++

17. Circuit Diagram for the Half AdderAB R SUM Q CARRY

18. Using the Half-Adder as a Building Block The Half-Adder circuit is more powerful than you might think. It operates as the key component in a divide-and-conquer approach to adding n digits. How does this work? The first leap of faith comes in realizing that three single digit numbers could be added two at a time: – First, add the first two digits together using a half-adder. – Then, add the result of this sum to the third number using another half-adder.

19. Full Adder Let’s try to add 3 single digit numbers by stringing together two half-adders: – We need 3 input lines, one for each number. – Two input lines feed into the first half adder; the third input line functions as a carry-in line. – The second half-adder has two input lines: the output sum from the first half-adder, plus the third, carry-in input line. The next slides show the truth table for the Full Adder and then the circuit diagram for the Full Adder …

20. Full Adder table Circuit Diagram for the Full Adder

22. Truth Table for the Full Adder INPUTSOUTPUTS AB C IN Q CARRY R SUM 00000 00101 01001 01110 10001 10110 11010 11111

23. Circuit Diagram for the Full AdderAB R SUM Q CARRY C IN

24. Shorthand Notation for the Full Adder The full adder is usually drawn in a shorthand notation: FULL ADDER AB C IN Q CARRY R SUM

25. Problem Solution Then, we add the first number of our problem (5 10 or 0101 2 ) to this two’s complemented number: 0101 2 + 1110 2 = 10011 2 If we throw away the carry line (the leftmost 1), our answer is correct: 0011 2 = 3 10 Is that legal? Why or why not? We are restricting ourselves here to 2, 4-bit numbers. We can choose to simply ignore the carry line (the 5 th bit).

26. Interpreting the Full Adder Circuit Do you see any patterns between the output lines and the input lines? If you look at the truth table, you can see that the Q CARRY output line is true if and only if exactly two or three input lines are high. Also, the R SUM output line is high when the XOR of A and (the sum of Input B and C IN ) is high? In other words, add the Input B and C IN values together, as though you were putting them through a half adder. Take the result, and XOR it with the value of the A input line. The result is the value for the R SUM output line

27. Summarizing the Full Adder By connecting two half-adders together, we can add 3 single digit binary numbers It shouldn’t come as a surprise to learn that by stringing even more half-adders together, we can add even more single digit numbers…

28. The N-Bit Adder The next leap of faith is to move from adding 3 (or more) single digit numbers to adding multi-bit numbers First, note that the sum of adding two N-bit numbers can be N + 1 bits This comes as a result of possibly obtaining a carry into the next column Adding N-bit numbers isn’t an abstract consideration – it’s a real computing operation - at a minimum, a modern computer would add 16 bit numbers, done with a 16-bit adder

29. The N-Bit Adder Here is the general rule: to add two N-bit numbers, N full adders are required, one for each column in the sum The first 2 inputs for each full adder come from the digits in the numbers to be added The carry-out produced for each column is used as the carry-in for the next column to the left At some point, you can have overflow as you exceed the adder circuitry capability.

30. 4-bit Full Adder

31. Getting Real The processing component of the ALU – the arithmetic logic unit – uses Adder circuitry to perform fundamental arithmetic operations. By connecting half adders to make a full adder, and multiple full adders in a chain, we now can work our way to simple addition. Other operations are a variation on this fundamental capability …

32. How to Subtract What other arithmetic operations do we need to be able to perform with circuits? The next logical step would be to tackle subtraction … We could come up with a separate subtraction circuit, and some clever scheme of knowing which circuit to activate based on what kind of math problem we encountered It turns out that the solution is more elegant -- it uses the approach we have seen repeatedly: turn a new problem into a problem we have already solved …

33. From Subtraction to Addition What we have is a circuit that can add numbers. We want to make this circuit also work for subtracting two numbers… Let’s rethink our problem, with a specific example: 5 - 2 = 3, or, put another way: 5 + (-2) = 3 Now, we have an addition problem, instead of subtraction. This means our existing circuit will work we could unlock how to represent negative numbers, like -2.

34. Two’s Complement Remember that we can represent negative numbers using a scheme called Two’s Complement. In two’s complement, to encode a negative number, you take its positive value, complement it, and then add one. The resulting number will be in two’s complement form.

35. Review of Two’s Complement 1.Convert each number from Base-10 to its Base-10 format (we’ll use four-bit numbers); for negative values, convert their absolute value: 5 10 = 0101 2 |-2 10 | = 0010 2

36. Review of Two’s Complement 2.Complement the negative value, changing all 1s to 0s and all 0s to 1s. The result is called a One’s Complement: 0010 2  1101 2

37. Review of Two’s Complement 3.Add 1 to the One’s Complement. The result is your original negative number in Two’s Complement form: -2 10  0010 2  1101 2 + 1 2 = 1110 2