Building Adders & Sub tractors

Dr Ahmed Telba

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:

Half Adder

nput B input A CARRYdigit

SUMdigit

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 0

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

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 …

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

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

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 …

Truth Table for Adding Two 1-Bit Numbers

INPUTS OUTPUTS

A B Q R

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 0

Breaking Down the Truth Table

• Notice the Q column (carry line) from the truth table? Recognize it? It’s an AND!:Q = A • B

INPUTS OUTPUTS

A B Q R

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 0

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 …

INPUTS OUTPUTS

A B Q R

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 0

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’)

Circuit Diagram

• Q = (~A • B) + (A • ~B)

Q

AA

BB

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 …

XOR Symbol

• The symbol for the XOR Gate:

A

B

Q

Q = A B++

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.

INPUTS OUTPUTS

A BQ R

A•B A B

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 0

++

Circuit Diagram for the Half Adder

A

B

RSUM

QCARRY

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.

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 …

Full Adder table

• Circuit Diagram for the Full Adder

Circuit Diagram for the Full Adder

Truth Table for the Full AdderINPUTS OUTPUTS

A B CIN QCARRY RSUM

0 0 0 0 0

0 0 1 0 1

0 1 0 0 1

0 1 1 1 0

1 0 0 0 1

1 0 1 1 0

1 1 0 1 0

1 1 1 1 1

Circuit Diagram for the Full Adder

A

B

RSUM

QCARRY

CIN

Shorthand Notation for the Full Adder

• The full adder is usually drawn in a shorthand notation:

FULLADDER

A

B

CIN

QCARRY

RSUM

Problem Solution

• Then, we add the first number of our problem (510 or 01012) to this two’s complemented number: 01012 + 11102 = 100112

• If we throw away the carry line (the leftmost 1), our answer is correct: 00112 = 310

• 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 5th bit).

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 QCARRY output line is true if and only if exactly two or three input lines are high.

• Also, the RSUM output line is high when the XOR of A and (the sum of Input B and CIN) is high? In other words, add the Input B and CIN 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 RSUM output line

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…

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

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.

4-bit Full Adder

• 4-bit Full Adder

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 …

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 …

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.

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.

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:510 = 01012

|-210| = 00102

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:00102 11012

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:-210 00102 11012 + 12 = 11102