In this set of slides, we present the two basic types of adders:

In order to understand the functioning of either of these circuits, we
must speak of arithmetic in terms that I learned in the second grade.

In the first grade, I learned by “plus tables”, specifically the sum of adding
any two one–digit numbers: 2 + 2 = 4, 2 + 3 = 5, etc.

digit each: 23 + 34 = 57, but 23 + 38 = 61.

We begin with two simple sums, each involving only single digits.

2 + 2 = 4, and 5 + 5 = 10.

If these are so, why do we write the following sum 25 + 25 as

25 + 25 = 50, and not as

25 + 25 = 4 10?         What digit is written in the unit’s column of the sum?

The reason that we do not do this is the idea of a carry from the unit’s
column to the ten’s column.

In the language of the second grade, we describe the addition as follows:

1.    5 + 5 is 0, with a carry–out of 1, which goes into the ten’s column.

2.    2 + 2 is 4, but we have a carry–in of 1 from the unit’s column,
so we say 2 + 2 + 1 = 5.  The sum digit in this column is a 5.

Positional Notation in Arithmetic

In standard decimal arithmetic, the number 25 is read as “twenty five”, and
represents 2
· 10 + 5 · 1: two tens plus five ones.

Remember that the only digits used in binary numbers are 0 and 1.

In binary arithmetic, the number 10 is read as “one zero”, avoiding the
names we use for decimal numbers.  It represents 1
· 2 + 0 · 1.

In binary arithmetic, the number 101 represents 1·22 + 0·21 + 1·1, or
1
·4 + 0·2 + 1·1 = 4 + 1 = 5.

In all arithmetics:                    0 + 0 = 0,
0 + 1 = 1, and
1 + 0 = 1.

In decimal arithmetic:             1 + 1 = 2.

In binary arithmetic what is 1 + 1?

The Sum 1 + 1 in Binary Arithmetic

We have just noted that the decimal number 2 is represented in binary as 10.

It must be the case that, in binary addition, we have the sum as

1 + 1 = 10

This reads as “the addition 1 + 1 results in a sum of 0 and a carry–out of 1”.

Recall the decimal sum 25 + 25.

1
2 5
2 5
5 0

The 1 written above the numbers in the ten’s column shows the carry–out
from the unit’s column as a carry–in to the ten’s column.

The half adder takes two single bit binary numbers and produces
a sum and a carry–out, called “carry”.

Here is the truth table description of a half adder.  We denote the sum A + B.

 A B Sum Carry 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1

Written as a standard sum, the last row represents the following:

01
+ 01
10

The sum column indicates the number to be written in the unit’s column,
immediately below the two 1’s.  We write a 0 and carry a 1.

In all arithmetics, including binary and decimal, the half adder represents
what we do for the unit’s column when we add integers.

There is no possibility of a carry–in for the unit’s column, so we do not design
for such.  Another way is to say that there is a carry–in; it is always 0.

The full adder in decimal arithmetic would be used for the other columns:
the ten’s column, the hundred’s column, and so on.

For these columns, a non–zero carry–in is a distinct possibility

Considered this way, we might write our sums table as follows.
2 + 2 = 4, if the carry–in is 0, and
2 + 2 = 5, when the carry–in is 1.

Admittedly, the complexities of decimal arithmetic suggest another way,
just add the values as three numbers, here 2 + 2 + 1 = 5.

Here again is the truth table for the half adder.

 A B Sum Carry 0 0 0 0 0 1 1 0 1 0 1 0 1 1 0 1

We need equations for each of the Sum and Carry.  Because we have used a
truth table to specify these functions, we consider Boolean expressions.

Note that the carry is the logical AND of the two inputs: Carry = A·B.

The sum can be given in two equivalent expressions.
The simplest expression uses the exclusive OR function: Sum = A
ĹB.
An equivalent expression in terms of the basic AND, OR, and NOT is:

Here are two slightly different circuit implementations of the half adder.

The circuit on the left implements the sum function as Sum = AĹB.

The circuit on the right implements the sum function as

Here we show the truth table for the sum of A and B, with carry–in of C.

 A B C Sum Carry 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1

To read this as decimal, pretend that the Sum is “Sum_Bit” and read as
the Decimal_Sum = Carry
·2 + Sum_Bit.

So   1 + 1 + 0 = 1·2 + 0 =        2 (decimal), and
1 + 1 + 1 = 1
·2 + 1 =        3 (decimal).

One Circuit for the Full Adder

The Full Adder with C = 0

The circuit above implements the following two expressions, where C is
the carry–in to the full adder.

Sum =   ··C + ·B· + A·· + A·B·C

Carry =  A·B + A·C + B·C

Suppose we let the carry–in C = 0.  Then  = 1.
What we have then is as follows.

Sum       = ··0 + ·B·1 + A··1 + A·B·0
=
·B + A·

Carry     = A·B + A·0 + B·0
= A
·B

As expected, a full adder with carry–in set to zero acts like a half adder.

When we build circuits with full adders or half adders, it is important to focus
on the functionality and not on the implementation details.

For this reason, we denote each circuit as a simple box with inputs and outputs.

The figure on the left depicts a full–adder with carry–in as an input.

The figure on the right depicts a half–adder with no carry–in as input.

The figure in the middle depicts a full–adder acting as a half–adder.

Here is a depiction of a four–bit full adder to add two binary numbers,
depicted as A3A2A1A0 and B3B2B1B0.

Note that the carry–out from the unit’s stage is carried into the two’s stage.
In general, the carry is propagated from right to left, in the same manner as
we see in manual decimal addition.  This is called a “ripple carry adder”.

Here is an example of its output.  The 4–bit sum is truncated to 1001.

1110
+ 1011
11001

Propagating the Carry Bits

Just as in standard arithmetic, when done by hand, the carry of one stage
is propagated as a carry–in to the next higher stage.

In order to convert a ripple–carry adder into a subtractor, we employ
the standard algebra trick: A – B is the same as A + (–B).

In order to subtract B from A, it is necessary to negate B to produce
–B, and then to add that number to A.

We now have to develop a circuit that will negate a binary value.

In order to do this, we must stipulate the method used to represent
signed integers.  As in earlier lecture, we use two’s–complement notation.

In this method, in order to negate a binary integer, it is necessary to
produce the two’s–complement of that value.

1.    First, take the one’s–complement of the binary integer, and then
2.    Add 1 to that value.

We have to develop a circuit to create the one’s–complement of a binary
integer.  We shall develop two circuits to do this.

The One’s Complement of a Binary Integer

In order to take the one’s–complement of an integer in binary form, just
change every 0 to a 1, and every 1 to a 0.  Here are some examples.

Original value           0110 0111              1010 0011
One’s complement
1001 1000              0101 1100

The circuit that does this conversion is the NOT gate.  The circuit below
would be used for four–bit integers.

If the input is B3B2B1B0 = 0110, the output is Y3Y2Y1Y0 = 1001.

This circuit can be extended to any number of bits required.

The XOR Gate as a NOT Gate

In order to make an adder/subtractor, it is necessary to use a gate that can
either pass the value through or generate its one’s–complement.

The exclusive OR gate, XOR, is exactly what we need.

This is controlled by a single binary signal: Neg.

Let B = 1011.  If Neg = 0, then Y = 1011.  If Neg = 1, then Y = 0100.

Some Notation for Bit–Wise Operations

Taking the one’s–complement of a binary integer involves taking the
one’s–complement of each of its bits.  We use the NOT notation to
denote the one’s–complement of the entire integer.

If B = B3B2B1B0 is a four–bit binary number, its one’s–complement is
.

Remember that to get the two’s–complement, one takes the one’s–complement
and adds one.  In two’s–complement arithmetic we have:  .

This suggests the use of an adder to produce the negative.  For B = B3B2B1B0,
we take the one’s complement to start the negation process.  The addition of 1,
necessary to take the two’s–complement, is achieved by setting the carry–in of the unit’s full–adder to 1.

This is the reason for using a full–adder in the unit’s position.  We can alternate
between an adder and subtractor.  More on this a bit later.

The Negator Circuit

Here is the circuit to produce the negative of a 4–bit number B = B3B2B1B0.

If Neg = 0, then the XOR gates pass the values B3B2B1B0 unchanged and
the ripple–carry adder adds 0 to that.  The result is Y = B.

If Neg = 1, then the XOR gates cause the one’s–complement of B = B3B2B1B0
to be computed.  Note that the carry–in of the full adder is set to 1, so that it
adds 1 to the one’s–complement, thus producing the negative: Y = – B.