Two’s Complement Arithmetic

We now address the issue of representing integers as binary strings in a computer.

There are four formats that have been used in the past; only one is of interest to us.

The four formats are

1.     Sign–Magnitude

2.     One’s–Complement

3.     Two’s–Complement

4.     Excess–N, where N is some positive integer; say 127 for Excess–127.

Sign–magnitude representation is obsolete.

Excess–127 representation is only used as part of floating point representations.
If discussed at all, it will be in the context of the IEEE–754 standard.

We focus on Two’s–complement, but discuss one’s–complement arithmetic as
a mechanism for generating the two’s complement.

Representation of Non–Negative Integers

In each of one’s–complement and two’s–complement arithmetic, no special steps
are required to represent a non–negative integer.

All conversions to the complement number systems begin with conversion to simple
binary representation, often from decimal notation.

The only issue with non–negative integers is to insure that the number is not too large
for the number of bits used in the representation.

There are two ways to define this size limitation in the complement numbers system.
1.     The most significant bit (leftmost) must be a “0” for non–negative numbers.
2.     For an N–bit representation, the number must be less than 2N–1.

The biggest signed 8–bit integer is    27 – 1 =        127.
The biggest signed 16–bit integer is  215 – 1 =   32767.

Three 8–bit examples:      100        =      0110 0100
22        =      0001 0110
0        =      0000 0000

When we talk about any number system that can represent signed integers, we must
specify the number of bits used to represent the number.

Each binary representation must have that number of bits.  This means padding

Thus, we could say           100        = 110 0100
22        = 1 0110
0        = 0.

But, when we discuss 8–bit numbers, we MUST SAY
100        =      0110 0100
22        =      0001 0110
0        =      0000 0000

The common integer sizes for signed arithmetic are 8–bit, 16–bit, and 32–bit.

I have worked with 12–bit, 18–bit, 30–bit, and 36–bit signed integers.
These are of interest only for assigning problems to students.

Taking the One’s–Complement

To take the one’s complement of a binary integer, just
Change every 0 to a 1
Change every 1 to a 0

One’s–complement arithmetic is that arithmetic that uses the one’s–complement of
a binary integer to represent its negative.

+ 100 in 8–bit one’s complement.                0110 0100.

– 100 in 8–bit one’s complement.
Convert 100 to 8–bit binary   0110 0100
Take the one’s complement   1001 1011
– 100 is represented as           1001 1011

+ 22 is 8–bit one’s complement            0001 0110

– 22 in 8–bit one’s complement
Convert 22 to 8–bit binary     0001 0110
Take the one’s complement   1110 1001
– 22 is represented as             1110 1001

We no longer use one’s–complement arithmetic.  It has two serious problems.

Problems with One’s–Complement Arithmetic

There are two serious problems with the use of one’s–complement arithmetic.  The first one will be described but not explained.

Problem 1:     It is a lot trickier to build a binary adder for one’s–complement numbers.
Just take my word for this one.

Problem 2:     Negative Zero

Consider 8–bit one’s–complement arithmetic

+ 0 is                                       0000 0000

Take the one’s complement   1111 1111

So – 0 is represented by          1111 1111

Most of us are uneasy about having a distinct – 0 in our number system.

Code such as the following is considered strange.
if ( ( x == 0) || (x == –0) )

This course will not cover one’s–complement arithmetic or use it directly for any purpose other than taking the two’s complement, discussed next.

Taking the Two’s Complement

There are two processes for taking the two’s complement of a number.  The one that students find the most natural involves first taking the one’s complement and then adding
1 to that complement.

We must then cover the addition of two binary bits, specifically mentioning each of the
sum bits and carry bits.  The analog in decimal arithmetic is to say that the sum of 5 and 5 is 0 with a carry of 1.  Yes, 5 + 5 = 10, generating two digits from 2 one–digit numbers.

Here is the binary addition table

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

Yes, 1 + 1 is really 2.  We denote this by saying that the sum 1 + 1 carries 1
into the two’s column.

Consider 5 + 7                5 =   0101
7 =   0111
1100        12

Taking the Two’s Complement (Part 2)

The recipe for taking the two’s–complement of a binary number is simple.

1.     Take the one’s–complement of the number
2.     Add 1 to that complement.

+ 100 in 8–bit two’s–complement binary           0110 0100

– 100 in 8–bit two’s–complement binary
Represent +100 as an 8–bit number
0110 0100
Take the one’s complement
1001 1011
1
– 100 is represented as
1001 1100

+ 22 in 8–bit two’s–complement binary             0001 0110

– 22 in 8–bit two’s–complement binary
Represent +22 as an 8–bit number
0001 0110
Take the one’s complement
1110 1001
1
– 22 is represented as
1110 1010

Taking the Two’s Complement (Part 3)

+ 0 in 8–bit two’s–complement binary          0000 0000

– 0 in 8–bit two’s–complement binary
Represent +0 as an 8–bit number
0000 0000
Take the one’s complement
1111 1111
1 Discard the high carry bit
– 0 is represented as
0000 0000

Claim: – 1 in 8–bit two’s–complement binary is 1111 1111.

– 127 and – 128 in 8–bit two’s–complement binary.

+ 127 in 8–bit two’s–complement binary                      0111 1111

– 127 in 8–bit two’s–complement binary
Represent +127 as an 8–bit number
0111 1111
Take the one’s complement
1000 0000
1
– 127 is represented as
1000 0001

– 128 is “one less” than – 127, so is represented as       1000 0000

a “carry in”.  For adding two binary numbers, the “carry in” is either 0 or 1.

In decimal arithmetic on integers,
a full adder would be used for adding the tens, hundreds, thousands (etc.)
columns as each of these can have a “carry in” from a column to the right.

The terms “half adder” and “full adder” apply only to binary arithmetic.

A full adder can be converted to a half adder by setting its “carry in” to 0.

Example From Unsigned Integer Arithmetic

Here we add 14 (binary 01110) and 11 (binary 01011) to get
25 (binary 11001). A    B  Sum  Carry                                A    B    C  Sum  Carry
0     0     0        0                                    0     0     0     0        0
0     1     1        0                                    0     0     1     1        0
1     0     1        0                                    0     1     0     1        0
1     1     0        1                                    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

We don’t really need a full adder to do two’s complement arithmetic, but
here it is.

Back to algebra.  We were taught that – ( – X) = X for all numbers.
Let’s test it out for two’s–complement arithmetic.

Double Negations (Part 1)

Remember that the range of 8–bit two’s–complement arithmetic is – 128 to 127.

+ 100 in 8–bit two’s–complement binary           0110 0100

– 100 in 8–bit two’s–complement binary
Represent +100 as an 8–bit number
0110 0100
Take the one’s complement
1001 1011
1
– 100 is represented as
1001 1100

+100 in 8–bit two’s–complement binary
Represent – 100 as an 8–bit number
1001 1100
Take the one’s complement
0110 0011
1
– 100 is represented as
0110 0100

This is the same as + 100.  So – ( – 100) is + 100.

Double Negations (Part 2)

Remember that the range of 8–bit two’s–complement arithmetic is – 128 to 127.

+ 22 in 8–bit two’s–complement binary             0001 0110

– 22 in 8–bit two’s–complement binary
Represent +22 as an 8–bit number
0001 0110
Take the one’s complement
1110 1001
1
– 22 is represented as
1110 1010

+22 in 8–bit two’s–complement binary
Represent – 22 as an 8–bit number
1110 1010
Take the one’s complement
0001 0101
1
+ 22 is represented as
0001 0110

This is the same as + 22.  So – ( – 22) is + 22.

Double Negations (Part 3)

Remember that the range of 8–bit two’s–complement arithmetic is – 128 to 127.

+ 127 in 8–bit two’s–complement binary                      0111 1111

– 127 in 8–bit two’s–complement binary
Represent +127 as an 8–bit number
0111 1111
Take the one’s complement
1000 0000
1
– 127 is represented as
1000 0001

– 128 is represented as                                                   1000 0000
Take the one’s complement
0111 1111
1
– (– 128) is represented as
1000 0000

( – 128) is the same as – 128 !!

What happened?   The answer is that  + 128 cannot be represented in
8–bit two’s–complement arithmetic.

Standard Ranges

Here is the standard Java implementation of Two’s–Complement arithmetic.

In general the range for N–bit two’s–complement arithmetic is

– 2N–1 to 2N–1 – 1

 Java Type Number of Bits Lower Limit Upper Limit byte 8 – 128 127 short 16 – 32768 32767 int 32 – 2 147 483 648 2 147 483 647 long 64 – 263 263 – 1

Log10(263) is about 0.30103 · 63 = 18.9649.

Log10(9) = Log10(32) = 2·Log10(3) = 2·0.47712 = 0.95424.

As 0.9649 > Log10(9), we conclude that 263 is bigger than 9·1018.

Standard Values in N–Bit Two’s Complement Arithmetic
(With Examples for 16–bit integers)

0      is represented by N zeroes             0000 0000 0000 0000

–1     is represented by N ones                1111 1111 1111 1111

– 2N–1  is represented by a 1
followed by (N – 1) zeroes            1000 0000 0000 0000

The student should memorize these patterns and not bother with
standard conversion techniques when obtaining them.

Technically speaking, the representation of – 2N–1 cannot be derived by
standard means, as 2N–1 cannot be represented.

Sign Extension

In two’s complement arithmetic, the leftmost bit is the sign bit.

It is        1 for negative numbers
0 for non–negative numbers

Sign extension is the process of converting a N–bit representation to
a larger format; e.g., a 16–bit number to a 32–bit number.

Example:

+ 100 in 8–bit two’s–complement binary   0110 0100

+ 100 in 16–bit two’s–complement binary 0000 0000 0110 0100

100 in 8–bit two’s–complement binary    1001 1100

100 in 16–bit two’s–complement binary  1111 1111 1001 1100

Rule – just extend the sign bit to fill the new “high order” bits.

Overflow: “Busting the Arithmetic”

The range of 16–bit two’s–complement arithmetic is

– 32,768 to 32,767

Consider the following addition problem: 24576 + 24576.
Now + 24,576 (binary 0110 0000 0000 0000) is well within the range.

0110 0000 0000 0000                        24576
0110 0000 0000 0000                        24576
1100 0000 0000 0000                     – 16384

What happened?

We had a carry into the sign bit.  This is “overflow”.  The binary representation
being used cannot handle the result.

NOTE:  This works as unsigned arithmetic.
24,576 + 24,576 = 49,152 = 32768 + 16384.

When we design a binary adder for N–bit integers (N > 1), we shall address
the standard method for detecting overflow.

Saturation Arithmetic

Saturation arithmetic is applied to unsigned N–bit integers.

The unsigned range is 0 … 2N – 1.

N = 8            0     through                   255
N = 16          0     through               65535
N = 32          0     through  4,294,967,295

Saturation arithmetic is commonly applied to 8–bit integers used in
computer graphics.

RULES:     In addition, if the sum is greater than 2N1, set it to 2N – 1.
In subtraction, if the number is less than 0, set it to 0.

Examples (8–bit saturation arithmetic)

100 + 100        = 200           128 + 128    = 255
128 + 127        = 255           200 + 200    = 255
255 + 255    = 255

Result: Brightening a bright scene does not accidentally make it dark.