Negative Numbers and Two’s Complement Arithmetic
We use negative numbers in everyday arithmetic to represent values that are less
than zero; for example, we may speak of a 
We begin with a general introduction to numbers and then discuss ways of representing negative numbers. Next, we look at two’s complement numbers and show how they work, how they used, and how they are generated. The most important point of this article is to demonstrate that two’s complement arithmetic effectively abolishes subtraction by converting subtraction into the adding of a complement.
Historically, negative numbers are a relatively recent invention. You can pluck 5
apples off a tree. But what does 
It wasn’t until the 17th Century that negative numbers gradually entered European mathematics. Indeed, negative numbers became respectable only after Leibniz introduced infinitesimal calculus.
Numbers
A number represents a means of counting objects or of measuring objects or quantities. We can categorize numbers as follows:
Natural Numbers
The natural numbers are used to count objects; for example: 1, 2, 3, 4, 5. These numbers are positive integers ranging from 1 to infinity. Today, the value 0 is also considered as a natural number and, therefore, the set of natural numbers is 0, 1, 2, 3 4 …
Integers
Integers are whole numbers with no fractional part. The set of integers includes
zero, natural numbers and negative natural numbers; that is …, 
Rational Numbers
Rational numbers are those that can be represented as one integer divided by another (i.e., as fractions); for example, ½ or ¾ If a rational number is represented as m/n, then it is greater than 1 is m < n, and is less than 1 if m < n.
Irrational Numbers
An irrational number is one that cannot be represented in the form m/n where m and n are integers. The constant π is irrational and that be represented as 3.14 in three digits or 3.14159265358979 in 15 digits. However, even an infinite number of digits cannot represent π exactly. Similarly, √2 is an irrational number. Note that numbers like 1/3 are not irrational because they can be represented by a fraction (i.e., 1/3) even though that cannot be represented in real decimal form by a finite number of digits..
Transcendental Numbers
I suppose you could regard a transcendental number as a sort of überirrational number.
A transcendental number is an irrational number that is not a root of a polynomial
equation with rational coefficients. For example, the square root of 2 is irrational
but it is not transcendental because it is a root of the equation x2 – 2 = 0. The
two most well
Real Numbers
Real numbers are the numbers that we use in everyday scientific or financial calculations. A real number may be positive or negative and it is represented by an integer part and a fractional part; for example, 12.34567. This representation is called positional notation using the base 10 with a decimal point between the integer and fractional parts.
Real numbers include positive and negative integers, rational numbers and irrational numbers. You can think of a real number as being a point on a line that extends continuously from minus infinity to positive infinity.
Negative Integers
A traditional way of describing negative numbers is to refer to a scale or continuum of numbers with (typically) negative numbers on the left, zero in the middle and positive numbers on the right. See figure 1a. Such a diagram also demonstrates that adding x to number y means stepping x places to the right, and subtracting x from y means stepping x places to the left. If x is greater than y then we step past 0 into the negative region of the figure.
Figures 1(b) to 1(d) illustrate addition and subtraction as moving left or right along the scale of integers.
Figure 1 Illustrating positive and negative integers
Numbers and Computers
Computers represents signed an unsigned integers in exact binary integer form. The
binary representation of an integer is exact and correct as long as there are sufficient
bits to represent the number; for example, you can represent any integer in the range
0 to 255 in eight bits. If you want’ to represent 256 as an 8
Real numbers are represented as floating
Representing Negative Numbers
How do we represent a negative number? We prefix it by a minus sign. The default
sign is positive and we use a plus sign to make a positive number explicitly positive;
for example 
Consider 
How can we represent negative numbers in a computer? There are three general techniques.
The first is called a biased representation and we actually remove negative numbers
by representing the most negative number by 0. Consequently, which means all other
numbers must be positive. For example, suppose we have numbers in the range 
Another way of representing negative numbers is called sign and magnitude. Here,
we borrow a bit from a number and call it a sign bit; exactly the same way as we
do in conventional arithmetic. By convention, a 0 bit is positive, and a 1bit is
negative. In five bits 00010 is +2 and 10010 is 
Finally, we come to the two’s complement representation of negative integers. Here,
the negativeness of a number is indeed encoded into the number and it is a true negative.
In other words, if you add a negative number you perform subtraction. In two’s complement
arithmetic, adding 3 and 1 gives you 4. Adding 3 and (
Ten’s Complement Arithmetic
Before we look at two’s complement (computer) arithmetic, we really need to demonstrate
that complementary arithmetic works in any base including the decimal base. A number
N is represented in its negative form by bm 
Suppose we want to calculate 73 – 25. Using subtraction we get 73 – 25 = 48.
Now consider adding the tens’s complement of 25 (i.e., 9,975) to 73.
73 + 99,975 = 10,048.
As we are using 4
Suppose we add 25 and 99,975. Here we are adding a number and its complement. We
get 10,000 or 0000 in four digits. This is what we would get if we added 25 + (
Note that the ten’s complement of a number is calculated by subtracting each of its
digits from 9 (the nine’s complement) and then adding 1 to the final result. For
example, in four digits the tens complement of 1234 is 9
Consider 3,231 – 1,234. Adding the ten’s complement gives 3231 + 8766 = 11,997. In four digits this is 1,997 which is the correct value of 3,231 – 1,234 = 2,997.
Ten’s complement arithmetic was used by some mechanical adding machines prior to the age of the digital computer. Typically, each key or slot had two labels: the positive number and the nine’s complement value; for example, the key labelled 7 would also have the number 2 (nine’s complement) by it. To perform subtraction, you added the nine’s complement using the secondary number scale and then added 1 to the total.
Now let’s look at two’s complement arithmetic.
Two’s Complement Arithmetic
Above, we said that a number N is represented in its negative form by bm
Let’s begin with m = 4 (4
In four bits the representation of 
The representation of 
The representation of 
The following table shows how all 16 binary values are interpreted as a two’s complement
value. Note that positive numbers are prefixed by 0 and negative numbers by 1. This
means that the most
0000 = 0
0001 = 1
0010 = 2
0011 = 3
0100 = 4
0101 = 5
0110 = 6
0111 = 7
1000 = 
1001 = 
1010 = 
1011 = 
1100 = 
1101 = 
1110 = 
1111 = 
Note how the negative values or representations appear backwards; that is, the higher they look the smaller they are.
Consider the following examples of 4
Example 1
Add 2 + 3.
0010 + 0011 = 0101
The result 0101 represents 5 which is the correct answer.
Example 2
Subtract 2 from 3 by adding the two’s complement representation of 
0011 + 1110 = 10001
The result in four bits is 0001 which is the correct answer
Example 3
Subtract 6 from 3 by adding the two’s complement representation of 
0011 + 1010 = 1101
The result is 1101. If we look at the table, this is interpreted as 
Example 4
Subtract 
1011 + 1110 = 11001
The result is 11001. In four bits this is 1001. If we look at the table, this is
interpreted as 
Comment
Note that two’s complement arithmetic works for all combinations of a+b, a
Example 5
Add 2 + 7.
0010 + 0111 = 1001
The result 1001 represents 
Example 6
Subtract 5 from 
1010 + 1011 = 10101
The result in four bits is 0101 which is +5 and is therefore the wrong answer.
Arithmetic Overflow
Examples 5 and 6 demonstrate that if we add together two positive values whose sum is greater than the maximum possible value we get an error. Similarly, if we add together two negative numbers whose value is more negative than the largest negative value, we also get an error.
In each of these cases the result, when interpreted as a two’s complement value,
has an incorrect sign. This out
It is impossible to get an out
How Does Two’s Complement Arithmetic Work?
Two’s complement arithmetic is not magic. Consider the representation of –p. This is defined as 2m – p. Now let’s subtract p from q; that is, we evaluate q – p.
To perform the subtraction, we add the two’s complement of p to get q + 2m – p =
q 
As you can see, we get the correct result q – p plus the 2m term. The value of 2m is the 1 in the m+1 bit position. This is the bit we discarded in the result.
Suppose we add together two negative numbers –p + –q. In two’s complement arithmetic we perform the addition of the complements; that is, 2m – p+2m – q. If we rearrange the terms, we get 2m + (2m – p – q). This is the 2m term as before plus the two’s complement representation of –p – q which is the expected result.
Forming two’s Complement Numbers
We have pointed out that subtraction can be performed by adding the complement of a number. However, apart from defining the two’s complement representation of N as 2m – N, we haven’t said how we calculate this value. Since we use complementary arithmetic to avoid subtraction, it seems strange that we have to do a subtraction to generate the complement in the first place.
Let’s look at 2m – N.
We are using m bits to represent N. The value of N which can be represented by the
m digits {nm
N = nm
In order for a number to be represented in its negative form, its most significant
but, nm
N = nm
We form the two’s complement of N by subtracting N from 2m. The value of 2m is 100 … 000, where there at m zeros following the leading 1. For example, in four bits 24 = 1000 and in 8 bits 28 = 100000000.
Now for the magic. Let’s rearrange 2m as 2m – 1 + 1. What does this look like? Well,
2m
The final step is to write down the two’s complement representation of N at the bit level as:
111 … 111 + 1 – (nm
Rearranging the bits, we get:
1 
That is, the two’s complement of N is calculated by subtracting each bit of N from 1 and then adding 1 to the final total.
If ni = 1 then 1 – ni = 0 and if ni = 0 then 1 – ni = 0.
In other words, the two’s complement of N is calculated by inverting all the bits of N and adding 1.
Let’s test this. Consider N = 5 in 4 bits.
N = 0101.
24 – 0101 = 10000 – 0101= 1011
Inverting the bits and adding 1 gives: 1010 becomes 0101. Add 1 gives 1011.
These two results are the same. Consequently, the two’s complement of N can be calculated very rapidly in binary arithmetic by inverting the bits of N (simply by passing each bit of N through an inverting gate) and then adding 1 to the result.
In practice, the operation of A – B is performed by inverting the bits of B, adding those to the bits of A with the carry in to the first stage of the adder set to 1 to perform the adding of the additional 1.
Properties of Two’s Complement Numbers
The range of signed values that an m
The two’s complement representation of zero is zero; that is, 000 … 00
The two’s complement is a true complement in the sense that –N + +N = 0
If a signed number in m bits is extended to m+1 bits, the most
Two’s Complement and Multiplication
Although two’s complement arithmetic works for addition and subtraction, it does not work for multiplication and division. Consider the product of P and –Q.
If we use two’s complement arithmetic, we get
P . (2m – Q) = 2mP – PQ.
What is the correct (expected answer)? When you multiply two m
What we get using two’s complement arithmetic is 2mP – PQ .
To obtain the correct answer we would have to add the correction factor 22m – 2mP to the result. In practice, there are specific algorithms that can be used to multiply signed values; for example, Booths algorithm.
When using assembly language, some computer architectures provide different multiplication for signed and unsigned multiplication and division; for example, if you are using the 68K, the signed multiplication for registers D0 and D1 is expressed ad MULS D0,D1 and the unsigned multiplication of D0 and D1 is represented by MULU D0,D1.
Consequences of Two’s Complement Arithmetic
In four bits, which is the greater value 0011 or 0001. These values represents 3 and 2, respectively, in decimal and therefore 0011 > 0001.
Now consider 1011 and 1001. Which is the greater? In this case, the answer is not
quite as simple. If we regard these are unsigned numbers, then 1011 > 1001 because
11 is greater than 9. However, if we regard these as signed numbers then 1001 > 1011
because 
Instruction set designers provide the same compare instructions for both signed and unsigned quantities, but provide different conditional branch operations to allow you to branch on, say, greater than for signed and unsigned comparisons.
Examples
What are the three most common ways of representing signed numbers in computer arithmetic?
Sign and magnitude representation
Biased representation
Two’s complement representation
Why is two’s complement representation used to handle signed integers in binary arithmetic?
Because two’s complement numbers are easy to generate (invert the bits and add 1).
In two’s complement arithmetic subtraction is performed by adding the complement
of a number. Consequently, separate adders and subtractors are not needed. The most
What are the disadvantages of two’s complement arithmetic?
Two’s complement arithmetic works for addition and subtraction. However, multiplying and dividing two’s complement values requires different algorithms for signed and unsigned operations. In turn, this means that the ALU requires different hardware for signed and unsigned operations. Because the ordering of signed and unsigned numbers is different, you have to provide different comparison and branch operations for signed and unsigned values. This is not hard to do in hardware, but it is a potential source of programmer error.
What range of values can be represented by unsigned integers? Give examples using
8
An m
An m
In 8 bits the range is 0 to 255 (unsigned) and 
In 16 bits the range is 0 to 65,535 (unsigned) and 
How is a two’s complement number calculated?
The negative representation of a number N in m bits is defined as 2m – N. In practice, the computer does not have to perform this calculation because inverting all the bits of N and adding 1 yields the same result.
In 8
The integer 42 is represented by 00101010.
The corresponding two’s complement is 11010101 + 1 = 11010110.
What is arithmetic overflow?
Because two’s complement arithmetic is valid only for numbers in the range –2m
When does arithmetic overflow occur during addition and subtraction?
If two positive values are added and the result is greater than 2m
How is a two’s complement value expanded from m bits to n bits where n > m?
The sign
42 = 00101010 in 8 bits and becomes 0000000000101010 in 16 bits

Example
Suppose you are using four digit ten’s complement arithmetic. Consider the following operation.
0073
+PQRS
X0048
What are the values of P, Q, R, and S. Does the value of X matter?