1.5: Integer Numbers (2's Complement)

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1.5.1 What is an Integer

Using only positive whole numbers is too limiting for any valid calculation, and so the concept of a binary negative number is needed. When negative values for the set of whole numbers are included with the set of whole number (which are positive), the resulting set is called integer numbers. Integers are non-fractional numbers which have positive and negative values.

When learning mathematics, negative numbers are represented using a sign magnitude format, where a number has a sign (positive or negative), and a magnitude (or value). For example -3 is 3 units (it's magnitude) away from zero in the negative direction (it's sign). Likewise, +5 is 5 units away from zero in a positive direction. Signed magnitude numbers are used in computers, but not for integer values. For now, just realize that it is excessively complex to do arithmetic using signed magnitude numbers. There is a much simpler way to do things called 2's complement. This text will use the term integer and 2's complement number interchangeably.

1.5.2 2's complement operation and 2's complement format

Many students get confused and somehow believe that a 2's complement has something to do with negative numbers, so this section will try to be as explicit here as possible. Realize that if someone asks, "What is a 2's complement?", they are actually asking two very different questions. There is a 2's complement operation which can be used to negate a number (e.g. translate 2 -> -2 or -5 -> 5). There is also a 2's complement representation (or format) of numbers which can be used to represent integers, and those integers can be positive and negative whole numbers.

To reiterate, the 2's complement operation can convert negative numbers to the corresponding positive values, or positive numbers to the corresponding negative values. The 2's complement operation negates the existing number, making positive numbers negative and negative numbers positive.

A 2's complement representation (or format) simply represents number, either positive or negative. If you are ever asked if a 2's complement number is positive or negative, the only appropriate answer is yes, a 2's complement number can be positive or negative.

The following sections will explain how to do a 2's complement operation, and how to use 2's complement numbers. Being careful to understand the difference between a 2's complement operation and 2's complement number will be a big help to the reader.

1.5.3 The 2's Complement Operation

A 2's complement operation is simply a way to calculate the negation of a 2's complement number. It is important to realize that creating a 2's complement operation (or negation) is not as simple as putting a minus sign in front of the number. A 2's complement operation requires two steps: 1 - Inverting all of the bits in the number; and 2 - Adding 1₂ to the number.

Consider the number 00101100₂. The first step is to reverse all of the bits in the number (which will be achieved with a bit-wise ! operation. Note that the ! operator is a unary operation, it only takes one argument, not two.

! (001011002) = 11010011₂

Note that in the equation above the bits in the number have simply been reversed, with 0's becoming 1's, and 1's becoming 0's. This is also called a 1's complement, though in this text we will never use a 1's complement number.

The second step adds a 1₂ to the number.

11010011₂

\(\ \underline{00000001_2}\)

11010100₂.

Thus the result of a 2's complement operation on 00101100₂ is 11010100₂ , or negative 2's complement value. This process is reversible, as the reader can easily show that the 2's complement value of 11010100₂is 00101100₂. Also note that both the negative and positive values are in 2's complement representation.

While positive numbers will begin with a 0 in the left most position, and negative numbers will begin with a 1 in the leftmost position, these are not just sign bits in the same sense as the signed magnitude number, but part of the representation of the number. To understand this difference, consider the case where the positive and negative numbers used above are to be represented in 16 bits, not 8 bits. The first number, which is positive, will extend the sign of the number, which is 0. As we all know, adding 0's to the left of a positive number does not change the number. So 00101100₂ would become 0000000000101100₂.

However, the negative value cannot extend 0 to the left. If for no other reason, this results in a 0 in the sign bit, and the negative number has been made positive. So to extend the negative number 110101002 to 16 bits requires that the sign bit, in this case 1, be extended. Thus 11010100₂ becomes 1111111111010100₂.

The left most (or high) bit in the number is normally referred to as a sign bit, a convention this text will continue. But it is important to remember it is not a single bit that determines the sign of the number, but a part of the 2's complement representation.

1.5.4 The 2's Complement (or Integer) Type

Because the 2's complement operation negates a number, many people believe that a 2's complement number is negative. A better way to think about a 2's complement number is that is a type. A type is an abstraction which has a range of values, and a set of operations. For a 2's complement type, the range of values is all positive and negative whole numbers. For operations, it has the normal arithmetic operations such as addition (+), subtraction (-), multiplication (*) , and division (/).

A type also needs an internal representation. In mathematics classes, numbers were always abstract, theoretical entities, and assumed to be infinite. But a computer is not an abstract entity, it is a physical implementation of a computing machine. Therefore, all numbers in a computer must have a physical size for their internal representation. For integers, this size is often 8 (byte), 16(short), 32(integer), or 64(long) bits, though larger numbers of bits can be used to store the numbers. Because the left most bit must be a 0 for positive, and 1 for negative, using a fixed size also helps to identify easily if the number is positive or negative.

Because the range of the integer values is constrained to the 2ⁿ values (where n is the size of the integer) that can be represented with 2's complement, about half of which are positive and half are negative, roughly 2^n-1 values of magnitude are possible. However, one value, zero, must be accounted for, so there are 1 less positive numbers than negative numbers. So while 2⁸ is 256, the 2's complement value of an 8-bit number runs from -128 ... 127.

Finally, as stated in the previous section, just like zeros can be added to the left of a positive number without effecting its value, in 2's complement ones can be added to the left of a negative number without effecting its value. For example:

0010₂ = 0000 0010₂ = 2₁₀

1110₂ = 1111 1110₂= -2₁₀

Adding leading zeros to a positive number, and leading ones to a negative number, is called sign extension of a 2's complement number.