Signed and Unsigned numbers

In mathematics, negative numbers in any base are represented in the usual way, by prefixing them with a "−" sign. However, on a computer, there are various ways of representing a number's sign. Now we discuss the following methods for representing signed numbers in a computer.

Sign-and-magnitude

The first approach for representing a number's sign is sign-and – magnitude. In this we allocate one sign bit to represent the sign: set that bit(often the most significant bit) to 0 for a positive number, and set to 1 for a negative number. The remaining bits in the number indicate the magnitude (or absolute value). Hence in a byte with only 7 bits represents the magnitude can range from 0000000 (0) to 1111111 (127). Thus you can represent numbers from −127₁₀ to +127₁₀ once you add the sign bit (the eighth bit). A consequence of this representation is that there are two ways to represent zero, 00000000 (0) and 10000000 (−0). Decimal −43 encoded in an eight-bit byte this way is 10101011.

This approach is directly comparable to the common way of showing a sign (placing a "+" or "−" next to the number's magnitude). Some early binary computers (e.g. IBM 7090) used this representation, perhaps because of its natural relation to common usage. (Many decimal computers also used sign-and-magnitude.)

One's complement

ones' complement can be used to represent negative numbers. The ones' complement form of a negative binary number is the bitwise NOT applied to it — the complement of its positive counterpart. Like sign-and-magnitude representation, ones' complement has two representations of 0: 00000000 (+0) and 11111111 (−0)..

As an example, the ones' complement form of 00101011 (43)  becomes 11010100 (−43).

The one’s complemenst form of 00101010 becomes 11010101

Two's complement

The problems of multiple representations of 0 and the need for the end-around carry are circumvented by a system called two's complement. In two's complement, negative numbers are represented by the bit pattern which is one greater (in an unsigned sense) than the ones' complement of the positive value.

In two's-complement, there is only one zero (00000000). Negating a number (whether negative or positive) is done by inverting all the bits and then adding 1 to that result. An easier method to get the two's complement of a number is as follows:



Related Topics : Computer Organization & Architecture