Number Representation in Binary

Different formats used for binary representation of both positive and negative decimal numbers include the sign-bit magnitude method, the 1’s complement method and the 2’s complement method.

Sign-Bit Magnitude 

In the sign-bit magnitude representation of positive and negative decimal numbers, the MSB represents the ‘sign’, with a ‘0’ denoting a plus sign and a ‘1’ denoting a minus sign. The remaining bits represent the magnitude. In eight-bit representation, while MSB represents the sign, the remaining seven bits represent the magnitude. For example, the eight-bit representation of +9 would be 00001001, and that for −9 would be 10001001. An n−bit binary representation can be used to represent decimal numbers in the range of −(2n−1 − 1) to +(2n−1 − 1). That is, eight-bit representation can be used to represent decimal numbers in the range from −127 to +127 using the sign-bit magnitude format. 

1’s Complement 

In the 1’s complement format, the positive numbers remain unchanged. The negative numbers are obtained by taking the 1’s complement of the positive counterparts. For example, +9 will be represented as 00001001 in eight-bit notation, and −9 will be represented as 11110110, which is the 1’s complement of 00001001. Again, n-bit notation can be used to represent numbers in the range from −(2n−1 − 1) to +(2n−1 − 1) using the 1’s complement format. The eight-bit representation of the 1’s complement format can be used to represent decimal numbers in the range from −127 to +127.

2’s Complement

In the 2’s complement representation of binary numbers, the MSB represents the sign, with a ‘0’

used for a plus sign and a ‘1’ used for a minus sign. The remaining bits are used for representing

magnitude. Positive magnitudes are represented in the same way as in the case of sign-bit or 1’s

complement representation. Negative magnitudes are represented by the 2’s complement of their

positive counterparts. For example, +9 would be represented as 00001001, and −9 would be written

as 11110111. Please note that, if the 2’s complement of the magnitude of +9 gives a magnitude of −9,

then the reverse process will also be true, i.e. the 2’s complement of the magnitude of −9 will give a

magnitude of +9. The n-bit notation of the 2’s complement format can be used to represent all decimal

numbers in the range from +(2n−1 − 1) to −(2n−1. The 2’s complement format is very popular as it is

very easy to generate the 2’s complement of a binary number and also because arithmetic operations

are relatively easier to perform when the numbers are represented in the 2’s complement format.