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.
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.
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