Signed numbers in digital world

In real world, it is easy to represent negative numbers as we can prefix a ‘+’ or ‘–’ sign to the number to denote if it is a positive or negative number. However, in digital world, we need to store all numbers as ‘0’ or ‘1’ and hence we need a representation of negative numbers in binary format.

Following  methods are commonly used to represent signed number in digital world:

Sign magnitude:

This is the easiest method to represent signed numbers in digital form. If a datatype stores signed number, then MSB (Most Significant Bit) represents the sign of number, and the remaining bits represent the magnitude of number. Usually value ‘0’ for MSB indicates a positive number and value ‘1’ for MSB indicates a negative number. For a signed char datatype, it can be pictorially represented as following:

E.g if we need to store a number in signed char datatype. i.e. 8 bits (1 byte), then

+3 = 00000011₂

-3 = 10000011₂

With this representation for signed numbers, the range of a datatype is

• For signed datatype
  1. Minimum value = -(2^n-1  – 1), where n is the size of datatype in bits
  2. Maximum value = (2^n-1 – 1), where n is the size of datatype in bits
• For unsigned datatype
  1. Minimum value = 0
  2. Maximum value = (2ⁿ – 1) where n is the size of datatype in bits.

Remember that you can know the size of any datatype on your system using sizeof operator. As you can notice, in this representation, integer 0 has two representation –  00000000₂ and 10000000₂, which are logically representing +0 and -0. This reduces the range that a datatype can have as integer zero occupies two representations. Additionally it makes the arithmetic operations complex and confusing.

1’s complement

1’s complement of a number is obtained by inverting all its digits. E.g. 1’s complement of 3 (= 00000011₂) is obtained by replacing binary digit 0’s with 1 and 1’s with 0, i.e. the 1’s complement for integer 3 is 11111100₂

With this representation, negative counterpart of a number is represented by it’s 1’s complement. Once again, the MSB denotes the sign of number.

+3 = 00000011₂

-3 = 10000011₂

Note that in this representation as well, the integer 0 has two representation – 00000000₂  and 11111111₂. Hence the problems that exist with Sign magnitude notation, persist in this representation as well. Also the range of numbers represented by a datatype remains the same, though the representation of negative numbers is different.

2’s complement

The most common method to represent negative numbers in digital world is 2’s complement. A negative number is represented as 2’s complement of it’s positive counterpart.

2’s complement of number = 1’s complement of number + 1.

e.g. 2’s complement of 3     = (1’s complement of 3) + 1

=  1’s complement of 00000011₂ + 1

= 11111100₂ + 1

= 11111101₂

With this representation, the negative counterpart of a number is represented by it’s 2’s complement. In this representation as well, the MSB denotes the sign of a number. Hence with 2’s complement representation for signed numbers-

+3 = 00000011₂

-3 = 11111101₂

As you can observe that now integer ‘0’ does not have different representation. 2’s complement of 0 (=00000000₂) too is 00000000₂

With 2’s complement, since the value ‘0’ for MSB denotes positive number, the maximum value for the datatype remains the same – 2^n-1 – 1. However the minimum value represented now is -1*2^n-1

e.g. +128 in binary notation is represented as 10000000₂

Hence with 2’s complement representation:

-128 = 1’s complement of 10000000₂ + 1

= 01111111₂+1

= 10000000₂

As you can observe that though +128 doesn’t fit in 8 bits, -128 very well fits in 8 bits, hence increasing the range of values that an 8 bit datatype can represent to -128 to + 127. This range was -127 to +127 in sign magnitude as well as 1’s complement representation.