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Negative quantities

Table of contents

  1. Signed and unsigned numbers
  2. Complements in binary system
    1. 1’s complement
    2. 2’s complement

Signed and unsigned numbers

Currently, we have just looked at unsigned numbers - they can only be positive, as there is no sign. However, sometimes we need to work with negative numbers too. To do this, we add a sign bit on the far left of the binary number, which indicates whether the number is positive (0) or negative(1).

For example, the number 10000011 would be 131 if the number is unsigned, but if the number is signed, the actual representation would be -3

  • The first bit 1 represents that the number is negative
  • The remaining bits 0000011 represent the actual number, 3

The downside to using a signed number is that it removes one bit from the actual number representation, halving the maximum value.

  • The minimum and maximum values for an unsigned 8-bit number would be 0 to 2<sup>8</sup>-1 (0 to 255)
  • The minimum and maximum values for a signed 8-bit number would be -2<sup>7</sup>-1 to 2<sup>7</sup>-1 (-127 to 127)

Complements in binary system

The Binary system has a base of r = 2. The base has r = 2 so the Binary system two types of complements. They are:

  1. 1’s complement
  2. 2’s complement.

1’s complement

To find the 1’s complement of a given number, you should change all the 0’s to 1’s and then all the 1’s to 0’s. This process is called as 1’s complement. Simply negate each digit present in the binary number. An Example for 1’s complement is as follows:

Given number        1  0  1  0  1
1's complement      0  1  0  1  0

2’s complement

In order to find 2’s complement firstly, evaluate 1’s complement of the number, and further add 1 to it. 2’s complement = 1’s complement + 1. An Example of 2’s complement is as follows:

Given number        1  0  1  0  1
1's complement      0  1  0  1  0

add 1               +           1
		   ---------------
2's complement      0  1  0  1  1
		   ---------------
  1. Using signed two’s complement notation, what is the decimal value of 00010001 ?
    • -47
    • +37
    • -149
      1. +17
  2. The greatest negative number which can be stored in a 8-bit register using 2’s complement arithemtic is ?
    • -256
    • -255
      1. -128
    • -127
  3. The two’s complement of the signed decimal number -78 is ?
    • 11001110
    • 01001110
      1. 10110010
    • 10110001
  4. The range of positive numbers possible in an eight-bit two’s complement system is ?
    • 0 to 64
      1. 0 to 127
    • 0 to 256
    • 0 to 100.