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# Signed binary integers

Signed integers are numbers with a “+” or “-“ sign. If n bits are used to represent a signed binary integer number, then out of n bits,1 bit will be used to represent a sign of the number and rest (n - 1)bits will be utilized to represent magnitude part of the number itself.

A real-life example is the list of temperatures (correct to nearest digit) in various cities of the world. Obviously they are signed integers like +34, -15, -23,and +17. These numbers along with their sign have to be represented in a computer using only binary notation orbits.

There are various ways of representing signed numbers in a computer −

Sign and magnitude

One's complement

Two's complement

The simplest way of representing a signed number is the sign magnitude(SM) method.

**Sign and magnitude** − The sign-magnitude binary format is the simplest conceptual format. In this method of representing signed numbers, the most significant digit (MSD) takes on extra meaning.

If the MSD is a 0, we can evaluate the number just as we would any normal unsigned integer. And also we shall treat the number as a positive one.

If the MSD is a 1, this indicates that the number is negative.

The other bits indicate the magnitude (absolute value) of the number. Some of signed decimal numbers and their equivalent in SM notation follows assuming a word size of 4 bits.

Signed decimal | sign-magnitude |
---|---|

+6 | 0110 |

-6 | 1110 |

+0 | 0000 |

-0 | 1000 |

+7 | 0111 |

-7 | 1111 |

## Range

From the above table, it is obvious that if the word size is n bits, the range of numbers that can be represented is from -(2^{n-1} -1) to +(2^{n-1} -1). A table of word size and the range of SM numbers that can be represented is shown in the following.

Word size | Range for SM numbers |
---|---|

4 | -7 to +7 |

8 | -127 to +127 |

16 | -32767 to +32767 |

32 | -2147483647 to +2147483647 |

Notice that the bit sequence 1101 corresponds to the unsigned number 13, as well as the number –5 in SM notation. Its value depends only on the way the user or the programmer interprets the bit sequence.

**One's complement** − This is one of the methods of representing signed integers in the computer. In this method, the most significant digit (MSD) takes on extra meaning.

- If the MSD is a 0, we can evaluate the number just as we would interpret any normal unsigned integer.
- If the MSD is a 1, this indicates that the number is negative.

The other bits indicate the magnitude (absolute value) of the number.

If the number is negative, then the other bits signify the 1's complement of the magnitude of the number.

Some signed decimal numbers and their equivalent in 1's complement notations are shown below, assuming a word size of 4 bits.

Signed decimal | 1’s complement |
---|---|

+6 | 0110 |

-6 | 1001 |

+0 | 0000 |

-0 | 1111 |

+7 | 0111 |

-7 | 1000 |

Range

From the above table, it is obvious that if the word size is n bits, the range of numbers that can be represented is from -(2^{n-1}- 1) to+(2^{n-1} -1). A table of word size and the range of 1's complement numbers that can be represented is shown.

Word size | Range for 1's complement numbers |
---|---|

4 | -7 to +7 |

8 | -127 to +127 |

16 | -32767 to +32767 |

32 | -2147483647 to +2147483647 ±2 × 10^{+9} (approx.) |

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