__Binary Representation of Numbers __

An integer is a whole number. It may be positive or negative. Ordinary numbers are called real numbers. This includes all integers and all numbers with a decimal point.

There are various ways in which integers can be represented using 1s and 0s.

The binary notation is a method of representing numbers using 1s and 0s (Fig 1), In a binary number each 1 represents a power of 2. The powers of two are the numbers 1, 2, 4, 8, 16, etc. (see Fig 2).

Decimal number | Binary equivalent |

0 | 0 |

1 | 1 |

2 | 10 |

3 | 11 |

4 | 100 |

5 | 101 |

6 | 110 |

7 | 111 |

8 | 1000 |

9 | 1001 |

Fig 1 Binary values for 0 to 9

__Example __

In the binary integer 110111, working from the left, the Is represent a 32, a 16, a 4, a 2 and a 1 (the zero indicates there is no 8).

The number is equal to (in decimal) 32+16+4+2+1=55.

If a small binary number is represented in a long storage location, the digits at the left are made zero.

__Worked question __

Express the decimal numbers 7 and 5 as six-bit binary numbers.

7_{10}=000111_{2}

5_{10}=000101_{2}

** Note**: A suffix is used to indicate the base or radix of the numbers-10 for decimal, 2 for binary.

__RELATIVE ADVANTAGES OF BINARY AND DECIMAL REPRESENTATION __

Advantages of the binary system:

1- A binary digit has only two possible states, 0 or 1, and so is easy to represent using electrical or magnetic devices.

2- The instructions and circuitry necessary to make a machine carry out arithmetic operations in binary are very simple.

Disadvantages of the binary system:

1- Numbers represented in binary have a larger number of digits.

2- Binary numbers are difficult to write down accurately and to remember.