We normally work with numbers in base 10. In this section we consider numbers in *base 2*, often called *binary numbers*.

In *base 10* we use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

In *base 2* we use only the digits 0 and 1.

Binary numbers are at the heart of all computing systems since, in an electrical circuit, 0 represents *no* current flowing whereas 1 represents a current flowing.

In *base 10* we use a system of place values as shown below:

1000 | 100 | 10 | 1 | ||

4 | 2 | 1 | 5 | 4 × 1000 + 2 × 100 + 1 × 10 + 5 × 1 | |

3 | 1 | 0 | 2 | 3 × 1000 + 1 × 100 + 2 × 1 |

Note that, to obtain the place value for the next digit to the left, we multiply by 10.

If we were to add another digit to the front (left) of the numbers above, that number would represent 10 000s.

In *base 2* we use a system of place values as shown below:

64 | 32 | 16 | 8 | 4 | 2 | 1 | ||

1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 × 64 = 64 | |

1 | 0 | 0 | 1 | 0 | 0 | 1 | 1 × 64 + 1 × 8 + 1 × 1 = 73 |

Note that the place values begin with 1 and are multiplied by 2 as you move to the left.

Once you know how the place value system works, you can convert binary numbers to base 10, and vice versa.

Convert the following binary numbers to base 10:

(a) | 111 | |

(b) | 101 | |

(c) | 1100110 |

Convert the following base 10 numbers into binary numbers:

(a) | 3 | |

(b) | 11 | |

(c) | 140 |

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