The ideas that we have considered can be extended to other number bases.

The table lists the digits used in some other number bases.

Base | Digits Used |

2 | 0, 1 |

3 | 0, 1, 2 |

4 | 0, 1, 2, 3 |

5 | 0, 1, 2, 3, 4 |

The powers of the base number give the place values when you convert to base 10.

For example, for base 3, the place values are the powers of 3, i.e. 1, 3, 9, 27, 81, 243, etc.

This is shown in the following example, which also shows how the base 3 number 12100 is equivalent to the base 10 number 144.

81 | 27 | 9 | 3 | 1 | ||

1 | 2 | 1 | 0 | 0 | (1 × 81) + (2 × 27) + (1 × 9) + (0 × 3) + (0 × 1) = 144 in base 10 |

The following example shows a conversion from base 5 to base 10 using the powers of 5 as place values.

625 | 125 | 25 | 5 | 1 | ||

4 | 1 | 0 | 0 | 1 | (4 × 625) + (1 × 125) + (0 × 25) + (0 × 5) + (1 × 1) = 2626 in base 10 |

Convert each of the following numbers to base 10:

(a) | 412 in base 6 | |

(b) | 374 in base 9 | |

(c) | 1432 in base 5 |

Convert each of the following base 10 numbers to the base stated:

(a) | 472 to base 4 | |

(b) | 179 to base 7 | |

(c) | 342 to base 3 |

Carry out each of the following calculations in the base stated:

(a) | 14 + 21 | base 5 | |

(b) | 16 + 32 | base 7 | |

(c) | 141 + 104 | base 5 | |

(d) | 212 + 121 | base 3 |

Check your answer in (a) by changing to base 10 numbers. |

Carry out each of the following multiplications in the base stated:

(a) | 141 × 23 | base 5 | |

(b) | 122 × 12 | base 3 | |

(c) | 512 × 24 | base 6 |

Check your answer to (b) by converting to base 10 numbers. |

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