Can you help me with a formula for converting base 10 numbers to other bases. It is needed for a JETS test next week.

Hi Paul:

Let's consider what is going on in a base 3 conversion to see the general algorithm (there's no nice `formula'). If we want to convert 204 base 10 to base 3, what we're doing is expressing 204 as a sum of powers of 3 -- instead of a 1's, 10's and 100's column we will have a 1's, 3's, 9's ... columns corresponding to the powers of 3. Consider the following table:

81 | 27 | 9 | 3 | 1 |

204 | ||||

68 | 0 | |||

22 | 2 | |||

7 | 1 | |||

2 | 1 |

- The top row keeps track of the powers of 3.
- In the 2nd row place 204 1's in the 1's column
- these 204 1's are the same as 68 3's and 0 1's which we record in row
three
- these 68 3's are the same as 22 9's and 2 3's which we record in row
four
- these 22 9's are the same as 7 27's and 1 9's which we record in row
five
- these 7 27's are the same as 2 81's and 1 27's which we record in row
six

49 | 7 | 1 |

204 | ||

29 | 1 | |

4 | 1 |

Of course what is going on in our base 3 example to get from one row to the next is that we are successively dividing by 3 and recording the result and remainders in the appropriate places; a shorter format once one has the proper understanding is shown below ( I don't recommend starting this way as students rarely understand why the method works if just shown the following).

3 | 204 | |

68 | 0 | |

22 | 2 | |

7 | 1 | |

2 | 1 | |

0 | 2 |

Hope this helps,

Penny Nom

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