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

What we then see is that 204 is 2x81 + 1x27 +1x9 + 2x3 + 0x1, that is, 204 base 10 = 21120 in base 3. There of course is nothing special about base 3; if we wanted base 7 you would have seen:
 49 7 1 204 29 1 4 1
so that 204 is 411 in base 7.

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
i.e. 3 into 204 is 68 and a remainder of 0; 3 into 68 is 22 and a remainder of 2; ...; until we get to 3 into 2 is 0 with a remainder of 2. When you're done you read the right hand column from the bottom up. That is 204 base 10 = 21120 base 3.

Hope this helps,
Penny Nom

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