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

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

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