   SEARCH HOME Math Central Quandaries & Queries  Question from Peter, a student: How do you change bases eg. 121 from base 3 to base 5? Hi Peter,

I'm going to illustrate by converting 4325 to base 8. I an going to do it in two steps. First convert 4325 to base 10 and then convert this base 10 number to base 8.

Converting 4325 to base 10 just requires remembering what base 5 means.

4325 = 4 × 52 + 3 × 5 + 2 = 4 × 25 + 15 + 2 = 117.

To put 117 into base 8 imagine you have 117 CDs. Put them is stacks of 8 disks each. You get 14 stacks of 8 disks with 5 remaining (that is 117 ÷ 8 = 14 with a remainder of 5). Now take the 14 stacks and put them in groups of 8. You get 1 group of 8 stacks and 6 stacks remaining ((that is 14 ÷ 8 = 1 with a remainder of 6). Thus you have

1 group of 8 stacks (1 × 8 × 8 disks)
6 stacks (6 × 8 disks) and
5 individual disks.

Hence 117 = 1 × 82 + 6 × 81 + 5 = 1658

Here, more compactly is what I did

117 ÷ 8 = 14, remainder of 5
14 ÷ 8 = 1, remainder 6
1 ÷ 8 = 0, remainder 1

Read the remainders from bottom to top, 1658

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.