I have to convert a number that looks like this 254.3212 base 6 to base 3. Now I can manage the decimal part (254) but I'm having trouble with the fraction (3212). How do I convert the fractional part ?

I have been trying this but it's not working out. I tried to multiply 3212 by 3 while calculating in base 6 but I think I am doing it wrong, so a very detailed (base oriented) explication would be much appreciated.

Here is what I'm unsure of: when I do this: 0.3212*3; 3*2 = 6 / 6 (divide by source base) = 1 r 0 ... then I keep the 0 and remember the 1 or am I doing it wrong ? then if I remember the 1 ... next step would be 3*1+1 / 6 = 4 / 6 = 0 r 4 remember the 0 keep the 4 so on and so forth. But somehow the result is nothing like Wolfram Alpha. I'm stumped.


Pi in base 10 is 3.1416... now to convert this to base let's say 26: 3 remains 3 that's easy, but the fractional part: 1416 × 26 = 3.6816 .6816 × 26 = 17.7216 .7216 × 26 = 18.7616 etc...

The number ends up being: 3.31718 ... BUT if we weren't starting from base 10 ... then the calculations (.6816 wouldn't be in base 10 either ... so they would look different, but different how ?)


Please no examples with base 10 because I am very unsure how this works and base 10 will confuse more than help.

I thank you in advance.

EDIT: If it works differently in any way when converting from smaller to bigger base (3 to 6) I would appreciate an example at least, thank you.

Also because of my bad English I may have used decimal in a wrong way ... By decimal I think I meant "not fractional". If some one could help me with the right word I'll edit it.

  • $\begingroup$ If you can do it for the "decimal" part, why don't you just convert your number into a purely decimal part, apply what you did and then convert it back? $\endgroup$ Dec 28 '11 at 13:18
  • $\begingroup$ I mentioned: "Here is what I'm unsure". It simply doesn't work for some reason and I don't get why. I'm not doing this at a mathematics course so the teacher just breezed trough these like they were nothing ... and I got nothing. $\endgroup$
    – Kalec
    Dec 28 '11 at 13:33

Here is what I'm unsure of ... when I do this: 0.3212*3; 3*2 = 6 / 6 (divide by source base) = 1 r 0 ... then I keep the 0 and remember the 1 or am I doing it wrong ?

No it's right. After that $3*1+1=4$ (in base 6) and so on :
$0.3212*3 = 1.4040$
$0.404*3 = 2.020$
... (warning : this will not end and you'll have a repetition of '1' at the end but this is not an error since nor 91/162 nor 1/2 admit a finite development in base 3)

Fine continuation!

  • $\begingroup$ Okay, think I'm getting it. Was very unsure if this is the right way or not. Now when I reform the number (use the "decimal" part of the different multiplications) I assemble them in order or reverse order ? $\endgroup$
    – Kalec
    Dec 28 '11 at 14:15
  • $\begingroup$ the first 'digit' is the most significative so that the fractional part will be .12001... $\endgroup$ Dec 28 '11 at 14:20

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