Converting Repeating Decimals from Base X to Base Y

Date: 10/02/2009 at 23:04:11
From: Evan
Subject: converting repeating decimals from base x to y 

How do I convert a repeating decimal from base 9 to base 6?  My
problem is 0.54444444... (base 9) to base 6. 

I don't know if the normal way to change a repeating decimal works in 
this situation.  Should I set .544444 as x and another number as 100x 
(base 9 or 10?)

I've tried thinking about it this way.  If I want to make 3 (base 9)
to 30 (base 9), I would have to multiply by 11 (base 10) which is 12
(base 9).  However, I don't know if this will always work.  Please
help!  Thank You.

Date: 10/03/2009 at 02:09:31
From: Doctor Greenie
Subject: Re: converting repeating decimals from base x to y

Hi, Evan --

The "normal way to change a repeating decimal" that I think you are 
referring to is a method for converting a repeating decimal to a 
common fraction in base 10.  I don't see that the method is 
applicable to converting a repeating decimal from one base to another 
since the beauty of it in base 10 is that multiplying the number by a 
power of 10 simply moves the decimal but does not change the digits 
or their order.

Note that, technically, the word "decimal" should only be used when 
we are talking about base 10 numbers.  However, in the discussion 
below, I informally use the word "decimal" to refer to both base 9 
and base 6 numbers.

Note that a repeating decimal in one base might well be a terminating
decimal in another base.  For example, in base 10, the fraction 1/3 is
0.333333...; but in base 3 it is simply 0.1.

Here is a link to a page in the Dr. Math archives which contains a 
description of a process for converting a decimal from base 10 to 
base 3, using base 10 arithmetic:

  Fraction/Decimal Conversion to Other Bases
    http://mathforum.org/library/drmath/view/55744.html 

You can use the process described there to convert from base a to 
base b--but the arithmetic has to be done in base a.  So to convert a 
number from base 9 to base 6, you need to do arithmetic in base 9.

It is, in general, far easier to convert a repeating decimal from 
one base to another by going through base 10.  Let's do that to find 
out what our answer should be for your base 9 repeating decimal 
0.544444..., converted to base 6.

To convert 0.544444... base 9 to a common fraction in base 10, we 
use an infinite geometric series:

  0.544444... base 9 =
  5/9 + 4/81 + 4/729 + ... =
  5/9 + (4/81)/(8/9) =
  5/9 + (4/81)/(72/81) =
  5/9 + 4/72 =
  40/72 + 4/72 =
  44/72 = 11/18

As it turns out, the base 6 decimal equivalent of this base 10 
fraction terminates:

  11/18 = 22/36

Since 36 is 6^2, the decimal equivalent of this fraction will 
terminate with two decimal places:

  22/36 = 18/36 + 4/36 = 3/6 + 4/36 = 0.34 base 6

Now we will use the process described on the referenced page to 
perform arithmetic in base 9 to perform the conversion of the 
repeating decimal 0.544444... base 9 to base 6.  The process is to 
multiply (in base 9) the given decimal fraction by 6 repeatedly, 
picking off the whole number part each time as the next digit in the 
converted decimal.

  0.544444...
       * 6
  ----------
  3.58888(6)...

In case you don't follow that base 9 arithmetic, here is a 
description of it....

We can't really multiply the entire repeating decimal by 6; however, 
we know that the digits of the product will repeat.  So we keep just 
a few of the repeating digits and look for the pattern in the 
product.

6 times 4 in base 10 is 24, which in base 9 is 26.  So the rightmost 
digit in our product is "6"; and we have a "carry" of 2.  (But since 
there are in fact digits in the repeating decimal to the right of 
the last one we kept, we know this last digit "6" will probably not 
be correct in the final result.)

6 times 4 in base 10 is again 24, plus the carry makes 26, which in 
base 9 is 28.  So the next digit of our product (which this time is 
probably the correct digit) is "8"; and again we have a carry of 2.

As long as we have digits "4" in our original base 9 decimal, we are 
going to continue getting digits 8 in our product.

Then, when we get to the leftmost digit in our base 9 decimal, we 
have 5 times 6 in base 10 is 30, plus the carry is 32; which in base 
9 is "35".  So the leftmost two digits of this product are "35".  
The "3" is the whole number part of the product; so the first digit 
of the base 6 equivalent of our number is "3".

The remaining digits of our product are

  .5888888...

But ".88888..." in base 9 is just like ".99999..." in base 10.  Just 
as 0.1999999... in base 10 is equivalent to 0.2, 0.58888... in base 
9 is equivalent to 0.6.

Now we again multiply (using base 9 arithmetic), the remaining 
decimal part, 0.6, by 6.  6 times 6 in base 10 is 36, which is 40 in 
base 9.  So this base 9 multiplication gives us

  0.6 * 6 = 4.0

This means the next digit of our converted base 6 decimal is "4"; 
and, since the remaining decimal part after this multiplication is 
0, the base 6 decimal terminates.

So the base 6 equivalent of the base 9 decimal 0.544444... is 0.34 --
as we determined earlier it should be.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/