Converting fractionals

To convert a fractional, such as 0·5, from base ten to another base, we multiply the fractional by the new base, and take the integer produced as the first digit in the new base. We then subtract the integer from the result of the multiplication and multiply the fractional left by the new base to produce the second digit of the fractional in the new base - and so on.

For example, changing one-fifth, or 0·2, from base ten to base twelve
calculationintegerresult so far
0·2 x 12 = 2·420·2
0·4 x 12 = 4·840·24
0·8 x 12 = 9·690·249
0·6 x 12 = 7·270·2497
and the sequence repeats, so 0·2 (base ten) is 0·2497 2497 ... (base twelve).

If you want to convert from some base other than ten to another base, using a basic calculator, you will need to convert to base ten on the way - but it's not as difficult as it might seem.

Recently, for example, I wanted to calculate pi in base seven. My basic calculator does not have pi built in, and I could only remember the first few digits: 3·14159..., but converting this would give few accurate digits in base seven. On the other hand I can remember the base twelve expression for pi: 3·1848 0949 ... because the pattern is easy to memorise (and I prefer to use base twelve anyway, though when you want to do any calculations in different bases you just have to do them yourself - without a calculator!)

To convert the decimal value to base seven I can use successive multiplications by 7 to produce the digits for the base seven version. To convert the base twelve expression to base seven I needed to adapt the method.

Firstly 0·1 (base twelve) is 1/12, so we can enter 1, divide by twelve, and then use the "multiply by 7" method to produce the digits for a base 7 version of one-twelfth.

Given more figures we can reduce the conversion to a convenient repeating pattern of instructions: for example, to convert the dozenal 0·46:

enter 6 and divide by twelve;
add 4 and divide by twelve.
This produces the base ten version 0·375. Now we just apply our usual method of multiplying by seven and using the integers produced for the digits in the base seven expression of 0·46: 0·242424....

To convert a fractional from one base to another, using the decimal calculator as a stepping stone:

  1. Enter the last digit of the fractional
  2. divide by the old base (the one you are converting from)
  3. add the next left-hand digit of the fractional and
  4. repeat steps 2 and 3 until you reach the point, running out of digits.
  5. multiply the result on the display by the new base
  6. write down the integer produced
  7. subtract this integer from the display and
  8. repeat steps 5, 6 and 7 until you have sufficient digits
Sufficient digits... or until accuracy becomes suspect ...

The base 7 value for pi, by the way, starting from the dozenal value quoted, comes out as 3·066365 ... , which is fairly easy to remember, should anyone want to memorize it, and, of course, can be abbreviated to 3·1 (i.e. the common abbreviation of 22/7).