Reduce a Vulgar Fraction
To a Decimal
The
method is to annex ciphers (0’s) to the numerator and divide by the
denominator. The decimal must consist of as many figures as there are ciphers
added to the numerator.
Examples:
(1)
Reduce to a decimal.
The
answer given without explanation is .078125.
(2)
Reduce to a decimal.
Macdonald
works this one out with longhand division.

318 

22 
70 


66 
3×22 

40 


22 
1×22 

180 


176 
8×22 

4 

I
have added the ciphers in red boldface. Three have been added so there are 3
figures in the decimal, i.e. Macdonald gives his result as .318. Notice that
the final subtraction yields the number 4 so that if we add the cipher to get
40 we are back to the line under 66. Consequently, the full answer (not given
by Macdonald is a repeating decimal .3181818 etc.
(3)
Reduce to a decimal.
The
answer given in the notebook is 0.03448275862068965517241379310. No details are
given. It is a good assumption that Macdonald just copied out the number and
did not work it through. It is a lot of work and cannot be checked with the
usual calculator or a program such Excel,
which does the calculation only to 16 decimal places, not 29. I did check the
division using the programming language R using function mpfr in the library Rmpfr,
which does calculation to any specified level of precision. The answer given is
correct. In writing out the next set of 28 decimals, it is also a repeating
decimal with blocks of 34482753448275862068965517241379310 repeating.