Sometimes a calculator is more complicated than it needs to be.  For example, the display of

7.33333333333

is almost certainly.  This is easy to recognise, but what about

0.09090909091

It should not take you too long to recognise this as

Problem 1    Rewrite the decimal numbers below as fractions:

                     a) 0.6666666667   b) 0.3636363636   c) 0.5555555556   d) 0.1428571429.

The first three are all relatively straightforward, but d) is a much more complicated recurring decimal.  Its cycle consists of 6 recurrent numbers (142857).  The number is in fact a decimal approximation to Check this on your calculator.

Now consider

0.0958904109589

You have to be very familiar with decimal approximations to spot this one!

There is, though, a method for finding the fraction equivalent.  We will illustrate it with this example.

Let                                               x = 0.0958904109589...

so that                                    10⁸ x = 9589041.09589...

Subtracting the first equation from the second gives

Problem 2   Cancel out common factors in the expression for x to find its fraction value in lowest terms.

Problem 3   Use the same procedure to find the fraction equivalent of

                    a) 0.5714285714...   b) 0.027027027...   c) 0.07692307692...

EXTENSION Use a computer, a calculator, or long division by hand, to find the recurring decimals for all fractions

Which fraction has the longest cycle?