Sometimes a calculator is more complicated than it needs to be. For example, the display of
is almost certainly. This is easy to recognise, but what about
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.
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?