Generating fraction

Viewed another way, a generating fraction is a way of expressing a decimal number. This, by means of an irreducible fraction, that is, where the numerator and the denominator do not have divisors in common, so that the fraction cannot be simplified into smaller numbers.

For example 6/8 is a reducible fraction because it is equivalent to 3/4, the latter being an irreducible fraction.

So, to make it clearer, the generating fraction of 0.25 would be 1/4, while the generating fraction of 0.15 is 3/20.

It should be remembered that a fraction is the division of a number into equal parts. It is made up of two numbers, both separated by a straight or inclined line (unless we are dealing with a mixed fraction). The number at the top is called the numerator, while the one at the bottom is called the denominator.

How to find the generating fraction

To know how to find the generating fraction we must distinguish three cases:

• When decimal number is exact: We take the number without the decimal point and divide it by ten raised to the number of decimal places, and then we simplify the fraction. That is, if we have, for example, 0.26, the conversion would be done as follows:

• When the decimal is pure periodic: We must remember that a pure periodic decimal is one that has one or more numbers in its decimal part that are repeated indefinitely. For example 0.1313131313…, so that 13 is repeated infinitely and can be expressed as follows:

So, to find the generating fraction of a pure repeating decimal, we must take the number without the decimal point, taking the period only once, and subtract the integer part from it. Then, we divide the result by a number that has as many nines as there are figures in the period, and finally we simplify until we find the irreducible fraction.

So, if we have 1.454545454545…, the conversion would be as follows:

• When the decimal is a mixed periodic decimal: A mixed periodic decimal is one whose decimal part is a periodic part and another is not, as in the following example: 3.456666666 … which can be expressed as

In these cases, to find the generating fraction we must take the number, without the decimal point and repeating the period only once. From that number we subtract the number made up of all the figures prior to the period. Finally, we divide the result by the number formed by as many nines as there are digits in the period and as many zeros as the decimal part that is not periodic (placing the nines before the zeros), and if possible the resulting fraction is simplified.

So, if we have the number 4.366666666…, the generating fraction would be: