Scheme of the continued fractions

Such fractions as shown here

are called “continued fractions”. In such fractions a fraction with numerator 1 is added to the number , a fraction with numerator 1 is added to the number of that fraction, and fraction with numerator 1 is added to the number of a fraction of that fraction and so on. Here the numbers are whole numbers (number can be equal to 0).

Fractions with the meaning in continued fractions are called making fractions. If makers are infinitely many, they are called infinite fractions, and if makers can be counted, they are called finite fractions.

The fraction shown above can be written in this form

().

Examples: this fraction we can be written in this form

Theorem1. Every finite continued fraction can be expressed in a form of common fraction with the value equal to the it.

Proof. Actually every fraction consists of adding and dividing operations, that is why if its making fractions are countable, it is possible every time to solve and finish adding and dividing operations shown among them. The solvation of those operations is certainly can be in a form of a common fraction.

for example: we have such fraction

Then

1) 2) 3) 4) 5)

This finite continued fraction can be expressed in a form of common fraction with the value equal to the it.

Theorem 2(converse theorem): Every common fraction with positive value can be expressed in a form of continued fraction with the value equal to it.

Proof. Let’s take a fraction with meaning . If we take whole numbers from it, we will get equation ,

where a – whole multiple, value of t would be equal to the value of dividing x by y. If there is no whole numbers in the fraction , the condition a = 0 is suitable and it is equation t = x.

Now if we divide both members of the fraction into t, so there will be equations

.

Here number is a whole multiple as a result of dividing y by t and is the difference.

Now if we divide both members of the fraction by , so we will get the equation:

Here the number is a whole multiple from dividing t by .

If we use this fraction line continueously for the following made fractions, it is proved that in the end given fraction

will make an expression in the above form. Hereof the theorem is solved.

In the above approach when we divide x by y, y by the first difference t, the first difference by the second difference, numbers are the resulting whole multiples. That is why they are also called multiples of the continued fractions. The mentioned approach is also used when finding the biggest new multiples of two numbers of value x and y.

Example:

It is necessary to convert the fraction into the infinite fraction. In this case it has the following expression:

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