A Class of Smaller Infinities

Taeyun Lee

A Class of Smaller Infinities

Abstract

A continued fraction is a fraction of the form , where a andb are traditionally integers, and the indicates that the fraction continues on forever. One of the interesting things about continued fractions, however, is that their values do not (usually) come out to , even though they continue for an infinite amount of time. Many aspects of continued fractions have been explored, such as the way that they can be used to represent square roots and other irrational values, approximating the values more closely with each iteration. In this paper a generalized formula for expressing the sum of twocontinued fractions as a single continued fraction will be explored, as well as the limitations of this formula and the ways it can be altered to fit new conditions.

Introduction

Continued fractions have been a commonly explored problem in mathematics for many years. It is hard to say exactly when they appeared as the latest big math problem, but many think that the origin of continued fractions can be placed around the time that Euclid’s Algorithm was created, due to their close relationship.

Since their discovery, many different mathematicians have explored the many different properties of the continued fraction. Aryabhata, Bombelli, Cataldi, Wallis, and Huygens all contributed greatly to the field of continued fractions.

The topic of continued fractions took off when Euler, Lambert, and Lagrange picked up continued fractions, followed by many other 19th century mathematicians, who jumped onto

the continued fraction bandwagon, resulting in a lot of interesting results.

In 1972, Bill Gosper solved the problem of general arithmetic with continued fractions, using an algorithm based off of the idea that infinite continued fractions return the most significant value first. His algorithm uses the basis of Euclid’s algorithm to express continued fractions and do basic arithmetic with them. The acknowledged problem with this algorithm is that it is possible for the scheme to go on forever without producing any results.

In this paper, different properties of continued fractions will be explored in hope of finding formulas for calculating the sum of continued fractions with a direct formula, rather than an algorithm. However, this formula has limitations separate from those of Gosper’s algorithm.

The Basics

Research on this topic startedsimple–infinite continued fractions of the form . A pattern soon emerged, after evaluations of continued fractions where a=1, then 2, then 3, and so on. It did not take long to prove said pattern using generalized simple algebra and the usual method of evaluating a continued fraction:

Theorem 1.1 and Proof:

Given, let=x.

(This algebraic equation is valid because of the infinite continuation of the continued fraction in question. If ais added to the inverse of a plus the inverse of a plus the inverse of a plus… for an infinite amount of time, then it can be said that the fraction is the process of inversion and addition, repeated an infinite number of times. But, the process of inverting and adding an infinite number of times is the same as inverting and adding an infinite number of times – 1, by the definition of infinity.)

So, for all values a, =. [1.1]

Theorem 1.2 and Quick Proof:

It is simple to expand this formula into one that works for fractions of the form , where b is any number instead of 1:

Similarly, when given , let=x.

Then, again, x = . Multiply by x and simplify the quadratic formula obtained from the resulting equation to get that =[1.2] for all values of a andq.

The two formulas are very similar, and it can and should be noted that if q=1, which was the case for the continued fraction in 1.1, then the formula found in 1.2 will be identical to that of 1.1, which supports the legitimacy of the two formulas.

Theorem 1.3 and Proof:

The formula for an infinite continued fraction with a period of 2 (of the form ) can also be found algebraically without too much trouble, though the algebra gets a little more complicated.

In the same fashion as before, given , let = x.

Ignoring the extraneous negative solution, = for all values a, b, q. [1.3]

The original intention was tocontinue to find algebraic formulas for infinite continued fractions of different period lengths with the hope of finding a general formula that would work for any infinite continued fraction of period length p. However, the formula for a periodic infinite continued fraction gets very complicated very quickly as n is increased. After finding (using the same methods shown above) that an infinite continued fraction of period 3 and numerator 1, of the form, could be expressed as, and starting to attempt to find the exponentially more complicated formula for an infinite continued fraction of period 4 and numerator 1, it was evident that the exploration of continued fractions of different periods would be much too complicated, especially considering there was no guarantee of a pattern and no evident trend among the formulas for p = 1, 2, and 3.

Picking Up Some Useful Things Along the Way to Sums

Theorem 2.1 and Proof:

Next, a different topic was pursued: finding a generalized formula for the sum of two infinite continued fractions. This problem initially seemed impossible, as infinite continued fractions could not be expressed without a square root, but then research uncovered the following helpful formula.

[2.1]

Neither proof of this formula, nor a reference to it anywhere outside of Wikipedia, was found, so the following is a simple algebraic proof that is valid.

Given , then there exists an x and y such that .

1:

2: , so, substituting into the expression above:

3: , since.

4:

5: so, again substituting into the expression above:

6:

So, so far:.

It is already known from 4 and 6 that

Sincecontinues to be the denominator, the fraction can be expressed as .

So .[2.2]

Theorem 2.3 and Proof:

In the process of trying to obtain a result for a formula for the sum of two continued fractions, another formula was found for the multiplication of an infinite continued fraction by a constant, through the evaluation of various examples. The following is a proof that for all constants r and all infinite continued fractions of the form

By 1.2, =.

So,

By distribution,

1.2 applies to ; here ra is a in 1.2, and qr2 is q in 1.2.

So, using 1.2, .

Therefore . [2.3]

Continued Fractional Addition

Theorem 3.1 and Proof:

Next, a new approach was tried: solving an equation instead of trying to simplify an expression. The following equation was solved for a (using a program to make the process simpler):

By 1.1, the above equation could be rewritten as .

When the equation was put into a program, it turned out that

So,

There is, of course, an obvious problem with this formula; it involves fractions within a fraction. Thanks to 2.3, this problem can be solved by multiplying the sum fraction by . However, since this is an algebraic equation, both sides must be multiplied by so that it remains true.

This gives us:

When expanded, turns out to be remarkably closely related to . Actually, is 4 more than twice .

So, can also be expressed as , where , for all values m and n. [3.1]But, of course, this formula only works for the sum of fractions in the form .

Getting Rid of Unwanted Square Roots:

It is clear that, with this formula, there will inevitably be a square root in the continued fraction expression of a sum of continued fractions, which goes against some definitions of a continued fraction that say they can only contain integers. This (to a certain extent) can be fixed, using the formula of 2.2, since the square root parts of the formula are already conveniently in the form . Thus:

by 2.3:

This removal of square roots is a bit counterproductive in that it results in the addition of continued fractions, which is what was started with. So, since the square root – continued fraction relationship is not the focus of the paper but rather just a small nuance worth noting, the formula will be kept in its original form.

Expanding the Sum Formula

Conjecture 4.1:

As a result of the way that the formula was derived, it is only effective for the addition of two continued fractions of the form . Originally, the goal was to generalize the formula to work for fractions with different period lengths, but as withthe previous attempt with exploring periodicity, little progress was made before the results became much too complicated to work with (as in, Wolfram Alpha stopped returning results). So, the results of altering the numerators were explored instead.

First, the following equation was solved, where theaddends had the same numerator (not equal to 1) and the resulting continued fraction had any numerator h:

is the same as by 1.2.

The solution is simpler if the resulting continued fraction has the same numerator as the summed continued fractions:

The disparity between the simplicities of the two formulas led to the conclusion that it is best for the sum fraction to have the same numerator as the two fractions being added together, though any numerator can be used. So, if the numerators of the two summed fractions are different, then what should the numerator of the resultant fraction be?

To answer this question, 3 different equations were solved to obtain expressions for a in different cases. In one, the numerator of the sum continued fraction was any number h. In the other two, the numerator of the sum continued fraction was the numerator of the first added continued fraction, or that of the second added continued fraction.

First case: Numerator is any separate number h. Then:

Second case: Numerator is same as that of first summed fraction (q)

Third case: Numerator is same as that of second summed fraction (s)

The terms in each formula were then examined to see if the formulas were interchangeable. It turns out that they are; there are some terms that are exactly the same in both formulas, some terms that require switching ofq and s, and some terms that require switching of not only q and s but also m and n. In the end, the formulas are the same in all ways that matter for the second and third cases. This is the expected result, considering that addition is supposed to be commutative and it would not make a lot of sense for the formulas to be different if the sum fraction had the numerator q rather than s, or vice versa. It can thus be concluded that it is of no importance which numerator of the summed fractions is used as the numerator for the resulting sum fraction, in terms of convenience or complexity of the final solution.

It can easily be seen that the formula for the first case is significantly longer and more complicated than formulas for the second and third cases.

Therefore,it can be concluded that, at least for the formulas, it is better to express the sum of two continued fractions with different numerators as a continued fraction with one of those two numerators (though it is of no importance which), rather than as a continued fraction with any other numerator. Of course, it is assumed that ‘better’ means ‘more convenient and less complicated’. [4.1]

Conclusion

In this paper, a semi-generalized formula to express the sum of two continued fractions as a continued fraction was covered. This formula is limited in that it will not hold for fractions with periodicities greater than 1, but has been expanded to include fractions with numerators other than one. In some ways, the formulas covered in this paper are more limited than Gosper’s algorithm, but they will always result insome answer.

Along the way, a couple of tangentially relevant though useful properties of continued fractions were explored and proved, including the expression of a square root as a continued fraction, and how to multiply a continued fraction by a given constant. These theorems, though simple to prove, were very useful later in the more complicated endeavor of obtaining generalized sum formulas.

Was a generalized formula for the continued fraction expression of the sum of two continued fractions? Yes. Unfortunately, this formula is not quite as generalized as it would be; ideally there could be one found for two continued fractions with any numerators and any period length. However, finding the formula of the sum of basic continued fractions of the form , as well as for continued fractions with varying numerators, is definitely a step forward in the direction of finding the ultimate formula that would work for continued fractions of any form.If research on this topic was continued, it would be especially interesting to try finding formulas that could apply to any one of the other 3 basic mathematical operations in the context of continued fractions. Alternatively, a better way to derive these types of formulascould be sought (better meaning that it will not cause Wolfram Alpha to stop returning answers), allowing for an exploration of addition with continued fractions of periodicities greater than one.

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