Diophantine Approximations: Examining the Farey Process an its Metho on Proucing Best Approximations Kelly Bowen Introuction When a person hears the phrase irrational number, one oes not think of anything clean an efinite, especially a mathematician Irrational numbers are typically estimate to a finite number of ecimal places, for example π 9, even though they have an infinite number of ecimal places There are vali methos that help give us closer approximations to irrational numbers in orer to obtain a more exact value Approximating an irrational number by a rational number is one topic in the fiel of iophantine approximation [] These approximations have been aroun since Diophantos of Alexanria who live in AD ; however, iophantine equations, which are polynomial equations such that the variables only represent integer values, have been examine even before Diophantos live One funamental problem of iophantine approximation is to estimate the value of an irrational number using a rational number of small enominator Farey fractions are an easily accessible tool for such approximations Farey fractions were stuie by British geologist John Farey Sr (-) These fractions specifically help estimate irrational numbers between an using rational fractions in irreucible form [] In this paper, we will be stuying the meiant, which is the funamental builing block of the Farey process, an its many properties We will also carefully efine the Farey process an how it can be use to estimate irrational numbers by forming a sequence of best left an best right approximations, in aition to using the Farey process in orer to etermine a best approximation to an irrational number Furthermore, we will en the paper with an application of the Farey process in an eucational setting First, let us efine the terms associate with the Farey process as carefully outline in the article Continue Fractions Without Tears written by Ian Richars [] Meiant Fractions We will begin with the funamental efinition of the theory behin the Farey process Definition The meiant of a pair of fractions a an c is the fraction forme by aing b the numerators an enominators, that is, Theorem shows how the meiant is orere among a b an c a+c b+ ()
Theorem Let a,b,c, > If a b < c, then a b < a+c b+ < c () Proof First, notice that c a bc a > Thus >, an so bc a >, since b > b b Now let us first prove that a b < a+c We have that b+ a+c b+ a b = (a+c)b a(b+) b(b+) = ab+bc ab a b(b+) = bc a b(b+) From our assumption, we know a,b,c, > an bc a > Thus an therefore bc a b(b+) >, a+c b+ > a b The proof for a+c b+ < c is similar Therefore a b < a+c b+ < c The Farey Process The meiant has more interesting properties when its ajacent fractions form a Farey pair Definition A Farey pair is a pair of two nonnegative fractions, say a b an c, with a b < c an bc a = A Farey interval is an interval, [a b, c ], such that the enpoints are a Farey pair Theorem an Definition form the basis of the Farey process The Farey process essentially starts with a Farey pair ( a b, c ), both of which are irreucible, an takes the meiant, a+c b+ This in turn creates two new Farey intervals [a b, a+c ] an [a+c b+ b+, c ] We prove this formally in Theorem The next step in the Farey process is to take the meiant of the Farey pairs ( a b, a+c ) an (a+c b+ b+, c ), which are a+(a+c) b+(b+ = a+c b+
an (a+c)+c (b+)+ = a+c b+, respectively One woul then continue taking the meiants of each new Farey pair, each time the enominators of the meiant are larger than either of the fractions in the Farey pair The Farey process was stuie in [] The following table is an example of the first five rows of the Farey process starting with the Farey pair (, ) Row Figure We can see from Figure that as the Farey process progresses, the number of fractions in each row increases as a result of forming the meiants between each Farey pair Several interesting observations can be note from examining the table These observations will be generalize for the entire Farey process an proven in the following section Observations There are several interesting patterns to notice that occur within the Farey process For any Farey pair ( a b, c ) it is true that c a b = bc a = b b, [] It follows irectly from the construction that for any row n, the istance between an the first fraction in the row is always Moreover, the ifference between any Farey pair n in row n is no greater than Aitionally, when the meiant is forme, the fraction pairs n ( a b, a+c ) an (a+c b+ b+, c ) are Farey pairs Thus the subintervals forme from the meiant are also Farey intervals The following theorem is foun in [] Theorem If a b an c are a Farey pair, then so are (a b, a+c ) an (a+c b+ b+, c ) That is, the two subintervals forme by inserting the meiant into a Farey interval are also Farey intervals
Proof Assume [ a b, c a+c ] is a Farey interval with meiant We will break the proof into b+ two parts, each examining one subinterval forme We will first examine [ a b, a+c b+ ] We alreay know a b < a+c We must prove b+ b(a+c) a(b+) = First, notethatsince[ a b, c ]isafareyinterval, weknowbc a = Well, b(a+c) a(b+) = ab+bc ab a Thus [ a b, a+c ] is a Farey subinterval b+ Now let us examine the subinterval [ a+c = bc a = b+, c must prove c(b+) (a+c) = Similarly, Thus [ a+c b+, c ] is a Farey subinterval c(b+) (a+c) = bc+c a c ] Again we alreay know a+c b+ < c We = bc a = We will now formally prove several of the observations mae Lemma If an c n are a Farey pair in row n, then either n or n, an their ifference c n = n Proof We will prove this lemma using inuction Thus, it is true for the first row: = Let n > an suppose the lemma is true for the (n ) th row That is, if a b an c are ajacent in row n, then c a b with either b n or n Also n suppose p q < r are ajacent in row n By how the Farey process is constructe, at least one s of p q an r s is the meiant of an ajacent Farey pair, say (a b, c ), from row n Without loss of generality, suppose r s = a+c b+ Then p q = a b an a b < a+c b+ < c are ajacent in row n Thus, by Theorem, a+c b+ a b = b(a+c) a(b+) = ab+bc ab a = bc a b(b+) b(b+) b(b+) = b(b+) = b +b By the inuction hypothesis, b = c a b <, an either b n or n, so n b+ (n )+ = n when n Also, b > n an we have b +b > +(n ) = n
Thus r s p q = a+c b+ a b = b +b < n As the Farey process progresses, the ifference between ajacent Farey fractions approaches zero Thus, for any real number x such that < x, the Farey process provies a sequence of fractions that approach x We will now prove that an irrational number < α < is the point of intersection Theorem Let α be an irrational number an < α < Let (, c n ) be the Farey pair in the n th step of the Farey process closest to α, with < α < c n As n, α an c n α Proof We know either or = + + an c n+ + < c n, < + + an c n+ + = c n Either way, [ +, c n+ ] [, c n ] By the neste interval property, there exists an α + + [, c n ] n N [] We know c n =, as prove in Lemma Thus as n n, c n It then follows that α is the only point in the intersection, so α an c n α as n Best Left an Best Right Approximations One of the immeiate consequences of the efinition of the Farey process is that it results in fining best left an best right approximations for an irrational number α The following efinition was first given in [] Definition Let α be an irrational number with < α < Then a fraction p q best left (respectively, best right) approximation to α if: (i) p q < α (respectively, p > α); an q is calle a (ii) There is no fraction x y between p q an α with a enominator y q We will now provie more notation to escribe the Farey process For < α <, let an c n be the fractions constructe by the Farey process in the n th step that are closest to
α on the left an right respectively So we have for n that < α < c n We will show later in Theorem that an c n are the respective best left an best right approximations Now in row n+ of the Farey process, either or + + = an c n+ + = +c n +, () + + = +c n + an c n+ + = c n () Suppose () hols, an let s be such that c n+k = c n for k s an that c n+s+ = +k +s+ +s +c n Thus we have the following sequence of approximations to α: +s + < + + < + + < < +s +s < α < c n+s+ +s+ < c n Note that + + = +c n +, + + = + +c n + +, an +s +s = +s +c n +s + Thus in general for k s Also note that, an so +k +k = +k +c n +k +, +s +c n +s + = c n+s +s > α, +s+ +s+ = +s +s Now suppose () hols Then there exists an s such that, similar to the first case, we will have c n+k = +c n+k an +k = for k s In this case we have the +k ++k +k following sequence of approximations to α: < +s+ +s+ < α < c n+s +s < < c n+ + < c n Example In orer to better illustrate this concept, we will provie an example examining an c n for the irrational number α =
n 9 9 c n One can see that keeps getting closer to the value of α in rows -, while c n stays at In row, remains the same an c n moves closer to α By row, our best left approximation is an our best right approximation is Demonstrating how the Farey Process Provies Best Left an Best Right Approximations In orer to be a best left or best right approximation, we nee to be sure that the an c n are closer to α on the left an right respectively than any fraction of lesser enominator The following theorem, foun in [], is the key to proving this Theorem 9 Among all fractions x y lying strictly between the Farey pair (a b, c ), the meiant is the one an only one with the smallest enominator
Proof Assume x y a+c b+ an a b < x y < c Note x,y > i) We will first show that y b+ Let = x y a b = bx ay Since x by y a b > an cy x = y bx ay >, the numerator bx ay so by Also, let = c x y Similarly cy x so y Thus Since ( a b, c ) are a Farey pair, we know + by + y = +b by = b+ b y Thus we have the following inequality: + = c a b = bc a b = b Multiplying both sies by b we have b = + b+ b y b+ y, an therefore y b+ ii) Now we will show y > b+ We alreay know the meiant a+c creates two Farey b+ subintervals Thus the meiants of these subintervals must have enominators greater than b+ Assume x y [a b, a+c ] By the previous proof of part one, we can see y b+(b+) = b+ b+ > b+ Thus y > b+ an the meiant has the smallest enominator The following theorem can be foun in [] Theorem Let α be an irrational number an let < α < Let a b an c be fractions given at some step of the Farey process that are closest to α on the left an right respectively Thus, a b an c are respective best left an best right approximations Proof ConsieranyFareypair a b an c thatareclosesttoαontheleftanrightrespectively, given at some step of the Farey process By Theorem 9 we know all fractions between a b an c have enominators that are greater than both b an Thus the fractions a b an c are clearly best left an best right approximations to any irrational number α lying between them
The following theorem can be foun in [] Theorem Every rational number p q written in lowest terms, with < p <, appears q at some stage in the Farey process Proof We will show this through a proof by contraiction Assume there exists a fraction p q, with < p <, that never appears in the Farey process This means at every stage in q the Farey process, p is somewhere between two ajacent fractions in the process We know q by Theorem that these fractions are always going to be a Farey pair Let n, with n being the row number We know at the n th step, < p q < c n, with an c n being the enpoints of the Farey interval in which p is an element For clarification we are not stating q p q is the meiant By Lemma, we know that at least one of the following is true: n or n By Theorem 9, we have q + n+ Thus since n is arbitrary, we have a contraiction The following theorem can be foun in [] Theorem Every best left or best right approximation to an irrational number α, with < α <, appears as the closest fraction to α on either the left or right at some step in the Farey process Proof By Theorem we know all fractions written in lowest terms appear at some stage in the Farey process Thus take any best left or best right approximation, say x, an examine y it the first time it appears in the Farey process By efinition, x y must be the meiant of its two neighbors, call them a b an c So we have a b < x y < c Consier the following two cases: i) Suppose a b < α < c Then either α [a b, x y ] or α [x y, c ] Either way x y occurs in the Farey process ii) Suppose α is not between a b an c Then α < a b or α > c Either way, one of a b or c is closer to α than x y However, a b or c have smaller enominators than x y, thus x is not y a best approximation to α Best Approximation Obtaining a best left an best right approximation is helpful in estimating irrational numbers, but what if we coul obtain a single rational number as a best approximation to an irrational number α? Definition A rational number p is consiere a best approximation to an irrational q number α if an only if α p q < α x whenever y q y 9
Notice it follows irectly from this efinition that every best approximation is either a best left or best right approximation, epening on which sie of α the fraction p is on We q are now intereste in whether the converse is true, that is, whether every best left or best right approximation is a best approximation Example In orer to better illustrate this concept, we will provie an example giving a best approximation to the irrational number α = e Let an c n enote the respective best left an best right approximations prouce by the n th step of the Farey process The first six values of an c n are provie in the following table Note the best approximations are bol-face n c n One can easily see that,,,, an are all best approximations Let us examine the best left approximation of an the best right approximation Clearly α 99 < α Thus no fractions of enominator less than exist that are closer to α since all fractions between an have enomiator at least However, when we reach row, is the first best right approximation that is not a best approximation because α 9 < α 9, but < This now raises the question: What conitions qualify a best one-sie approximation to be a best approximation? The following theorem takes into consieration the cases where the best approximation is a best left or best right approximation Theorem Let an c n be the sequence of respective best left an best right approximations to α prouce by the Farey process For a given n, let s be the largest nonnegative integer for which c n = c n+s Then +k is a best approximation if s < k s Similarly, if +s +k
s is the largest nonnegative integer such that = +s +s, then c n+k +k is a best approximation if s < k s Proof We will first consier the case where the best approximation is a best left approximation Suppose first that s is the largest nonnegative integer such that c n = c n+s Then +s the following orer is true: < + + < < +k +k < < +s +s < α < c n+s+ +s+ < c n () We are intereste in etermining when +k is a best approximation Since +k an +k +k c n form a Farey pair, there are no fractions between +k an c n of enominator less +k than +k Thus, if α +k < α c n, then +k will be a best approximation, since +k +k +k = +k + Let k be such that s < k s Then we nee to etermine when α a n+k < c n+s+ a n+k +s+ < c n+s+ c n +s+ < α c n, +k +k that is, we want to show +k is closer to α than c n +k It is clear from that α a n+k < c n+s+ +k +s+ It is also clear that +k +k c n+s+ c n +s+ < α c n Our goal therefore is to show c n+s+ a n+k +s+ < c n+s+ c n +s+ () Recall, +k + = +c n + = ( +c n )+c n = +c n + = ( +c n )+c n = +c n +k = ( +(k )c n )+c n = +kc n +s = ( +(s )c n )+c n = +sc n
an + = + + = ( + )+ = + + = ( + )+ = + +k = ( +(k ) )+ = +k +s = ( +(s ) )+ = +s Also, c n+s+ = + (s + )c n an +s+ = + (s + ) If k s, we have the following: c n+s+ +k +s+ +k = +kc n+s+ +k +s+ +k +s+ = ( +k )( +(s+)c n ) ( +(s+) )( +kc n ) +k +s+ Expaning the numerator an simplifying gives us: c n+s+ +k +s+ +k = k +(s+) c n (s+) k c n +k +s+ = k( c n )+(s+)( c n ) +k +s+ = k( )+(s+)() +k +s+ k +s+ = () +k +s+ Also, c n+s+ c n +s+ = c n+s+ c n +s+ +s+ = ( +(s+)c n ) c n ( +(s+) ) +s+ = +(s+)c n c n (s+)c n +s+ = c n +s+ = () +s+
By an, to show when hols we must etermine when is true if k +s+ +k +s+ < multplying both sies by +s+ we obtain Multiplying both sies by +k we have +s+, k +s+ +k < k +s+ < +k = +k, k +s+ +k +s+ < +s+ This which implies Rearranging gives an iviing by gives Since < we have an so k +s+ < +k s+ < k, s + < k <, < < Thus, if k > s +, then hols If k > s, then k s + > s + an k +s+ therefore hols Thus < is true if k > s +k +s+ +s+ So if s < k s, then +k +k is best The proof for if = +s, then c n+k is a best approximation is similar +s +k Example We will now emonstrate this concept with an example examining the irrational number α = + 9 The fractions an c n provie in the following table are the respective best left an best right approximations for α, while the best approximations are bole Aitionally, k is provie in orer to better illustrate the concept of which best one-sie approximations are best approximations base on the conition of s < k s
n k s = c n We can see for n, the meiant of an c n is on the right of α Thus when n =, s =, an s = So for k, we have c +k is a best approximation to α by +k Theorem Specifically, c, c, c are all best approximations One can verify by checking α c +k < α for k +k Now we will examine when c n+s+ +s+ = c n+s +s an +s +s = +s +c n +s +
n k 9 s = 9 9 c n n k 9 9 c n 9 Now we may start the process over with n = Then for n, the meiant of an c n is on the left Since s =, s = an so for k, we have a +k is b +k a best approximation to α by Theorem Specifically, a, a, a, a, a are all best b b b b b approximations One can verify by checking α a +k < α for k Note b +k Theorem states s < k s, while in this example we have s k s This is allowe since Theorem is not an if an only if proof This specific situation is consiere as an area for further research
Applications for the Classroom Every stuent is taught that when aing two fractions together, one cannot just a the numerators an enominators together, one must first fin a common enominator between the two fractions at han Once the fractions have a common enominator, then they are allowe to a the numerators Thus the Farey metho for aing two fractions together will at first seem foreign to a stuent; however, the Farey process can be incorporate into lessons within the classroom as a ifferent, but useful metho for problem solving For example, when a Farey interval is ivie into two subintervals when the meiant is forme, the ratio of the length of the two intervals is b [] Notice, a+c b+ a b c a+c b+ ba+bc ab a b(b+) = bc+c a c (b+) = bc a b bc a = b One source iscusse various problems that stuents can solve using the Farey process [] Example Farmer Fre has chickens an cows, an he knows that they have a total of heas an legs How many chickens an how many cows oes he have []? We know that the ratio between heas an legs for a cow is, while the same ratio for a chicken is We know from the problem that the ratio between heas an legs for the total number of chickens an cows that Farmer Fre has is, thus this woul be consiere the meiant of the two fractions we are trying to obtain since < < Note that has to be equal to the ratio of chicken legs to cow legs Using one of our previous observations that state the ratio between the length of two ajacent Farey intervals is, we can etermine b just how many chickens an cows Farmer Fre has = Thus the ratio of chicken legs to cow legs is, so out of the total of legs, of the total are chicken legs an of the total are cow legs Thus there are chicken legs an cow legs, making Farmer Fre have chickens an cows [] This is just one of many problems that eucators can apply in the classroom as another approach to problem solving One coul also apply the Farey process to chemistry problems involving fining out the amount one nees of two solutions in orer to make one solution of a certain concentration, in aition to problems involving figuring out istance travele base on how fast a person was going at various points of the trip
9 Conclusion In this paper on iophantine approximations, we specifically examine the Farey process which uses irreucible fractions between an in orer to prouce best left an best right approximations for an irrational number < α < We then further researche the process in orer to iscover the conition in which a best one-sie approximation is a best approximation Areas for further research woul inclue examining if a best approximation coul occur uner the conition s k s when s is a whole number Aitionally, it woul be interesting to examine if the Farey process properties stay consistent through other intervals of length one, or if any new properties arise Furthermore, we may compare the Farey process to other methos of estimating irrational numbers using rational numbers to etermine which metho is more accurate References [] Diophantine Approximation an Abelian Varieties Es A Dol, B Eckmann, F Takens Berlin: Springer-Verlag, 99 [] Mihaila, Ioana FAREY SUMS an Unerstaning RATIOS The Mathematics Teacher,9: October, - [] Niven, Ian, Herbert S Zuckerman, Hugh L Montgomery An Introuction to the Theory of Numbers Fifth E New York: John Wiley & Sons, Inc, 99 [] Richars, Ian Continue Fractions Without Tears Mathematics Magazine,: September 9, - [] Richarson, Leonar F Avance Calculus: An Introuction to Linear Analysis New Jersey: John Wiley & Sons, Inc,