Ja 4, 007 USING THEON S LADDER TO FIND ROOTS OF QUADRATIC EQUATIONS Thomas J. Osler Mathematics Departmet Rowa Uiversit Glassboro, NJ 0808 Osler@rowa.edu. Itroductio Theo of Smra (circa 40 A. D.) described a remarkabl simple wa to calculate ratioal approximatios to. (See [], [3], ad [5].) It has become kow as Theo s ladder ad is show below. 3 5 7 7 9 4 M M Each rug of the ladder cotais two umbers. Call the left umber o the th rug x ad the right umber. We see that x = x + ad that = x + x. So the ext rug of the ladder is 9 + 4 = 70, ad 70 + 9 = 99. The ratio of the two umbers o each rug, / x gives us successivel better approximatios to. / =.00000L 3 3/ =.50000L 5 7 7 / 5 =.40000L 7 7 / =.4666L 9 4 4/ 9 =.4379L 70 99 99 / 70 =.448L 69 39 39 /69 =.440L
Notice that the umbers are alteratel above ad below =.44L. The covergece of / x to is slow. From the above calculatios, it appears that we gai a extra decimal digit i after calculatig aother oe or two rugs of the ladder. I [], Theo s ladder was geeralized so that a square root could be calculated, ad i [4] it was geeralized to allow for the calculatio of a cube, fourth, fifth, etc., root. I this paper we exted Theo s ladder so that we ca fid the roots of a arbitrar quadratic equatio. This paper is almost etirel precalculus mathematics. I the fial sectio we discuss the details of the calculatio of a limit, ad this requires a more advaced backgroud. If this last sectio is omitted, the remaider of the paper should be easil followed b precalculus studets.. Theo s ladder for quadratic equatios Suppose we wish to fid ratioal approximatios to the roots of the quadratic equatio () r br c = 0. We will show that the recursio relatios () x = ( bx ) +, ad (3) = cx + achieve this ed. (There will be restrictios o b ad c, but we will fid them later.) For example, to fid a root of r r = 0 we use x = ad = x +. We alwas start with the first rug of. We get
3 / =.00000L / =.00000L 3 3/ =.50000L 3 5 5/ 3 =.66666L 5 8 8/ 5 =.60000L 8 3 3/ 8 =.6500L 3 /3 =.6538L Notice that the Fiboacci umbers appear i this ladder, ad the root beig approximated is the golde sectio ( + 5 )/. We ow show wh the recursio relatios () ad (3) lead to ratioal approximatios of a root of (). Our examiatio i this sectio is simple, but ot rigorous, sice we are required to assume that lim x exists. Later, (sectio 3), idepedet of this sectio, we will prove that this limit exists. Dividig () b () gives us cx + = x ( bx ) + Dividig the umerator ad the deomiator o the right had side b x we get x c + x = ( b) + x. Assumig that the limit exists, we let r = lim. The we have x c+ r r =, which b + r reduces to r br c = 0. Thus we see that the ladder ca be expected (at least i some cases), to gives ratioal approximatios to oe of the roots of the quadratic ().. Coectio to ( ). r
4 We ow show that the rugs of our ladder x ca be geerated b powers of simple biomials ( ) (4) rx = r. We will prove (4) b iductio. Notice that (4) is true whe =. Assume (4) is true for N+ N ( r) = ( r) ( r) = ( x r)( r) = ( x r + ) ( x + ) N N N N N N x ( br + c) + x + r = cx + ( b) x + ) r. ( ) ( ) ( ) ( N N N N N N N N But from the recursio relatios () ad (3), xn+ = ( bx ) N + N ad N + = cx N + N, we ow have (5) ( r) N + = N+ xn+ r. This completes our iductive proof. r = 3. A rigorous proof that the ladder coverges to r Earlier, i sectio, we gave a simple o rigorous demostratio that our exteded ladder coverges to r. Now we are able to give a rigorous demostratio. Suppose b < 0 ad 0 < c, the from () ad (3), x+ = ( bx ) + ad + = cx +, we see that both x ad are strictl icreasig series, ad that both are alwas positive. Thus the positive root of the quadratic equatio (6) r ( b ) c b = + 4 + / r br c = 0 which is is the ol root that our ladder could hope to approach. Dividig (5) b x we have (7) r r =. x x
5 Sice x icreases to ifiit, if we ca show that (8) r, the (7) will prove that our ladder coverges. To this ed, we seek restrictios o b ad c that make (8) true. From (6) we have b + 4c + b r =. Thus if (8) is true we have + 4 + ad so b c b b + 4c + b. Addig to this iequalit we get ad so b + 4c + b 0, b + 4c 4 b. Squarig ad simplifig we get (9) c+ b 4. Thus we have show that uder the restrictios b < 0, 0 < c ad c+ b 4, the iequalit (8) is true ad thus b (7) our ladder coverges to a root of the quadratic equatio. These restrictios are sufficiet for covergece, but the are ot ecessar. The ladder does coverge for a wider choice of b ad c, ad the reader might wish to explore this extesio.
6 The recursio relatio s () ad (3) that geerate the ladder are ot the ol pair that produce these umbers. We leave it to the reader to explore other possibilities. Refereces [] Giberso, Shau ad Osler, Thomas J., Extedig Theo's ladder to a square root, The College Mathematics Joural, 35(004), pp. -6. ISSN 0746-834 [] Heath, Sir Thomas, A Histor of Greek Mathematics, Vol., origiall prited b Oxford at the Claredo Press, ad reprited b Dover Publicatios, New York, 98, pp. 9-93. ISBN: 048640738 [3] Newma, James R., The World of Mathematics, Vol., origiall prited b Simo ad Schuster, 956, ad reprited b Dover, New York, 003, pp. 97-98. ISBN 048643688 [4] Osler, T. J., Wright, M. ad Orchard, M., Theo's ladder for a root, Iteratioal Joural of Mathematical Educatio i Sciece ad Techolog, 36(005), pp. 389-398. ISSN 000-739X [5] Vedova, G. C., Notes o Theo of Smra, Amer. Math. Mothl, 58(95), pp. 675-683. ISSN 000-9890