| Sažetak | Prvi put spominjani i uvedeni još u 17. stoljeću, verižni razlomci su brzo pronašli svoju uporabu u kompleksnom svijetu matematike. Definiraju se kao funkcije oblika:
\( \left [ a_{0},a_{1},a_{2},\cdots a_{n} \right ]=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{\ddots +\frac{1}{a_{n}}}}}\)
Verižne razlomke dijelimo na konačne i beskonačne. Konačni verižni razlomci su dani kao razvoj racionalnih brojeva u verižni razlomak i oni se mogu zapisati na točno dva načina, s parnim ili neparnim brojem članova. Pokazali smo neka svojstva konačnih verižnih razlomaka, primjerice da se članovi dobivaju rekurzivno. Ako bismo verižni razlomak \([a_{0}, a_{1}, a_{2}, . . . , a_{n}] \) gledali do nekog člana, npr. \(a_{k}\), k ∈ N, onda izraz nazivamo k-ta konvergenta od
\([a_{0}, a_{1}, a_{2}, . . . , a_{n}] \). Konvergenta verižnog razlomka ima brojne primjene, primjerice kod rješavanja linearnih diofantskih jednadžbi i kod aproksimacije iracionalnih brojeva. Beskonačni verižni razlomci su verižni razlomci oblika \([a_{0}, a_{1}, a_{2}, . . . , a_{n}] \) . Pokazali smo da se svaki iracionalni broj može zapisati u obliku beskonačnog verižnog
razlomka i da je taj zapis jedinstven, a nakon toga smo pokazali neka njegova svojstva. Uveli smo pojam periodskog verižnog razlomka, vrsta beskonačnog verižnog razlomka kod kojeg se nakon nekog člana, niz prethodnih članova ponavlja u
pravilnim intervalima, te smo pokazali povezanost kvadratne iracionalnosti, rješenja kvadratne jednadžbe oblika \(ax_{2} + bx + c = 0, a\neq 0\), i periodskog verižnog razlomka.
Na kraju smo se bavili primjenama verižnih razlomaka, odnosno Pellovom jednadžbom i linearnom diofantskom jednadžbom oblika \(ax + by = (−1)^{n}\) , za a, b ∈ Z, b > 0,(a, b) = 1, koja je usko povezana s konvergentama verižnog razlomka. |
| Sažetak (engleski) | First mentioned and introduced as early as the 17th century, continued fractions
quickly found their utility in the complex realm of mathematics. They are defined
as functions of the form:\( \left [ a_{0},a_{1},a_{2},\cdots a_{n} \right ]=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{\ddots +\frac{1}{a_{n}}}}}\) .
We divide continued fractions into finite and infinite. Finite continued fractions are given as the expansion of rational numbers into continued fractions, and they can be written in exactly two ways, with an even or odd number of terms. We
have shown some properties of finite continued fractions, for example, that the terms are obtained recursively.
If we were to consider the continued fraction \([a_{0}, a_{1}, a_{2}, . . . , a_{n}] \) up to a certain term, for example, ak
, k ∈ N, then the expression \([a_{0}, a_{1}, a_{2}, . . . , a_{n}] \) is called the
k-th convergent of \([a_{0}, a_{1}, a_{2}, . . . , a_{n}] \). Convergents of continued fractions have numerous applications, such as in solving linear Diophantine equations and in approximating irrational numbers.
Infinite continued fractions are continued fractions of the form \([a_{0}, a_{1}, a_{2}, . . . , a_{n}] \)
We have shown that every irrational number can be represented as an infinite continued fraction, and this representation is unique. Afterward, we demonstrated some of its properties. We have introduced the concept of a periodic continued
fraction, a type of infinite continued fraction where, after a certain term, the sequence of previous terms repeats at regular intervals. Additionally, we showed the connection between quadratic irrationalities, solutions of quadratic equations of
the form \(ax^{2} + bx + c = 0, a \neq 0\), and periodic continued fractions. Finally, we explored the applications of continued fractions, namely the Pell equation and the linear Diophantine equation of the form \(ax + by = (−1)^{n}\)
, where a, b ∈ Z, b > 0, (a, b) = 1, which is closely related to convergents of continued fraction. |
