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# 数学代写|数论代写Number Theory代考|Continued Fractions

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## 数学代写|数论代写Number Theory代考|Continued Fractions

We will conclude this chapter on divisibility by introducing a topic that has a very old and rich history, continued fractions. Recall that the Euclidean algorithm produced the following sequences of equations to find the greatest common divisor for 30 and 72 :
$$72=2 \cdot 30+12,$$
$$30=2 \cdot 12+6,$$
$$12=2 \cdot 6+0 .$$
We can rewrite the first equation to produce an equation for the fraction $\frac{72}{30}$ :
$$\frac{72}{30}=2+\frac{12}{30}=2+\frac{1}{\frac{30}{12}} .$$
In turn, we can rewrite the second equation to produce an equation for the fraction $\frac{30}{12}$ :
$$\frac{30}{12}=2+\frac{6}{12}=2+\frac{1}{\frac{12}{6}},$$

which we can now substitute into the above equation for $\frac{72}{30}$ to get
$$\frac{72}{30}=2+\frac{1}{2+\frac{1}{\frac{12}{6}}} .$$
Finally, we rewrite the last equation from the Euclidean algorithm to write the fraction $\frac{12}{6}$ as 2 , and then express the fraction $\frac{72}{30}$ as
$$\frac{72}{30}=2+\frac{1}{2+\frac{1}{2}}$$

## 数学代写|数论代写Number Theory代考|The Arithmetica

Although many of the early discoveries in number theory were made in various regions throughout the world, often quite independently, modern number theory began when the remarkable work of the midthird-century Greek mathematician Diophantus was rediscovered in the West in the early seventeenth century by Pierre de Fermat.

Greek mathematics-completely lost to Europe for roughly seven hundred years-began to be known again in Europe only by the twelfth and thirteenth centuries because of Latin translations that became available along trade routes into Europe from the Islamic East. Islamic mathematicians and translators had been preserving and extending Greek mathematics, as well as absorbing influences from India, all these years in their great centers of learning such as Cairo, Baghdad, and Damascus.

The Arithmetica of Diophantus was originally a work consisting of thirteen books, though only ten of these survive-six books in Greek have long been known, but only recently four more books in Arabic translation have also been found. Much of the content of the Arithmetica was known in Europe by the fifteenth century largely due to the efforts of the most influential mathematician of that century, Johann Müller, who is usually known by the name Regiomontanus (which is Latin for “from Königsberg”).

By 1575 , the Arithmetica had been translated into Latin, and subsequently into French, and Rafael Bombelli had substantially revised his Algebra before its publication in 1572 as a result of reading a manuscript of Diophantus in the Vatican library and even included 143 problems taken directly from the Arithmetica. But the most famous translationand certainly the most important, at least until a definitive translation appeared in the late nineteenth century-was the 1621 edition by Claude Gaspard Bachet de Méziriac. In 1612, Bachet had published an extremely successful book on recreational mathematics, Problémes plaisants et delectables qui se font par les nombres, a collection of mathematical puzzles involving numbers. Bachet’s lifelong fascination with numbers made it inevitable that in his translation of Diophantus the commentary would emphasize that which interested Bachet the most, namely, questions involving numbers.

## 数学代写|数论代写Number Theory代考|Continued Fractions

$$72=2 \cdot 30+12,$$
$$30=2 \cdot 12+6,$$
$$12=2 \cdot 6+0 .$$

$$\frac{72}{30}=2+\frac{12}{30}=2+\frac{1}{\frac{30}{12}} .$$

$$\frac{30}{12}=2+\frac{6}{12}=2+\frac{1}{\frac{12}{6}},$$

$$\frac{72}{30}=2+\frac{1}{2+\frac{1}{\frac{12}{6}}} .$$

$$\frac{72}{30}=2+\frac{1}{2+\frac{1}{2}}$$

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