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matqkks
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What is the most motivating way to introduce continued fractions? Are there any real life applications of continued fractions?
Continued fractions are a way of representing numbers as a sequence of fractions. They are important because they can provide a more accurate and efficient approximation of irrational numbers compared to other methods, and they have many real-life applications in fields such as mathematics, physics, and finance.
Regular fractions, also known as simple fractions, are a ratio of two integers. Continued fractions, on the other hand, are a sequence of fractions that are nested within each other, with each fraction in the sequence representing a partial sum of the original number.
Continued fractions have many practical applications, some of which include solving Diophantine equations, finding best rational approximations, and analyzing the behavior of certain mathematical functions. They are also used in fields such as signal processing, cryptography, and number theory.
In finance, continued fractions are used to calculate the value of financial assets, such as bonds and options. They are also used in risk management and forecasting, as they can provide a more precise representation of market trends and fluctuations compared to other methods.
No, continued fractions can only represent numbers that are irrational or have an infinite decimal expansion. Rational numbers, which have a finite decimal expansion, can be represented using regular fractions.