Continued fractions and nested radicals are mathematical expressions that involve an infinite number of fractions or square roots nested within each other. They are used to represent irrational numbers in a concise and elegant way.
Continued fractions can be converted into nested radicals and vice versa. This means that they are essentially two different ways of representing the same number. Nested radicals are often considered simpler and more intuitive, while continued fractions are more useful for calculations.
A simple continued fraction is one in which all the numerators are equal to 1, while a regular continued fraction has a fixed pattern in the numerators. For example, a simple continued fraction would be 1 + 1/(1 + 1/(1 + 1/...)), while a regular continued fraction could be 1 + 1/(2 + 1/(3 + 1/...)).
Continued fractions and nested radicals have many practical applications in fields such as number theory, physics, and engineering. They can be used to solve equations, approximate irrational numbers, and even optimize algorithms.
Yes, there are many famous examples of continued fractions and nested radicals. One of the most well-known is the golden ratio, which can be represented as both a continued fraction and a nested radical. Another famous example is the square root of 2, which has a simple continued fraction that converges to its exact value.