Continued Fractions: General Statement & Evidence

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In summary, a continued fraction is a way to represent a number as a sequence of fractions, with the numerator of each fraction being 1 and the denominators being positive integers. It has various applications in mathematics, such as solving equations and approximating irrational numbers. The general statement for continued fractions states that any real number can be represented in this form, supported by mathematical proofs and its use in different areas of math. However, continued fractions have limitations, such as not being able to represent complex numbers and the potential for very large representations.
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Pirate21
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Please help with the following question:

http://img161.imageshack.us/img161/691/continuousfraction5az5.gif

By considering other values of k, determine a generalized statement for the exact value of any such continued fraction. For which values of k does the generalised statement hold true? How do you know? Provide evidence.

Thanks.
 
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Any attempt at a solution? The trick may be slightly hard to see at first, but it's fairly standard. One method to see the answer might be to cover your hand over everything above the second highest 1 in the formula. What do you see?
 
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FAQ: Continued Fractions: General Statement & Evidence

What is a continued fraction?

A continued fraction is a representation of a number as a sequence of fractions, where the numerator of each fraction is 1 and the denominators are positive integers. It is written as [a0; a1, a2, a3, ...], where a0 is the whole number part and a1, a2, a3, ... are the partial quotients.

What is the general statement for continued fractions?

The general statement for continued fractions states that any real number can be represented as a continued fraction, with a unique sequence of partial quotients. This means that continued fractions are an alternative way to express real numbers, similar to decimal or binary notation.

How are continued fractions used in mathematics?

Continued fractions have various applications in mathematics, including number theory, algebra, and calculus. They can be used to solve equations, approximate irrational numbers, and study the properties of real numbers. Continued fractions also have connections to other mathematical concepts, such as infinite series and continued fractions.

What evidence supports the general statement for continued fractions?

The evidence for the general statement for continued fractions comes from various mathematical proofs, including the Euclidean algorithm and the Lagrange theorem. These proofs show that any real number can be expressed as a continued fraction and that the sequence of partial quotients is unique. Additionally, the use of continued fractions in different areas of mathematics further supports their validity.

Are there any limitations to the use of continued fractions?

While continued fractions have many applications in mathematics, they have some limitations. For example, continued fractions can only represent real numbers and cannot represent complex numbers. Additionally, the length of a continued fraction representation can grow very large for certain numbers, making it difficult to work with in some cases.

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