Fortsetzungsbrüche

Fortsetzungsbrüche, also known as continued fractions, are a way of representing a real number as a sequence of integers. Instead of a simple fraction, a continued fraction takes the form of an expression where each term is an integer, and the subsequent terms appear in the denominator of the fraction that follows. The general form can be written as a0 + 1/(a1 + 1/(a2 + 1/(a3 + ...))).

The numbers a0, a1, a2, and so on are called the partial quotients. For a simple continued fraction, these partial quotients are positive integers, except for a0, which can be any integer. Every rational number can be represented by a finite continued fraction, meaning the sequence of partial quotients terminates. Irrational numbers, on the other hand, are represented by infinite continued fractions where the sequence of partial quotients continues indefinitely.

One of the key properties of continued fractions is their ability to provide excellent rational approximations for real numbers. The finite truncations of an infinite continued fraction, called convergents, yield successively better approximations of the original number. These convergents are themselves rational numbers and have the remarkable property that no other rational number with a smaller or equal denominator is closer to the original number. This makes continued fractions particularly useful in areas like number theory, Diophantine approximation, and the study of quadratic irrationals.