Continued fraction

A continued fraction is a mathematical expression where the denominator contains a sum involving another fraction. This process can repeat many times, creating a nested structure. If this repetitive process stops, the result is a finite continued fraction. If it continues without end, it is an infinite continued fraction. A special version is the simple continued fraction. In this case, all the numerators are equal to one. All the denominators must also be positive integers.

To understand how they work, we look at their specific parts. The expression begins with a leading term called the integer part. The numbers within the nested layers are called partial numerators and partial denominators. As we add more layers, we create a sequence of values called convergents. These convergents are simple fractions that represent the value of the expression at each step. We can find these values using fundamental recurrence formulas. These formulas use the numerators and denominators of previous steps to calculate the next ones.

There are different ways to classify these expressions depending on the math field. In number theory, the term continued fraction usually means a simple continued fraction. Other versions are called generalized continued fractions. In complex analysis or numerical analysis, the term continued fraction is often used for the general case. The numerators and denominators can even be sequences of numbers or functions. Some fractions may converge, meaning the convergents approach a single limit. Others may diverge by oscillation or by producing infinite zero denominators.

The history of these ideas began with the Euclidean algorithm. This procedure finds the greatest common divisor of two natural numbers. It works by dividing to extract a remainder and then dividing by that remainder repeatedly. Nearly two thousand years passed before mathematicians used these ideas for quadratic equations. In 1613, Pietro Cataldi introduced the first formal notation for generalized continued fractions. He used dots to show where the next fraction would be placed. Later, in the late seventeenth century, John Wallis introduced the term "continued fraction" into mathematical literature.

Many famous mathematicians added to this field. In 1748, Leonhard Euler published a theorem linking certain continued fractions to infinite series. His work remains a basis for modern proofs of convergence. In 1761, Johann Heinrich Lambert used a continued fraction to prove that pi is an irrational number. An irrational number cannot be written as a simple fraction. In the late eighteenth century, Lagrange used these tools to solve Pell's equation. This equation had fascinated people for over a thousand years. Brahmagupta had previously conducted a systematic study of this equation between 598 and 670.

Continued fractions are vital in several areas of study. They are especially useful in number theory and for solving Diophantine equations. These are equations where mathematicians look for whole number solutions. In 1813, Gauss derived continued fractions from complex-valued hypergeometric functions. These are known as Gauss's continued fractions. They can express many elementary and advanced functions, such as Bessel functions. These fractions are often rapidly convergent, meaning they reach a value very quickly in the complex plane.

A fascinating example is the golden ratio. When expressed as a simple continued fraction, its numerators are all one. Using recurrence formulas, we find that its numerators and denominators are the Fibonacci numbers. This connection shows how deeply these structures are linked to other mathematical patterns. Even when a fraction diverges by oscillation, we can study its even and odd parts separately. This allows mathematicians to find two different limits from a single expression.