Continued fraction

A continued fraction is a special kind of mathematical expression that looks like a fraction inside another fraction, and sometimes that fraction is inside yet another one! This can go on forever or stop after a few steps. When it stops, we call it a finite continued fraction. If it goes on without ending, it is an infinite continued fraction.

One common type is called a simple (or regular) continued fraction. In this type, all the numbers on top are one, and all the numbers on the bottom are whole numbers that are positive. Every positive rational number — which means a number that can be written as a simple fraction — can be shown as a finite simple continued fraction. And every positive irrational number — a number that cannot be written as a simple fraction, like the square root of 2 — can be shown as an infinite simple continued fraction.

Different parts of mathematics use different words and ways to write continued fractions. For example, in number theory, when people just say "continued fraction," they usually mean the simple kind. But in complex analysis and numerical analysis, the term "continued fraction" often means the more general kind. The numbers in these fractions can be simple lists of numbers or even functions.

Formulation

A continued fraction is a special way of writing a number. It looks like a fraction inside another fraction, and that pattern can go on forever. In simple terms, it starts with a whole number and then adds a series of fractions on top of each other.

When we calculate these fractions step by step, we get numbers called convergents. These convergents can either settle down to a single value, which we call a convergent continued fraction, or they might keep jumping around without settling, which we call a divergent continued fraction.

History

The story of continued fractions starts with the Euclidean algorithm, a way to find the largest number that can divide two other numbers evenly. This method uses repeated division to find new remainders.

Much later, in the mid-1500s, Bombelli (1579) used continued fractions to help estimate the answers to certain math problems. Soon after, in 1613, Pietro Cataldi created the first way to write continued fractions using special symbols.

In the 1600s, John Wallis gave these fractions their name. Later, important math ideas like Newton's and Leibniz's calculus helped people use continued fractions more.

In 1748, Euler showed how a type of continued fraction relates to endless math sums. And in 1761, Johann Heinrich Lambert used continued fractions to prove that the number π (pi) cannot be written as a simple fraction. These fractions are also useful in studying number theory and solving hard math puzzles.

Notation

Mathematicians have created different ways to write continued fractions to make them easier to read and print. One common way is to put each part of the fraction on the same line, using plus signs to show how they are connected. Another method uses special symbols to make the fractions look neat.

A famous mathematician named Carl Friedrich Gauss used a special symbol, like the one for infinite products ∞, to write continued fractions in a compact form. This method is very neat but not commonly used in English books because of printing limits.

Some elementary considerations

Continued fractions are special kinds of mathematical expressions where a fraction has another fraction in its bottom part, and that fraction might have another fraction inside it, and so on. If this pattern stops after a few steps, we call it a finite continued fraction. If it goes on forever, it’s an infinite continued fraction.

One special type is called a “simple” or “regular” continued fraction. In this type, all the top numbers (called numerators) are just one, and all the bottom numbers (called denominators) are whole numbers that are positive.

These ideas help mathematicians study and understand numbers in deeper ways.

A n − 1 B n − A n B n − 1 = ( − 1 ) n a 1 a 2 ⋯ a n = ∏ i = 1 n ( − a i ) {\displaystyle A_{n-1}B_{n}-A_{n}B_{n-1}=\left(-1\right)^{n}a_{1}a_{2}\cdots a_{n}=\prod _{i=1}^{n}(-a_{i})}

Linear fractional transformations

A linear fractional transformation is a special kind of mathematical function. It looks like this: w = f(z) = (az + b) / (cz + d), where z is a complex number, and a, b, c, d are constants with cz + d not equal to zero.

This type of transformation has interesting properties. If c is not zero, the transformation has one or two fixed points, which are values of z where f(z) equals z. If ad is not equal to bc, the transformation can be reversed, meaning there's another function that undoes it.

When you combine two of these transformations, you get another transformation of the same type. If a is zero, the transformation simplifies to w = b / (cz + d), which has one special point called a pole.

The continued fraction as a composition of LFTs

We can build a continued fraction by combining simple transformations. Each transformation is like adding a piece to the fraction. When we put them together, we get a continued fraction. For example, starting with z, we might add b0, then divide by b1 plus z, and so on.

A geometric interpretation

Thinking of a continued fraction as a transformation helps us see it in a new way. If the continued fraction settles down to a value, the transformations map small and large values of z close to that value. For values in between, the transformations also pull them close to the settled value.

For continued fractions that don’t settle, there are different behaviors. Sometimes they jump between two values, sometimes they include infinity, and sometimes they move around without settling.

Euler discovered a way to connect continued fractions with other math expressions, which helps us understand how continued fractions behave.

Examples

Transcendental functions and numbers

Here are some continued fractions that can be built using special math rules. For example, the math constant e raised to the power x can be written as a continued fraction. Similarly, the logarithm of (1 + x) can also be expressed this way.

There are also more complex continued fractions for functions like arctangent and others. These show how continued fractions can be used in advanced mathematics.

π

Here are three well-known continued fractions for the number π. The first one, called the Leibniz formula, adds and subtracts fractions to get closer to π. Another one, created by Nilakantha Somayaji, also uses a pattern of adding and subtracting. A third method converges faster, giving more accurate digits of π with fewer steps.

Roots of positive numbers

The _n_th root of any positive number can be shown as a continued fraction. For example, the square root of a number can be written in a special continued fraction form. This method can be adjusted to find cube roots, fifth roots, and other roots quickly.

Example 1

The cube root of two (about 1.259921) can be calculated in two different ways using continued fractions. One method gives a slower but steady approach, while the other method with chosen values converges much faster.

Example 2

Pogson's ratio (about 2.511886), which is 100 raised to the power 1/5, can also be expressed as a continued fraction. This shows how useful continued fractions are for finding roots of numbers.

Example 3

The twelfth root of two (about 1.059463) can be shown as a continued fraction. This helps in understanding how these fractions work for finding roots.

Example 4

For equal temperament in music, the perfect fifth interval (about 1.498307) can be calculated using continued fractions. One way gives a steady result, while another special method converges very quickly.

More details on this technique can be found in General Method for Extracting Roots using (Folded) Continued Fractions.

Higher dimensions

A generalized continued fraction can also mean extending ideas to more than two dimensions. For example, the simple continued fraction for a special kind of number relates to how points on a grid lie around a straight line. We can wonder about similar patterns in three or more dimensions.

Studying this helps us understand surprising connections in math and may help solve special problems. Many mathematicians, like Felix Klein, Georges Poitou, and George Szekeres, have worked on building this theory.