Continued fraction

A continued fraction is a special kind of mathematical expression that looks like a fraction inside another fraction. Sometimes this pattern continues inside more fractions! 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 to write a number. It looks like a fraction inside another fraction, and this pattern can go on forever. It starts with a whole number and then adds a series of fractions on top of each other.

When we work out these fractions step by step, we get numbers called convergents. These convergents can either settle down to one value, called a convergent continued fraction, or they might keep jumping around without settling, called 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.

In the mid-1500s, Bombelli (1579) used continued fractions to help estimate answers to math problems. 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, 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 math sums. In 1761, Johann Heinrich Lambert used continued fractions to show that the number π (pi) cannot be written as a simple fraction. These fractions are also useful in studying number theory.

Notation

Mathematicians have created different ways to write continued fractions so they are 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 connect. 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 neat but not commonly used in English books because of printing limits.

Some elementary considerations

Continued fractions are special kinds of math expressions. In these expressions, a fraction has another fraction in its bottom part. This can keep going, with each new fraction having another fraction inside it.

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. 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 type of math rule. It looks like this: w = f(z) = (az + b) / (cz + d), where z is a complex number, and a, b, c, d are fixed numbers with cz + d not equal to zero.

This kind of rule has some neat features. If c is not zero, the rule has one or two special points where f(z) equals z. If ad is not equal to bc, we can reverse the rule, meaning there is another rule that undoes it.

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

The continued fraction as a composition of LFTs

We can create a continued fraction by joining simple rules together. Each rule is like adding a piece to the fraction. When we join them, 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

Seeing a continued fraction as a rule helps us picture it in a new way. If the continued fraction settles to a value, the rules move small and large values of z close to that value. For values in between, the rules also pull them close to the settled value.

For continued fractions that don’t settle, they can behave in different ways. Sometimes they jump between two values, sometimes they include infinity, and sometimes they move around without settling.

Euler found a way to link continued fractions with other math ideas, which helps us learn about how continued fractions act.

Examples

Transcendental functions and numbers

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

There are more complex continued fractions for functions like arctangent. These show how continued fractions are 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 be used in more than two dimensions. For example, the simple continued fraction for a special kind of number shows how points on a grid lie around a straight line. We can look at similar patterns in three or more dimensions.

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