Continued fractions offer a unique way to represent real numbers using sequences of integers. They're particularly useful for approximating irrational numbers and solving certain equations. This method provides insights into number properties and often yields more accurate approximations than other techniques.
Continued fractions can be finite or infinite, with finite ones representing rational numbers and infinite ones representing irrationals. Their depends on the behavior of partial denominators. Applications include Diophantine approximation, solving quadratic equations, and approximating irrational numbers with high accuracy.
Definition of continued fractions
Continued fractions provide an alternative representation of real numbers using a sequence of integers
Express numbers as a sum of a whole number and a reciprocal of another number, which itself is a sum of a whole number and a reciprocal, and so on
Useful for approximating irrational numbers and solving certain types of equations
Representation in mathematical notation
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General form: a0+a1+a2+a3+⋯111, where a0,a1,a2,… are integers
Compact notation: [a0;a1,a2,a3,…]
Examples:
2=[1;2,2,2,…]=1+2+2+2+⋯111
73=[0;2,3]=0+2+311
Finite vs infinite continued fractions
Finite continued fractions have a limited number of terms and represent rational numbers
Infinite continued fractions have an unlimited number of terms and can represent irrational numbers
Truncating an infinite continued fraction yields a rational approximation of the original number
Convergence of continued fractions
Convergence refers to the behavior of a continued fraction as the number of terms increases
Determines whether the continued fraction approaches a specific value or diverges
Conditions for convergence
For a continued fraction [a0;a1,a2,a3,…] to converge, the sequence of partial denominators a1,a2,a3,… must not contain any zeros
If ∑n=1∞an diverges, the continued fraction converges
Example: The continued fraction of e=[2;1,2,1,1,4,1,1,6,1,…] converges because the partial denominators are non-zero and their sum diverges
Convergence vs divergence
Convergent continued fractions approach a specific value as the number of terms increases
Divergent continued fractions do not approach a specific value and may oscillate or grow without bound
Example: The continued fraction [1;1,−1,1,−1,…] diverges because the partial denominators alternate between 1 and -1
Rate of convergence
The rate at which a continued fraction converges depends on the growth of the partial denominators
Continued fractions with rapidly growing partial denominators converge faster than those with slowly growing partial denominators
Example: The continued fraction of π=[3;7,15,1,292,1,1,1,2,…] converges slowly because the partial denominators grow slowly
Continued fraction expansions
Process of expressing a number as a continued fraction
Provides insights into the properties and structure of the number
Rational numbers as continued fractions
Every rational number has a finite
The continued fraction expansion of a rational number is unique
Example: 75=[0;1,2,2]
Irrational numbers as continued fractions
Irrational numbers have infinite continued fraction expansions
The continued fraction expansion of an irrational number is unique
Example: 3=[1;1,2,1,2,1,2,…]
Periodic continued fractions
Some irrational numbers have periodic continued fraction expansions, where a sequence of partial denominators repeats indefinitely
Quadratic irrationals (irrational roots of quadratic equations with integer coefficients) always have periodic continued fraction expansions
Example: The golden ratio ϕ=21+5=[1;1,1,1,…] has a periodic continued fraction expansion
Applications of continued fractions
Continued fractions have various applications in number theory, approximation theory, and computational mathematics
Diophantine approximation
Continued fractions provide a systematic way to find rational approximations of irrational numbers
The convergents of a continued fraction (rational numbers obtained by truncating the continued fraction) are the best rational approximations of the original number
Example: The convergents of 2=[1;2,2,2,…] are 11,23,57,1217,…, which are the best rational approximations of 2
Solving quadratic equations
Continued fractions can be used to solve quadratic equations of the form ax2+bx+c=0
The roots of the equation can be expressed as periodic continued fractions
Example: The equation x2−3x−1=0 has roots 23+13=[2;1,1,1,1,…] and 23−13=[1;6,6,6,…]
Approximating irrational numbers
Continued fractions provide a way to approximate irrational numbers with rational numbers
The approximations obtained from continued fractions are often more accurate than those obtained from other methods, such as decimal expansions or power series
Example: The continued fraction approximation of π to 5 terms is 113355=3.14159292…, which is more accurate than the decimal approximation 3.14159
Computation of continued fractions
Practical aspects of working with continued fractions, including truncation, approximation error, and computational efficiency
Truncation of infinite continued fractions
Infinite continued fractions must be truncated to a finite number of terms for practical computation
Truncation introduces an approximation error, which decreases as more terms are included
Example: Truncating the continued fraction of e=[2;1,2,1,1,4,1,1,6,1,…] to 5 terms yields 71193≈2.71830, which is an approximation of e
Approximation error
The approximation error of a truncated continued fraction depends on the number of terms included and the properties of the original number
The error can be estimated using the difference between successive convergents
Example: The approximation error of truncating the continued fraction of 2 to 4 terms is ∣2−1217∣≈0.00035
Efficiency vs accuracy trade-off
Including more terms in a truncated continued fraction improves the accuracy of the approximation but increases the computational cost
The optimal number of terms to include depends on the desired accuracy and available computational resources
Example: Approximating π to 10 decimal places requires including 4 terms of its continued fraction, while approximating it to 100 decimal places requires including 25 terms
Relationship to other approximation methods
Continued fractions can be compared and contrasted with other approximation methods, such as power series and Padé approximants
Continued fractions vs power series
Power series represent functions as infinite sums of terms involving powers of a variable
Continued fractions often provide better approximations than power series for functions with singularities or poles
Example: The power series of tan(x)=x+31x3+152x5+⋯ converges slowly near x=2π, while the continued fraction tan(x)=[0;x1,−3x,5x,−7x,…] converges rapidly
Continued fractions vs Padé approximants
Padé approximants are rational function approximations of functions, obtained by matching the coefficients of the Taylor series expansion
Continued fractions and Padé approximants are closely related, as the convergents of a continued fraction are Padé approximants
Example: The Padé approximant of ex of order (2, 2) is 1−21x+121x21+x+21x2, which is also the 4th convergent of the continued fraction of ex
Advantages and disadvantages of continued fractions
Advantages:
Provide accurate approximations with relatively few terms
Work well for functions with singularities or poles
Yield the best rational approximations (convergents) of irrational numbers
Disadvantages:
Can be more difficult to compute than power series or Padé approximants
May require more terms to achieve the same accuracy as other methods for some functions
Convergence can be slow for some numbers or functions
Key Terms to Review (18)
Analytic functions: Analytic functions are complex functions that are locally given by convergent power series. These functions are defined in a neighborhood of every point in their domain, meaning they can be differentiated infinitely many times and retain their differentiability. A significant property of analytic functions is that they can be represented by their Taylor series, connecting them closely to concepts such as holomorphic functions and uniform convergence.
Approximation of real numbers: Approximation of real numbers refers to the process of finding numbers that are close to a given real number, typically represented in a simpler or more manageable form. This concept is essential in many mathematical contexts where exact values are not feasible or necessary, allowing for practical calculations and analyses. It often involves techniques such as rounding, truncating, or using fractions to express numbers in a way that is easier to work with while maintaining a reasonable level of accuracy.
Best Approximation: Best approximation refers to the closest or most accurate representation of a function or signal within a given set of functions, minimizing the difference between them. This concept is crucial in various areas of mathematics and engineering, as it allows for efficient modeling and analysis of complex systems. The best approximation can often be expressed in terms of specific properties like uniform convergence or minimizing errors in specific norms, linking it to various approximation techniques.
Continued fraction expansion: Continued fraction expansion is a way of expressing a real number as an infinite series of fractions, where each fraction's denominator contains another fraction. This representation allows for more accurate approximations of irrational numbers and provides insights into their properties. Continued fractions can reveal the best rational approximations to a number and are often used in number theory, particularly in understanding quadratic irrationals and Diophantine equations.
Convergence: Convergence refers to the process of a sequence or function approaching a limit or a desired value as the number of iterations or data points increases. This concept is critical across various approximation methods, as it indicates how closely an approximation represents the true function or value being estimated, thereby establishing the reliability and effectiveness of the approximation techniques used.
Diophantine Equations: Diophantine equations are polynomial equations that seek integer solutions. Named after the ancient Greek mathematician Diophantus, these equations often take the form of linear or quadratic equations and are significant in number theory, especially when examining relationships between integers.
Dirichlet's Theorem: Dirichlet's Theorem is a fundamental result in number theory that asserts there are infinitely many prime numbers in any arithmetic progression where the first term and the common difference are coprime. This theorem connects deeply to concepts of continued fractions, as they can be used to study the distribution of primes, and it also has implications for trigonometric interpolation, where primes relate to sampling frequencies and polynomial roots.
Euclidean Algorithm: The Euclidean Algorithm is a method for finding the greatest common divisor (GCD) of two integers by repeatedly applying the principle that the GCD of two numbers also divides their difference. This algorithm is efficient and forms the basis for various mathematical concepts, including continued fractions, where it is used to derive the coefficients that represent rational numbers as infinite sequences.
General Continued Fraction: A general continued fraction is a representation of a number as an infinite sequence of fractions, where each fraction's numerator is typically a constant and the denominators form a sequence. This format allows for various ways to express real numbers and can converge to irrational numbers, providing a powerful method in approximation and number theory.
Irrational numbers representation: Irrational numbers representation refers to the ways in which irrational numbers can be expressed, particularly through methods such as continued fractions. These representations provide insight into the properties of irrational numbers and allow for better approximations in numerical analysis, demonstrating how these non-repeating, non-terminating decimals can be systematically understood and used.
Joseph-Louis Lagrange: Joseph-Louis Lagrange was an influential mathematician and astronomer in the 18th century, known for his contributions to various fields, including calculus, number theory, and approximation theory. He developed important concepts such as Lagrange interpolation, which forms the foundation of Hermite interpolation, and made significant advancements in the use of continued fractions for approximating functions and solving equations. His work laid the groundwork for future developments in these mathematical areas.
Khinchin's Theorem: Khinchin's Theorem states that for almost all real numbers, the continued fraction expansion is infinite and can be expressed in a specific way that relates to the measure of their frequency. This theorem connects continued fractions to the properties of real numbers, demonstrating that the average length of the terms in the continued fraction representation approaches a limit, which has implications for number theory and approximation.
Leonhard Euler: Leonhard Euler was an 18th-century Swiss mathematician and physicist renowned for his contributions to various fields of mathematics, including calculus, graph theory, and number theory. His work laid the foundation for modern mathematical analysis, and he introduced numerous concepts and notations that are still in use today, particularly in the study of continued fractions.
Number Theory Applications: Number theory applications involve using the principles and techniques of number theory to solve problems and analyze structures in various mathematical fields, particularly in relation to algorithms, cryptography, and approximation theory. These applications often focus on properties of integers and rational numbers, revealing connections between numbers and their representations.
Partial Quotients: Partial quotients are the terms generated during the process of expressing a real number as a continued fraction. Each partial quotient represents the integer part of the fraction at each stage, contributing to the overall approximation of the number. This method reveals how close the continued fraction gets to the actual value, allowing for a systematic approach to approximating irrational numbers and understanding their properties.
Pointwise convergence: Pointwise convergence occurs when a sequence of functions converges to a limit function at each individual point in its domain. This means that for every point, the value of the function sequence approaches the value of the limit function as you consider more and more terms of the sequence. It is a crucial concept in understanding how functions behave under various approximation methods and plays a significant role in the analysis of series, sequences, and other mathematical constructs.
Simple continued fraction: A simple continued fraction is a way of expressing a real number as an infinite sequence of fractions, where each fraction is formed by an integer part and a reciprocal of another fraction. This representation is particularly useful for approximating real numbers and helps in understanding the relationships between numbers through their convergents. Simple continued fractions can offer insights into number theory and have connections to other mathematical concepts like Padé approximation.
Uniform Convergence: Uniform convergence refers to a type of convergence of a sequence of functions where the rate of convergence is uniform across the entire domain. This means that for every positive number, there exists a point in the sequence beyond which all function values are within that distance from the limit function, uniformly for all points in the domain. It plays a crucial role in many areas of approximation, ensuring that operations such as integration and differentiation can be interchanged with limits.