Continued fractions | Mathematical analysis
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers are called the coefficients or terms of the continued fraction. It is generally assumed that the numerator of all of the fractions is 1. If arbitrary values and/or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction. When it is necessary to distinguish the first form from generalized continued fractions, the former may be called a simple or regular continued fraction, or said to be in canonical form. Continued fractions have a number of remarkable properties related to the Euclidean algorithm for integers or real numbers. Every rational number / has two closely related expressions as a finite continued fraction, whose coefficients ai can be determined by applying the Euclidean algorithm to . The numerical value of an infinite continued fraction is irrational; it is defined from its infinite sequence of integers as the limit of a sequence of values for finite continued fractions. Each finite continued fraction of the sequence is obtained by using a finite prefix of the infinite continued fraction's defining sequence of integers. Moreover, every irrational number is the value of a unique infinite regular continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values and 1. This way of expressing real numbers (rational and irrational) is called their continued fraction representation. The term continued fraction may also refer to representations of rational functions, arising in their analytic theory. For this use of the term, see Padé approximation and Chebyshev rational functions. (Wikipedia).
Continued Fraction Expansions, Pt. III
A fascinating generalization linking sequences, continued fractions, and polynomials. Email: allLogarithmsWereCreatedEqual@gmail.com Subscribe! https://www.youtube.com/AllLogarithmsEqual
From playlist Number Theory
The first part develops the fraction from a simple equation or statement with a single unknown variable and demonstrates the recursive, iterative procedure. Possibly as simple and straightforward as it is possible for me to do. The second part still confuses me and amounts to no mare than
From playlist Number Theory
Infinite Continued Fractions, simple or not?
Start learning today, click https://brilliant.org/blackpenredpen/ to check out Brillant.org. First 200 people to sign up will get 20% off your annual premium subscription! What Are Continued Fractions? Continued Fractions, Write sqrt(2) as a continued fraction, infinite simple continued
From playlist [Math For Fun] Brilliant Problems
Keith Conrad (University of Connecticut) — January 28, 2015
From playlist Number Theory
Arithmetic With... Continued Fractions?? #SoME2
Arithmetic! On continued fractions! It's possible, but not well known or widely used in practice. This video explores the basics of this underappreciated area of math. This is my submission for SoME2 (https://www.youtube.com/watch?v=hZuYICAEN9Y&t=0s) SOURCES & FURTHER READING: Continued
From playlist Summer of Math Exposition 2 videos
This video defines what a simplified fraction is and explains how to simplify fractions using prime factors and division.
From playlist Simplifying Fractions
Continued Fraction Expansions Part II: Example Calculations
Complete with calculations and exercises! Let me know what you come up with. NEW (2016 Season): See Episode 3 here! https://youtu.be/4U9z5qoiDNQ ----- Sources for additional information: Very useful informative article (no surprise, it's from MathWorld): http://mathworld.wolfram.com/Con
From playlist Number Theory
Continued fractions | Lecture 17 | Fibonacci Numbers and the Golden Ratio
What is a continued fraction, and why is the golden ratio considered to be the most irrational of the irrational numbers? Join me on Coursera: https://www.coursera.org/learn/fibonacci Lecture notes at http://www.math.ust.hk/~machas/fibonacci.pdf Subscribe to my channel: http://www.youtu
From playlist Fibonacci Numbers and the Golden Ratio
Graphing Continued Fractions of Quadratic Irrationals
http://demonstrations.wolfram.com/GraphingContinuedFractionsOfQuadraticIrrationals The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. Let x= ( a ) / ( b ) + ( c ) / ( d ) SqrtBox[S], a,b,c,d,S??. The continued fraction
From playlist Number Theory
Unusual Decimal-to-Fraction Conversion that Just WORKS
Continued fractions prove to be an invaluable tool in converting decimals back to their fractions when you have limited information. They can also be used to find good fractional approximations for irrational numbers and in finding original square roots based solely off of their decimal ex
From playlist Summer of Math Exposition Youtube Videos
Laura Capuano: An effective criterion for periodicity of p-adic continued fractions
Abstract: It goes back to Lagrange that a real quadratic irrational has always a periodic continued fraction. Starting from decades ago, several authors proposed different definitions of a p-adic continued fraction, and the definition depends on the chosen system of residues mod p. It turn
From playlist Women at CIRM
Strategic Math S1 | Lecture 02 Same idea with Continued Fractions
★Before watching this lecture, I encourage everyone to try their own hands on this problem: Here are the thinks that you can use: 1. Use this thorough introduction of CF to learn more: 🔗 http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html 2. Use this calculator to c
From playlist Summer of Math Exposition Youtube Videos
Infinite fractions and the most irrational number
NEW: Follow-up video with puzzle solution is here: https://youtu.be/leFep9yt3JY In this video the Mathologer uses infinite fractions to track down the most irrational of all irrational numbers. Find out about how the usual suspects root 2, e, and pi stack up against this special number and
From playlist Recent videos
How did Ramanujan solve the STRAND puzzle?
Today's video is about making sense of an infinite fraction that pops up in an anecdote about the mathematical genius Srinivasa Ramanujan. 00:00 Intro 04:31 Chapter 1: Getting a feel for the puzzle 08:27 Chapter 2: Algebra autopilot 12:37 Chapter 3: Infinite fraction 17:51 Chapter 4: Roo
From playlist Recent videos
2023 Number Challenge: Rational Approximation for Square Root 2023
In this video, we look at how to use continued fraction to approximate square root of 2023, which is an irrational number. sqrt(2023) ≈ 44.9777722277722277 It can be approximated by two forms of continued fraction: 44 + 88/ (87 + 88/ (87 + 88/ (87 .... )))) Or in a simple continued fr
From playlist Math Problems with Number 2023
Summary for simplifying complex fractions
👉 Learn how to simplify a complex fraction. A complex fraction is a fraction with another fraction or fractions in the numerator and/or in the denominator. To simplify a complex fraction is to reduce the fraction in such a way as there is only one numerator and denominator. In doing that,
From playlist How to Simplify Complex Fractions | Learn About