Egyptian fractions | Greedy algorithms | Number theory | Integer sequences

Greedy algorithm for Egyptian fractions

In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a representation of an irreducible fraction as a sum of distinct unit fractions, such as 5/6 = 1/2 + 1/3. As the name indicates, these representations have been used as long ago as ancient Egypt, but the first published systematic method for constructing such expansions is described in the Liber Abaci of Leonardo of Pisa (Fibonacci). It is called a greedy algorithm because at each step the algorithm chooses greedily the largest possible unit fraction that can be used in any representation of the remaining fraction. Fibonacci actually lists several different methods for constructing Egyptian fraction representations . He includes the greedy method as a last resort for situations when several simpler methods fail; see Egyptian fraction for a more detailed listing of these methods. As Salzer (1948) details, the greedy method, and extensions of it for the approximation of irrational numbers, have been rediscovered several times by modern mathematicians, earliest and most notably by J. J. Sylvester; see for instance and . A closely related expansion method that produces closer approximations at each step by allowing some unit fractions in the sum to be negative dates back to . The expansion produced by this method for a number x is called the greedy Egyptian expansion, Sylvester expansion, or Fibonacci–Sylvester expansion of x. However, the term Fibonacci expansion usually refers, not to this method, but to representation of integers as sums of Fibonacci numbers. (Wikipedia).

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How to integrate by partial fractions

Free ebook http://bookboon.com/en/learn-calculus-2-on-your-mobile-device-ebook How to integrate by the method of partial fraction decomposition. In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is a fraction such that the numerator

From playlist A second course in university calculus.

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Solving Algebraic Fractions | Algebra | Maths | FuseSchool

Algebraic fractions are simply fractions with algebraic expressions either on the top, bottom or both. We treat them in the same way as we would numerical fractions. In part 1 we saw how to simplify, and add and subtract algebraic fractions. We discovered that algebraic fractions follow th

From playlist MATHS

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Number Systems Ancient to Modern 1: the Egyptians | Sociology and Pure Mathematics | N J Wildberger

Some of the difficulties with modern arithmetic have their origins already in two great ancient civilizations: the Egyptians and the Mesopotamian. Their approaches to arithmetic were quite different: the Egyptians relied on a base 10 system with a theory of fractions, while the Sumerians,

From playlist Sociology and Pure Mathematics

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Learn to use summation notation for an arithmetic series to find the sum

👉 Learn how to find the partial sum of an arithmetic series. A series is the sum of the terms of a sequence. An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of n terms of an arithmetic sequence is given by Sn = n/2 [2a + (n - 1)d], where a is

From playlist Series

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Oxford MAT 2018 Question 6 (Maths Admissions Test)

These video solutions and more are also now available in this free online course: https://courses.mathsaurus.com/courses/mat 2019 Playlist https://www.youtube.com/playlist?list=PLUuwecyIK3fh6ia74lkh-pPIQWJ38ctIF 2018 Playlist https://www.youtube.com/playlist?list=PLUuwecyIK3fhz00OVD3PhwLG

From playlist Oxford MAT 2018 - Maths Admissions Test

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Ex: Setting Up Partial Fraction Decomposition

This video provides several examples of how to set up the fractions in order to perform partial fraction decomposition. Site: http://mathispower4u.com Blog: http://mathispower4u.wordpress.com

From playlist Performing Partial Fraction Decomposition

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Pre-Calculus - Partial fraction decomposition

This video will show you how the method of partial fraction decomposition can break down a fraction into smaller fractions. Remember to carefully identify your factors as either linear or quadratic, this will really help you identify what to put in the numerator of each fraction. To find

From playlist Pre-Calculus

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Seffi Naor: Recent Results on Maximizing Submodular Functions

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From playlist HIM Lectures 2015

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Multiplying Fractions

👉 Learn how to multiply fractions. To multiply fractions, we need to multiply the numerator by the numerator and multiply the denominator by the denominator. We then reduce the fraction. By reducing the fraction we are writing it in most simplest form. It is very important to understand t

From playlist How to Multiply Fractions

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Multiplying fractions

👉 Learn how to multiply fractions. To multiply fractions, we need to multiply the numerator by the numerator and multiply the denominator by the denominator. We then reduce the fraction. By reducing the fraction we are writing it in most simplest form. It is very important to understand t

From playlist How to Multiply Fractions

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Greedy Algorithm | What Is Greedy Algorithm? | Introduction To Greedy Algorithms | Simplilearn

This video on the Greedy Algorithm will acquaint you with all the fundamentals of greedy programming paradigm. In this tutorial, you will learn 'What Is Greedy Algorithm?' with the help of suitable examples. And finally, you will also discover few important applications of greedy algorithm

From playlist Data Structures & Algorithms [2022 Updated]

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1. Introduction, Optimization Problems (MIT 6.0002 Intro to Computational Thinking and Data Science)

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From playlist MIT 6.0002 Introduction to Computational Thinking and Data Science, Fall 2016

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Matthias Poloczek: New Approximation Algorithms for MAX SAT Simple, Fast, and Excellent in Practice

Matthias Poloczek: New Approximation Algorithms for MAX SAT Simple, Fast, and Excellent in Practice We present simple randomized and deterministic algorithms that obtain 3/4-approximations for the maximum satisfiability problem (MAX SAT) in linear time. In particular, their worst case gua

From playlist HIM Lectures 2015

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Rico Zenklusen: The Submodular Secretary Problem Goes Linear

During the last decade, the matroid secretary problem (MSP) became one of the most prominent classes of online selection problems. The strong interest in MSPs is due to both its many applications and the fact that matroid constraints have useful properties for the design of strong online a

From playlist HIM Lectures 2015

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Finding the sum or an arithmetic series using summation notation

👉 Learn how to find the partial sum of an arithmetic series. A series is the sum of the terms of a sequence. An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of n terms of an arithmetic sequence is given by Sn = n/2 [2a + (n - 1)d], where a is

From playlist Series

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Dynamic Programming I - Lecture 11

All rights reserved for http://www.aduni.org/ Published under the Creative Commons Attribution-ShareAlike license http://creativecommons.org/licenses/by-sa/2.0/ Tutorials by Instructor: Shai Simonson. http://www.stonehill.edu/compsci/shai.htm Visit the forum at: http://www.coderisland.c

From playlist ArsDigita Algorithms by Shai Simonson

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Knapsack, Bandwidth Min. Intro: Greedy Algorithms - Lecture 14

All rights reserved for http://www.aduni.org/ Published under the Creative Commons Attribution-ShareAlike license http://creativecommons.org/licenses/by-sa/2.0/ Tutorials by Instructor: Shai Simonson. http://www.stonehill.edu/compsci/shai.htm Visit the forum at: http://www.coderisland.c

From playlist ArsDigita Algorithms by Shai Simonson

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General Method for Integer Power Sum Formula

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From playlist *** The Good Stuff ***

Related pages

Odd greedy expansion | Perfect number | Fibonacci number | Fibonacci | Engel expansion | On-Line Encyclopedia of Integer Sequences | Sylvester's sequence | Irreducible fraction | Irrational number | Mathematics | Rational number | Unit fraction | Group theory | Golden ratio | Greedy algorithm | Erdős–Straus conjecture | Egyptian fraction