Mathematics
Grade 10
15 min
Decompose fractions into unit fractions
Decompose fractions into unit fractions
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1
Introduction & Learning Objectives
Learning Objectives
Define a unit fraction and an Egyptian fraction.
Apply the Greedy Algorithm (Fibonacci-Sylvester method) to decompose any proper fraction into a sum of distinct unit fractions.
Prove that any positive rational number can be expressed as a finite sum of distinct unit fractions.
Decompose a given fraction into unit fractions using algebraic manipulation and factorization.
Analyze why Egyptian fraction representations are not unique.
Solve basic Diophantine equations related to unit fraction decomposition.
Ever wonder how ancient Egyptians performed complex calculations without decimals or modern notation? š They used a unique system of 'unit fractions' to represent any rational number!
This tutorial explores the fascinating world of Egyptian fra...
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Key Concepts & Vocabulary
TermDefinitionExample
Unit FractionA rational number written as a fraction where the numerator is 1 and the denominator is a positive integer.1/2, 1/5, and 1/148 are all unit fractions.
Egyptian FractionA finite sum of distinct unit fractions. The denominators must all be different.2/3 can be written as the Egyptian fraction 1/2 + 1/6. The denominators 2 and 6 are distinct.
Proper FractionA fraction where the absolute value of the numerator is less than the absolute value of the denominator.4/7 is a proper fraction because 4 < 7. The decomposition methods we study apply to proper fractions.
Greedy Algorithm (Fibonacci-Sylvester Method)An iterative method for finding an Egyptian fraction representation. In each step, it chooses the largest possible unit fraction that is smaller than the...
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Core Formulas
Greedy Algorithm Formula
Given a fraction \( \frac{x}{y} \), the first unit fraction in the decomposition is \( \frac{1}{n_1} \) where \( n_1 = \lceil \frac{y}{x} \rceil \). The next step is to decompose the remainder: \( \frac{x}{y} - \frac{1}{n_1} \).
This is the core iterative step of the Fibonacci-Sylvester method. You repeatedly apply this rule to the remainder until the remainder itself is a unit fraction.
Splitting Identity
\( \frac{1}{n} = \frac{1}{n+1} + \frac{1}{n(n+1)} \)
This identity is useful for creating alternative decompositions or for breaking down a unit fraction into two smaller, distinct unit fractions. It proves that decompositions are not unique.
General Decomposition Form
\( \frac{a}{b} = \sum_{i=1}^{k} \frac{1}{n_i} = \frac{1}{n_1} + \frac{1}{n...
4 more steps in this tutorial
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Challenging
Find all positive integer pairs (x, y) with x < y that solve the Diophantine equation 1/x + 1/y = 1/6.
A.(7, 42) and (8, 24) and (9, 18)
B.(7, 42), (8, 24), (9, 18), and (10, 15)
C.(7, 42), (8, 20), and (10, 15)
D.(1, 5), (2, 4), (3, 3)
Challenging
For any odd integer n > 1, the fraction 2/n can be decomposed into a two-term Egyptian fraction. Which of the following is a general formula for this decomposition?
A.2/n = 1/n + 1/n
B.2/n = 1/((n+1)/2) + 1/(n(n+1)/2)
C.2/n = 1/n + 1/(2n)
D.2/n = 1/((n-1)/2) + 1/(n(n-1)/2)
Challenging
The Greedy Algorithm for x/y produces a sequence of denominators nā, nā, nā, ... . A known (but not provided in the tutorial) property is that nāāā > nā(nā - 1). Given the decomposition of 4/5 is 1/2 + 1/4 + 1/20, which part of this sequence demonstrates this property?
A.nā > nā(nā - 1) because 4 > 2(2 - 1) = 2
B.nā > nā(nā - 1) because 20 > 4(4 - 1) = 12
C.Both A and B are correct demonstrations.
D.Neither A nor B are correct as the property does not hold.
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