how to find the sum of unit fractions

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From Wikipedia

Unit fraction

A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/n. Examples are 1/1, 1/2, 1/3, 1/4 etc.

Elementary arithmetic

Multiplying any two unit fractions results in a product that is another unit fraction:

\frac1x \times \frac1y = \frac1{xy}.

However, adding, subtracting, or dividing two unit fractions produces a result that is generally not a unit fraction:

\frac1x + \frac1y = \frac{x+y}{xy}
\frac1x - \frac1y = \frac{y-x}{xy}
\frac1x \div \frac1y = \frac{y}{x}.

Modular arithmetic

Unit fractions play an important role in modular arithmetic, as they may be used to reduce modular division to the calculation of greatest common divisors. Specifically, suppose that we wish to perform divisions by a value x, modulo y. In order for division by x to be well defined modulo y, x and y must be relatively prime. Then, by using the extended Euclidean algorithm for greatest common divisors we may find a and b such that

\displaystyle ax + by = 1,

from which it follows that

\displaystyle ax \equiv 1 \pmod y,

or equivalently

a \equiv \frac1x \pmod y.

Thus, to divide by x (modulo y) we need merely instead multiply by a.

Finite sums of unit fractions

Any positive rational number can be written as the sum of unit fractions, in multiple ways. For example,

\frac45=\frac12+\frac14+\frac1{20}=\frac13+\frac15+\frac16+\frac1{10}.

The ancient Egyptians used sums of distinct unit fractions in their notation for more general rational numbers, and so such sums are often called Egyptian fractions. There is still interest today in analyzing the methods used by the ancients to choose among the possible representations for a fractional number, and to calculate with such representations. The topic of Egyptian fractions has also seen interest in modern number theory; for instance, the Erdős–Graham conjecture and the Erdős–Straus conjecture concern sums of unit fractions, as does the definition of Ore's harmonic numbers.

In geometric group theory, triangle groups are classified into Euclidean, spherical, and hyperbolic cases according to whether an associated sum of unit fractions is equal to one, greater than one, or less than one respectively.

Series of unit fractions

Many well-known infinite series have terms that are unit fractions. These include:

  • The harmonic series, the sum of all positive unit fractions. This sum diverges, and its partial sums
\frac11+\frac12+\frac13+\cdots+\frac1n
closely approximate ln n + Î³ as n increases.

Matrices of unit fractions

The Hilbert matrix is the matrix with elements

B_{i,j} = \frac1{i+j-1}.

It has the unusual property that all elements in its inverse matrix are integers. Similarly, Richardson defined a matrix with elements

C_{i,j} = \frac1{F_{i+j-1}},

where Fi denotes the ith Fibonacci number. He calls this matrix the Filbert matrix and it has the same property of having an integer inverse.

Unit fractions in probability and statistics

In a uniform distribution on a discrete space, all probabilities are equal unit fractions. Due to the principle of indifference, probabilities of this form arise frequently in statistical calculations. Additionally, Zipf's law states that, for many observed phenomena involving the selection of items from an ordered sequence, the probability that the nth item is selected is proportional to the unit fraction 1/n.

Unit fractions in physics

The energy levels of the Bohr model of electron orbitals in a hydrogen atom are proportional to square unit fractions. Therefore the energy levels of photons that can be absorbed or em

Egyptian fraction

An Egyptian fraction is the sum of distinct unit fractions, such as \tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{16}. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The sum of an expression of this type is a positiverational numbera/b; for instance the Egyptian fraction above sums to 43/48. Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including 2/3 and 3/4 as summands, were used as a serious notation for rational numbers by the ancient Egyptians, and continued to be used by other civilizations into medieval times. In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern number theory and recreational mathematics, as well as in modern historical studies of ancient mathematics.

Ancient Egypt

For more information on this subject, seeEgyptian numerals, Eye of Horus, and Egyptian mathematics.

Egyptian fraction notation was developed in the Middle Kingdom of Egypt, altering the Old Kingdom's Eye of Horus numeration system. Five early texts in which Egyptian fractions appear were the Egyptian Mathematical Leather Roll, the Moscow Mathematical Papyrus, the Reisner Papyrus, the Kahun Papyrus and the Akhmim Wooden Tablet. A later text, the Rhind Mathematical Papyrus, introduced improved ways of writing Egyptian fractions. The Rhind papyrus was written by Ahmes and dates from the Second Intermediate Period; it includes a table of Egyptian fraction expansions for rational numbers 2/n, as well as 84 word problems. Solutions to each problem were written out in scribal shorthand, with the final answers of all 84 problems being expressed in Egyptian fraction notation. 2/n tables similar to the one on the Rhind papyrus also appear on some of the other texts. However, as the Kahun Papyrus shows, vulgar fractions were also used by scribes within their calculations.

Notation

To write the unit fractions used in their Egyptian fraction notation, in hieroglyph script, the Egyptians placed the hieroglyph D21 (er, "[one] among" or possibly re, mouth) above a number to represent the reciprocal of that number. Similarly in hieratic script they drew a line over the letter representing the number. For example:

The Egyptians had special symbols for 1/2, 2/3, and 3/4 that were used to reduce the size of numbers greater than 1/2 when such numbers were converted to an Egyptian fraction series. The remaining number after subtracting one of these special fractions was written using as a sum of distinct unit fractions according to the usual Egyptian fraction notation.

The Egyptians also used an alternative notation modified from the Old Kingdom and based on the parts of the Eye of Horus to denote a special set of fractions of the form 1/2k (for k = 1, 2, ..., 6), that is, dyadic rational numbers. These "Horus-Eye fractions" were used in the Middle Kingdom in conjunction with the later notation for Egyptian fractions to subdivide a hekat, the primary ancient Egyptian volume measure for grain, bread, and other small quantities of volume, as described in the Akhmim Wooden Tablet. If any remainder was left after expressing a quantity in Eye of Horus fractions of a hekat, the remainder was written using the usual Egyptian fraction notation as multiples of a ro, a unit equal to 1/320 of a hekat.

Calculation methods

Modern historians of mathematics have studied the Rhind papyrus and other ancient sources in an attempt to discover the methods the Egyptians used in calculating with Egyptian fractions. In particular, study in this area has concentrated on understanding the tables of expansions for numbers of the form 2/n in the Rhind papyrus. Although these expansions can generally be described as algebraic identities, they do not match any single identity; rather, different methods were used for prime and for composite denominators, and more than one method was used for numbers of each type:

  • For small odd prime denominators p, the expansion 2/p = 2/(p + 1) + 2/p(p + 1) was used.
  • For larger prime denominators, an expansion of the form 2/p = 1/A + (2A-p)/Ap was used, where A is a number with many divisors (such as a practical number) in the range p/2 < A< p. The remaining term (2A-p)/Ap was expanded by representing the number 2A-p as a sum of divisors of A and forming a fraction d/Ap for each such divisor d


From Yahoo Answers

Question:How to express a fraction as a sum of unit fractions. Example: 2/3 = 1/2 + 1/6. How were they able to get 1/2 and 1/6? express 2/3 as a sum of unit fractions? where do you begin?

Answers:2/3 = 1/2 + x 2/3 - 1/2 = x 2/3(2/2) - 1/2(3/3) = x 4/6 - 3/6 = x 1/6 = x so 2/3 = 1/2 + 1/6

Question:Unit fractions are those with the numerator of 1. Such as 1/2, 1/3, 1/4, 1/5, and so on. The sum cannot be a mixed number only a whole number.

Answers:1/2+1/3+1/6=1

Question:what are seven different unit fractions whose sum is 1? all of the fractions have to have a numerator of one. find 7 ways.

Answers:pick a big number 90 find all divisible numbers 1 2 3 5 6 9 10 15 18 30 45 (1+3+6+15+45+2+18)/90 = 1 (1/90+1/30+1/15+1/6+1/2+1/45+1/5)=1

Question:I already know what's continued fraction but NOT discontinued fraction! also I don't understand what's sum of unlike unit fractions... :(

Answers:I am not sure what you mean by a "discontinued fraction"....thats not a term I hear very often. It would also help to know what level of math you are in, in order to put my answer in proper context. Anyways, as far as the second part goes, the wonderful thing about fractions is that they can always be written as the sum of two smnaller fractions. My guess is that you are looking for something that looks like: 1/a + 1/b = 5/12 right? These would bot the to unique fractions. So, lets look at it at the most basic of levels...what gives us 5/12? well 1/12 and 4/12...2/12 and 3/12 these are both "good sets" as they will all reduce down into unit fractions. 1st set: 1/12 + 4/12 is the same as 1/12 + 1/3 = 5/12 2ns set: 2/12 + 3/12 is the same as 1/6 + 1/4 = 5/12 Personally, I would go with the second set, as all three fractions have a different denominator. Hope this helps a bit Good luck

From Youtube

Fractions of a Sum Word Problem (S19)+- :Learn how to solve a fractions GMAT word problem with help from Dr. Steve.

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