Expert answer:Ib MATH IA (IB diploma)

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Introduction
The Egyptian fraction is a notation that was developed in Egypt in the middle kingdom,
giving an alteration to the old kingdom’s numeration of the eye of Horus. The five early texts
that the Egyptian fractions appear include:
•
Egyptian Mathematical Leather Roll
•
Moscow Mathematical Papyrus
•
Reisner Papyrus
•
Kahun Papyrus
•
Akhmim Wooden Tablet
•
The Rhind Mathematical Papyrus later introduced an improvement in the ways of
writing Egyptian fractions.
Ahmes wrote the Rhind papyrus dating from the Second Intermediate Period. The Rhind
papyrus includes a table of Egyptian fraction expansions for the rational numbers (2/n) and the
84-word problems. The solutions to each of the issues are written out in scribal shorthand. The
final answers are expressed in Egyptian fraction notation.
Notation
EGYPTIAN FRACTIONS
2
The hieroglyph was written on top of the number to imply a reciprocal of the figure.
Similarly, in the hieratic script, they drew a line over the letter representing the number.
The Egyptians had special symbols for 1/2, 2/3, and 3/4 that were used to reduce the size
of numbers higher than 1/2 after the conversion to the series of an Egyptian fraction. The
remainder after subtracting one of the appropriate portions got written incorporating the sum of
distinct unit fractions according to the usual Egyptian portion notation. Egyptians also used a
modified notation from the Old Kingdom as an alternative to denote a particular set of portions
in the form: 1/2k (for k = 1, 2, and 6) and their sums which are necessarily dyadic rational
numbers. These have been called “Horus-Eye fractions” after a theory that they were based on
the parts of the Eye of Horus symbol.
The sums were used in the Middle Kingdom in conjunction with the following notation
for the Egyptian fractions to subdividing a hekat. The hekat is the primary ancient Egyptian
volume for which the Egyptians used to measure for grains and other small quantities of mass, as
described in the Akhmim Wooden Tablet. If any remainder was left after expressing an amount
in Eye of Horus fractions of a hekat, the rest got written through the usual Egyptian proportion
notation as ro multiples, to which a unit is equal to 1/320 of a hekat.
Calculation methods
Modern mathematical historians, in the attempt to discover the techniques the Egyptians
used in calculating with Egyptian fractions they have studied the Rhind papyrus and other
ancient sources. In particular, the study in this area has been concentrated on the understanding
of the tables of expansions for numbers with the form 2/n in the Rhind papyrus. The methods
EGYPTIAN FRACTIONS
3
used by the Egyptians do not correspond directly to the identities although the extensions get
described as algebraic identities.
Also, the expansions from the table match none of the identities, but rather, varying
identities do match the expansions for the prime and the composite denominators. Many
identities fit the numbers of each type:
•
For the small odd prime denominators p, the expansion 2/p = 2/(p + 1) + 2/p(p + 1)
was put into use.
• An expansion 2/p = 1/A + (2A − p)/Ap got used for larger prime denominators, where
‘A’ is a practical number with many divisors between p/2 and p. The rest of the term (2A − p)/Ap
got expanded through the representation of the number (2A − p)/Ap as a summation of divisors
of ‘A’ and formed a fraction d/Ap for each of such divisors of d in this sum. For example, the
Ahmes’ expansion; 1/24 + 1/111 + 1/296 for 2/37 fits the pattern with the equation: A = 24 and
the other (2A − p)/Ap = 11 = 3 + 8, because 1/24 + 1/111 + 1/296 = 1/24 + 3/(24 × 37) + 8/(24 ×
37).
Different expansions may exist of this type for a given p. However, the observation by K.
S. Brown, the expansion that got chosen by the Egyptians was the one that caused the biggest
denominator to be much small as possible, of all expansions that fit the pattern.
• For the composite denominators, that are factored as p×q, and can be expanded to 2/pq
through the use of the identity 2/pq = 1/aq + 1/apq, where a = (p+1)/2. In the application for this
method for PQ = 21 gives p = 3, q = 7, and a = (3+1)/2 = 2, giving a produce to the expansion as
2/21 = 1/14 + 1/42 as derived from the Rhind papyrus.
EGYPTIAN FRACTIONS
4
Some of the authors preferred to write the expansion as being 2/A × A/PQ, such that A =
p+1. Replacing the second term of the product by p/PQ + 1/pq, the application of the distributive
law to the product, and then simplifying the issue leads to an expression that is equivalent to the
first expansion. The method appeared to be used in the Rhind papyrus, for the many composite
numbers, with the exceptions of 2/35, 2/91, as well as 2/95.
• 2/pq can also be expanded as, 1/pr + 1/qr, where r = (p+q)/2. For instance, the
expansion by Ahmes of 2/35 = 1/30 + 1/42, such that p = 5, q = 7, and that r = (5+7)/2 = 6.
According to Eves 1953, the scribes used the general forms of the expansion, n/PQ = 1/pr + 1/qr,
such that r = (p + q)/n, that often works in the case when p + q gets to be a multiple of n.
• For some other composite denominators, the expansion for 2/pq has the form of
expansion for 2/q along with each denominator multiplied by ‘p.’ For example, 95=5×19, and
2/19 = 1/12 + 1/76 + 1/114, can can also be determined through the method for prime numbers
with A = 12, and thus 2/95 = 1/(5×12) + 1/(5×76) + 1/(5×114) = 1/60 + 1/380 + 1/570. The
expression would be simplified as 1/380 + 1/570 = 1/228, but the Rhind papyrus uses the
simplified form.
•
The Rhind papyrus’ final prime expansion, 2/101, do not fit any of the forms but
rather uses the expansion 2/p = 1/p + 1/2p + 1/3p + 1/6p that can get applied regardless of the pvalue, That is, 2/101 = 1/101 + 1/202 + 1/303 + 1/606. A related expansion got incorporated in
the Egyptian Mathematical Leather Roll for various cases.
An Egyptian fraction is a sum of definite, distinct unit proportions. The famous Rhind
papyrus is dated to around 1650 BC and contains a table of representations of Egyptian portions
EGYPTIAN FRACTIONS
5
for odd between 5 and 101. The reason the Egyptians chose this method for representing
fractions is not apparent though André Weil named the decision as “a wrong turn” according to
Hoffman 1998.
The different fraction was 2/3 that the Egyptians did not represent using unit fractions
(Wells 1986, p. 29). Egyptian portions are always required to exclude repeated terms, since their
representations are trivial. Any rational number has representations as an Egyptian fraction with
arbitrarily many conditions, and with considerable denominators, for a given fixed number of
terms, there are only finitely many. Fibonacci proved that any portion could get represented as a
summation of distinct unit proportions (Hoffman 1998). Identity can be used to construct an
infinite chain of unit fractions.
According to Martin 1999, he demonstrated that for every positive rational figure, an
Egyptian fraction exists and whose largest denominator is at most and the denominators form a
definite proportion of the integers up to for sufficiently large. Each portion with odd has an
Egyptian fraction in which each denominator is odd (Breusch 1954; Guy 1994, p. 160).
There is no algorithm yet known for producing representations of a unit fraction, which
have the least number of terms or smallest possible denominator in accordance to Hoffman 1998,
p. 155. However, there are some algorithms (including the binary remainder method, continued
fraction unit fraction algorithm, generalized remainder method, greedy algorithm, reverse greedy
algorithm, multiple small ways, and splitting algorithm) for decomposing an arbitrary fraction
into unit fractions.
EGYPTIAN FRACTIONS
6
In the year 1202, a construction of unit fraction representations an algorithm got
developed by Fibonacci, and the algorithm was subsequently rediscovered by Sylvester
(Hoffman 1998, p. 154; Martin 1999).
The number of terms in the representations include 1, 1, 2, 1, 1, 2, 1, 2, 2, 3, 1, … (OEIS
A050205). The least denominators for the solo representation are provided by; 2, 3, 2, 4, 2, 2, 5,
3, 2, 2, 6, 3, 2, … (OEIS A050206), while the highest denominators are; 2, 3, 6, 4, 2, 4, 5, 15, 10,
20, 6, 3, 2, … (OEIS A050210).
THEOREM
Every rational number gets regarded as an Egyptian number. The current proof of the
Theorem got discovered in the year 1880. The European’s know to computing the Egyptian
numbers since the times of Fibonacci in the 12th century. Before the exhibition of the rule, we
make a convention. Given a non-integer ‘r,’ let ‘r’ stand for the least integer > r. And suppose p/q
< 1 gets written in the lowest terms then there is an Egyptian fraction with at most p terms and whose sum is p/q. Proof: Let r/s be equal to p/q -1/[q/p]. And so p/q = 1/[q/p] + r/s. If r=1, examples include: Consider that p/q = 4/23. And since 23/4 = 5.65, [23/4] = 6. Computing 4/23 - 1/6 as a fraction, gets 1/138, Thus, 4/23 = 1/6 + 1/138. Consider that 5/22. [22/5] = 5. 5/22 - 1/5 = 3/110. The problem at hand [110/3] = 37, though 3/110 - 1/37 = 1/4070, and so 5/22 = 1/5 + 1/37 + 1/4070. EGYPTIAN FRACTIONS 7 With less ingenuity, we can determine various Egyptian proportions whose sum is 5/22. For example, starting with 1/6 in the place of 1/5, 5/22 - 1/6 = 2/33. [33/2] = 17 and 2/33 - 1/17 = 1/561 and thus, 5/22 = 1/6 + 1/17 + 1/561. In the final expression for 5/22, keep 1/6 but exchange 1/17 for 1/22; This implies that we compute 2/33 - 1/22 = 1/66. Thus, 5/22 = 1/6 + 1/22 + 1/66. The expression is more satisfactory since the denominators are not as large as in the preceding two cases. Proof of the theorem It is noticed that when 0 < p < q are integers with only one as a standard divisor that is, p/q < 1 is written in lowest terms. The construction in the 2nd paragraph after the theorem gives, via mathematical induction, the proof of the theory, since for p[q/p]-q < p. Despite the existence of the theorem, there have been fewer interests in Egyptian fractions neither the modernized version of the continued portions today, but only the infinite series. A topic of the elementary calculus would imply their passage, (recalling from the calculus e-2 = 1/2 + 1/6 + 1/24 + 1/120 + 1/720 + ...). However, there exist exciting and relevant problems. The Egyptians wrote that 1/3 + 1/15 for our 2/5; in fact, the Ahmes scroll contains a table of decomposed odd fractions of the form 2/n where n ranges from 3 to 101. Some of the examples include: 2/5 = 1/3 + 1/5 2/7 = 1/4 + 1/28 2/9 = 1/6 + 1/18 EGYPTIAN FRACTIONS 8 2/15 = 1/10 + 1/30 2/17 = 1/12 + 1/51 + 1/68 2/101 = 1/101 + 1/202 + 1/303 + 1/606 Any fraction that has an odd denominator can get represented as a finite sum of unit fractions, each having a bizarre denominator (Starke 1952, Breusch 1954). Graham proved that infinitely many portions with a specific range could be represented as a sum of unit’s fractions with square denominators (Hoffman 1998, p. 156). Paul Erdős and E. G. Straus have conjectured that the Diophantine equation always can be solved, an assertion is sometimes known as the Erdős-Straus conjecture, and Sierpiński (1956) speculated that could be addressed (Guy 1994).The harmonizing number is never an integer except for. This result was proved in 1915 by Leisinger and the more general results that Kürschák demonstrated any number of consecutive terms not necessarily starting with one never sum to an integer in 1918 (Hoffman 1998, p. 157). In the year 1932, Erdős determined that the summation of the reciprocals of any number of equally spaced integers is never reciprocal. The nontrivial sets of integers are known whose reciprocals sum too small integers. For example, there exists a collection of 366 positive integers (with maximum 992) whose sum of reciprocals is precisely 2 (Mackenzie 1997; Martin). EGYPTIAN FRACTIONS 9 Reference Aguilar, M., Cavasonza, L. A., Alpat, B., Ambrosi, G., Arruda, L., Attig, N., ... & Barrau, A. (2016). Antiproton flux, antiproton-to-proton flux ratio, and properties of elementary
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Anshel, Michael M.; Goldfeld, Dorian (1991), “Partitions, Egyptian fractions, and free products
of finite abelian groups”, Proceedings of the American Mathematical Society, 111 (4):
889–899, doi:10.1090/S0002-9939-1991-1065083-1, MR 1065083
Beeckmans, L. (1993), “The splitting algorithm for Egyptian fractions”, Journal of Number
Theory, 43 (2): 173–185, doi:10.1006/jnth.1993.1015, MR 1207497
LaMotte, L. R., Roe, A. L., Wells, J. D., & Higley, L. G. (2017). A statistical method to
construct confidence sets on carrion insect age from development stage. Journal of
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Wilder, R. L. (2013). Evolution of mathematical concepts: An elementary study. Courier
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