The Counting System in Ancient Egypt - Case Study Example

Egyptian Fractions al Affiliation The Egyptian counting system The counting system in ancient Egypt was based on decimals. The decimals were not positional, but they could be used to solve problems dealing with large numbers. The Egyptians had no apparent way of constructing arbitrary large numbers. Their numbering system contained unique symbols for 1, 10, 100, 1,000, 10,000, 100,000, and 1,000,000. The Egyptians added numbers by grouping and regrouping them. They multiplied or divided numbers basing on binary values. They wrote fractions in the form of unit fractions except 2/3 and 3/4. The Egyptian geometry revolved around the areas of objects, volumes and similarity (Mathworld.com). Ancient Egyptian numerals Numeral Value | 1 || 2 ||| 3 || || 4 ||| || 5 ||| ||| 6 |||| ||| 7 |||| |||| 8 ||| ||| ||| 9 10 20 (Kubarski 2012) Egyptian fractions An Egyptian fraction is any fraction that can be expressed as the sum of separate unit fractions. The Egyptian fractions are of the form  where n is an integer. The ancient Egyptians used to write these fractions in specially-designed papyrus reeds. They used to squash and press the reeds into long scrolls of paper that could easily be carried. The scrolls were then preserved by drying in the sun. The ancient Egyptians prepared most of the scrolls available today as early as 2000 BCE. Historians predict that the preparation of these scrolls coincided with the construction of some of the largest Egyptian pyramids. The coincidence, therefore, is a likely indication that the Egyptians might have required some mathematical principles in the construction of the pyramids. The intense heat of the sun in Egyptian deserts helped in preserving the scrolls for long periods (Clawson, 1999; Knott, n.d). One of the ancient Egyptian papyrus scrolls is available in the British Museum, in London. The scroll is popularly known as Rhind Mathematical Papyrus (RMP). The scroll was named after Alexander Henry Rhind, a Scotsman who brought it from Egypt in 1858. Rhind bought the papyrus scroll at a market in Luxor, Egypt. The Egyptian sellers of the scroll had discovered it at the tomb in Thebes. The scroll later found its way to the British Museum after the death of Rhind. The Egyptian fractions consist of separate units that can be written as. It is imperative to take into account that the numerator is always 1. The denominators will vary depending on the sum of the denominators of individual unit fractions. The examples below illustrate this description. ,  and  are some of the unit fractions. In order to write a fraction such as  in Egyptian form, the unit fractions, and  are added, i.e.  To write another fraction such as  in Egyptian form, the unit fractions,  and  are added, i.e.  (Kubarski 2012) The Egyptian fractions in most cases required the exclusion of repeated terms. For example,  cannot be expressed as  in terms of the unit fractions. Instead,  should be represented as  . In order to write a fraction in terms of Egyptian fractions, the following identity is used.  Decomposition of fractions The RMP contains a table of fractions of the form  where n represents odd integers between 3 and 101. The Egyptians used the table to decompose the fraction  into its respective unit fractions. They chose to have the decomposition of only odd numbers because the numerator, 2, simplifies an even denominator. The ancient Egyptians did not have a system of rational fractions in the current form . They represented the division of p by q as the product of p and . The Egyptians then wrote p as sums of 2s and 1. For example,  is written as follows. 5 = 2 + 2 + 1, and  is the product of (2 + 2 + 1) and  i.e.   and can then be decomposed to its particular fractional units as follows.  The Egyptian fraction in terms of rational fractions, thus, takes the following form.  . The ancient Egyptians used to decompose large ordinary fractions into smaller unit fractions. In most cases, they preferred using halves in the decomposition. The following decomposition illustrates this explanation (Mathbuffalo.edu; Knott, n.d).  There is no researcher who has ever come up with a satisfactory explanation of the criteria used in decomposing fractions. There is, however, a widely-accepted formula that can be used to determine the values in the Rhind Table.  There is a difficulty that arises when the formula above is used to decompose ordinary fractions. The formula fails to decompose fractions into three or more distinct unit fractions (Knott, n.d). Attempts of finding satisfactory decomposing criteria Some researchers have been trying to come up with other methods that can explain better the criteria used by Egyptians to decompose fractions. One notable researcher in this field is Dorsett. Dorsett published details of finding the decompositions of 2/n into unit fractions. Although the method does not give a uniquely determined result, the end decompositions match with the Rhind Table. The procedure is to find an odd number o and another number p. The two numbers o and p must satisfy the following requirements. n + o = 2p and  When the chosen values of o and p satisfy the two conditions, o is then decomposed into decreasing divisors of the chosen p. Some ancient and modern uses of Egyptian fractions The ancient Egyptians used the fractions to perform practical divisions of quantities in real life. An example is the division of five loaves of bread among eight people. In a rational fraction, the division can be represented as . If an electronic calculator is used to divide the two quantities, the result is 0.625. To obtain 0.625 of a loaf of bread requires sophisticated measuring instruments. The total mass of all the loaves of bread is obtained. The total mass is then divided by eight so that everyone gets an equal share. In ancient times, sophisticated measuring instruments were not available. The Egyptians, therefore, might have developed the system of fractions to solve fractional problems. To divide five loaves among eight people is simple. One needs to proceed as follows. Step1. Divide each of the four loaves into two equal parts. Step2. Divide the remaining loaf into eight equal parts. In the Egyptian system of fractions  is written as follows.  But  Thus,  If the Egyptian fraction system is used in dividing the loaves, each person gets  and an extra  of the loaves of bread (Mathworld.com). The ancient Egyptian fractions can be used to compare two fractions. For example, when comparing  and , the usual method is to convert the two fractions into decimals. When the two fractions are converted to decimals, one obtains 0.75 and 0.7778 respectively. It verifies that  is greater than . In the ancient Egyptian system,  and . It is clear from this illustration that  is greater than  by . Multiple representations of a fraction In the Egyptian fraction system, there are different ways in which the same fraction can be represented. Consider a fraction such as . In the Egyptian fractions system,  can be represented as in the following way.  … (1) Consider another equation  When equation (2) is divided by 4, the result is  … (3) Substituting equation (3) in (1) gives:  … (4) When equation (3) is divided by 6, the following equation is obtained.  … (5) When equation (5) is substituted in equation (4), the fraction  changes to:  … (6) It is evident from the computations above that  can be written in infinite different ways. To obtain a new form, the last term of the Egyptian equation is substituted with its equivalent unit fractions. Greedy algorithms All ordinary fractions can be written in the form of Egyptian unit fractions. The procedure is shown below. Consider an ordinary fraction such as . Divide 1050 by 521 The highest fraction that can be taken from  must have a denominator that is greater than 2.0153. The most appropriate value, therefore, is 3. A third is, therefore, taken from  i.e.  To get the value of R,  is subtracted from  i.e.  Divide 350 by 57 The most appropriate value to be taken from  is  because 7 > 6.1403 The new fraction becomes  Subtract  by   Divide 350 by 7 The next fraction that can be taken out of  is  i.e.  Therefore,  It is therefore true that any ordinary fraction can be expressed in terms of unit fractions of Egyptian fraction forms (Mathworld.com). Proof of Greedy algorithms Any ordinary fraction usually has the following Egyptian form.  Where u1 4.25 Therefore  To get the value of R,  is subtracted from   85 is divided by 3 in order to get the next value of R, i.e. 85 ÷ 3 = 28.333 The most appropriate value to be taken from  is  because 29 > 28.333 Thus, the new fractional equation becomes  A different value, such as , is chosen instead of choosing  i.e.  To get the value of R,  is subtracted from   Thus,  becomes  The fraction, , has been written in shortest Egyptian form. The fraction can also be simplified further as shown below.  can be expressed as  (Mathworld.com). The smallest number of cuts Consider 10 loaves of bread that should be divided among 12 people. Each person will get 10/12 of a loaf. In order to obtain 10/12 of a loaf, each of the 10 loaves must be divided into equal pieces. The pieces can be obtained by practically cutting the loaves equally. The total number of cuts will be 11 x 10 = 110. To divide a loaf into 12 equal parts, 11 cuts are needed. 11 cuts x 10 loaves = 110 cuts. In the Egyptian fraction system, the approach is slightly different. Six out of the ten loaves are cut into two equal parts each. The remaining four loaves are cut into three equal parts each. The total number of cuts will be 6 + (2 x 4) = 14. The cutting can be expressed mathematically as follows. To divide a loaf into 2 equal pieces, 1 cut is needed. 1 cut x 6 loaves = 6 cuts Similarly, to divide a loaf into 3 equal pieces, 2 cuts are needed. 2 cuts x 4 loaves = 8 cuts The total number of cuts will be 6 + 8 = 14 Consider a general situation where there are N people. There are a large number of loaves that should be divided so that each person gets p/q of a loaf. The total pieces of loaves that will be distributed are computed as follows. Total number of pieces = Total number of people x the fraction that an individual can get i.e. Total pieces = N x p/q If the Egyptian fractional approach is used, all the loaves will be grouped into smaller categories. All the loaves from each category will be cut into q1 parts, q2 parts, and qn distinct parts depending on the category. From the first category, an individual will get 1/q1 of a loaf. In the second category, the individual will get 1/q2 of a loaf. Similarly, in the nth category, an individual will get 1/qn of a loaf. First category:  of a loaf Second category:  of a loaf nth category:  of a loaf In order to divide any loaf into qi equal pieces, (qi – 1) cuts should be made. The total number of cuts will be obtained from the following equation.  But  The total number of cuts, thus, can be determined by the following equation.  The average number of cuts per person will be determined by the following equation.  (Kosheleva and Kreinovich, n.d). Erdös-Straus Conjecture and the Egyptian fractions for 4/n Fractions such as 4/13 and 4/5 cannot be written in terms of two unit fractions. Instead, they are written as the sum of three distinct unit fractions. In a general fraction such as 4/n, there are some clues for predicting the nature of the fractional forms. If n is an even number, the following formula is true.  If n is a multiple of 3, the following generalization is true.  The following formula is also true for all multiples of 3.  (Mathworld.com). One of the problems associated with these equations is the repetition of values. In most cases, the ancient Egyptians used to avoid cases of repetitive values of unit fractions. Nevertheless, the three equations are useful because they are easy to manipulate. Consider a situation where the solution of 4/n has been found. In order to obtain another solution of 4/nk, the value of k is just multiplied by n. The only values that cannot be predicted are those of prime numbers. Designing an auxiliary algorithm  An algorithm  is built over n by induction. For n = 1, consider the following equation.  The only conditions that can make the equation above to be true is when  and . It means that the only fractions having  are of the type . Consider another algorithm that can be used to confirm if . In order to construct an algorithm , the interpretation of  should be analyzed. The condition  means that any given fraction  can be represented as the sum of  unit fractions.  for some  It can also be assumed that  Therefore  for all  and hence  But   becomes  Also  The numbers of integers  are finite in the interval  Also   can be represented as . The algorithms above can produce the corresponding representation of r and also compute the value of  (Kosheleva and Kreinovich). Determining the smallest number of cuts In order to determine the smallest number of cuts using algorithms, consider the following algorithm in the form of a fraction.  Applying the auxiliary algorithm , it is clear that 5 does not divide 4 without a remainder. It implies that  i.e.  To check whether , algorithm  is used. The following inequality must be satisfied.  The inequality  reduces to . The only integer between 1.25 and 2.5 is 2. The value of  is 2, i.e. . Using algorithm,     But  is not a unit fraction  and hence  To check whether, algorithm  is used. The following inequality must be satisfied by.  The inequality  reduces to. The only integers between 1.25 and 3.75 are 2 and 3. The corresponding integers can be represented as  and. When algorithm is used, the following results are obtained.   When, the difference  When algorithm  is applied to this difference, the following inequality must be satisfied. , This inequality reduces to. The only integers within the range are 4, 5 and 6. When 4 is chosen,  =  But  is a unit fraction It implies that. The corresponding representation is. But The representation above implies that. The fraction  can then be represented in the following way.  (Kosheleva and Kreinovich). Modern applications of Egyptian fractions Some of the most difficult things in elementary and middle school mathematics are fraction concepts. Students should master fractions by avoiding numerous misconceptions that arise in the learning of fraction concepts. In order to increase students’ interest in mathematics, researchers have developed interesting ways of teaching fractions. A commonly used strategy is adopting the Egyptian fractional system. The Egyptian structure has easy-to-understand strategies. Learners of mathematics, particularly fractions, find these techniques simple. When analyzing the Egyptian fractions, an individual can manipulate a simple arithmetic to obtain complex algorithms. The system can be applied in future computer programming because of the common similarities. The unit fractions can help in finding faster algorithms that can be applied in programming. A computer solves an equation such as p/q as the product of p and 1/q, i.e. the computer calculates 1/q then multiplies the result by p.  The computation above is similar to the ancient Egyptian fraction computation. A new approach to programming, therefore, might develop from the study of unit fractions. The Egyptian fractions were characterized by a unit numerator and denominator with a different value other than 1. Because of these restrictions on the choice of the numerator, the Egyptian fractions system did not evolve a lot (Clawson 1999). The ancient Egyptians applied their fractions mostly in solving practical problems in real life. There was no theoretical advancement or analysis of mathematical concepts among the Egyptians. For example, they calculated the areas of land under cultivation along the Nile Valley. The ancient Egyptians determined the areas of those because of taxation purposes. The Egyptians might have also applied the knowledge of fractions in dividing quantities in markets accurately. References Clawson, C. C. (1999). Mathematical sorcery: revealing the secrets of numbers. New York: Plenum. Egyptian Fraction. (n.d.). Wolfram MathWorld. Retrieved May 14, 2014, from http://mathworld.wolfram.com/EgyptianFraction.html Knott, R. (n.d.). Egyptian Fractions. MathsSurrey. Retrieved May 14, 2014, from http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html Kosheleva, O., & Kreinovich, V. (n.d.). Egyptian fractions revisited. University of Texas. Retrieved May 14, 2014, from http://www.cs.utep.edu/vladik/2005/tr05-01.pdf Kubarski, J. (2012, August 28). Egyptian fractions and their modern continuation. Topology Atlas. Retrieved May 14, 2014, from http://at.yorku.ca/cgi-bin/abstract/cbfr-84 Rhind 2/n table - Mathematicians of the African Diaspora. (n.d.). MathBuffalo. Retrieved May 14, 2014, from http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptroll2-n.html Read More

n + o = 2p and  When the chosen values of o and p satisfy the two conditions, o is then decomposed into decreasing divisors of the chosen p. Some ancient and modern uses of Egyptian fractions The ancient Egyptians used the fractions to perform practical divisions of quantities in real life. An example is the division of five loaves of bread among eight people. In a rational fraction, the division can be represented as . If an electronic calculator is used to divide the two quantities, the result is 0.625. To obtain 0.

625 of a loaf of bread requires sophisticated measuring instruments. The total mass of all the loaves of bread is obtained. The total mass is then divided by eight so that everyone gets an equal share. In ancient times, sophisticated measuring instruments were not available. The Egyptians, therefore, might have developed the system of fractions to solve fractional problems. To divide five loaves among eight people is simple. One needs to proceed as follows. Step1. Divide each of the four loaves into two equal parts. Step2. Divide the remaining loaf into eight equal parts.

In the Egyptian system of fractions  is written as follows.  But  Thus,  If the Egyptian fraction system is used in dividing the loaves, each person gets  and an extra  of the loaves of bread (Mathworld.com). The ancient Egyptian fractions can be used to compare two fractions. For example, when comparing  and , the usual method is to convert the two fractions into decimals. When the two fractions are converted to decimals, one obtains 0.75 and 0.7778 respectively. It verifies that  is greater than .

In the ancient Egyptian system,  and . It is clear from this illustration that  is greater than  by . Multiple representations of a fraction In the Egyptian fraction system, there are different ways in which the same fraction can be represented. Consider a fraction such as . In the Egyptian fractions system,  can be represented as in the following way.  … (1) Consider another equation  When equation (2) is divided by 4, the result is  … (3) Substituting equation (3) in (1) gives:  … (4) When equation (3) is divided by 6, the following equation is obtained.

 … (5) When equation (5) is substituted in equation (4), the fraction  changes to:  … (6) It is evident from the computations above that  can be written in infinite different ways. To obtain a new form, the last term of the Egyptian equation is substituted with its equivalent unit fractions. Greedy algorithms All ordinary fractions can be written in the form of Egyptian unit fractions. The procedure is shown below. Consider an ordinary fraction such as . Divide 1050 by 521 The highest fraction that can be taken from  must have a denominator that is greater than 2.0153. The most appropriate value, therefore, is 3.

A third is, therefore, taken from  i.e.  To get the value of R,  is subtracted from  i.e.  Divide 350 by 57 The most appropriate value to be taken from  is  because 7 > 6.1403 The new fraction becomes  Subtract  by   Divide 350 by 7 The next fraction that can be taken out of  is  i.e.  Therefore,  It is therefore true that any ordinary fraction can be expressed in terms of unit fractions of Egyptian fraction forms (Mathworld.com). Proof of Greedy algorithms Any ordinary fraction usually has the following Egyptian form.

 Where u1

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