set b egyptian mathematics

7/30/2019 Set B Egyptian Mathematics

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Set B

Egyptian mathmatics


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History of Egyptian mathmatics

Egyptian mathematics was developed and used in Ancient Egypt from ca. 3000 BCto ca. 300 BC.

Written evidence of the use of mathematics dates back to at least 3000 BC withthe ivory labels found at Tomb Uj at Abydos. These labels appear to have beenused as tags for grave goods and some are inscribed with numbers. [

Further evidence of the use of the base 10 number system can be found on for

instance the Narmer Macehead which depicts offerings of 400,000 oxen,1,422,000 goats and 120,000 prisoners.[

The evidence of the use of mathematics in the Old Kingdom (ca 26902180 BC) isscarce, but can be deduced from for instance inscriptions on a wall neara mastaba in Meidum which gives guidelines for the slope of the mastaba.[Thelines in the diagram are spaced at a distance of one cubit and show the use ofthat unit of measurement.[

The earliest true mathematical documents date to the 12th dynasty (ca 1990

1800BC). The Moscow Mathematical Papyrus, theEgyptian Mathematical Leather Roll,the Lahun Mathematical Papyri which are a part of the much larger collectionofKahun Papyri and the Berlin Papyrus all date to this period. The RhindMathematical Papyrus which dates to the Second Intermediate Period (ca 1650BC) is said to be based on an older mathematical text from the 12th dynasty.
http://en.wikipedia.org/wiki/Ancient_Egypthttp://en.wikipedia.org/wiki/Abydos,_Egypthttp://en.wikipedia.org/wiki/Abydos,_Egypthttp://en.wikipedia.org/wiki/Narmer_Maceheadhttp://en.wikipedia.org/wiki/Narmer_Maceheadhttp://en.wikipedia.org/wiki/Old_Kingdomhttp://en.wikipedia.org/wiki/Mastabahttp://en.wikipedia.org/wiki/Meidumhttp://en.wikipedia.org/wiki/Mastabahttp://en.wikipedia.org/wiki/Meidumhttp://en.wikipedia.org/wiki/Cubithttp://en.wikipedia.org/wiki/Ancient_Egyptian_units_of_measurementhttp://en.wikipedia.org/wiki/Cubithttp://en.wikipedia.org/wiki/Ancient_Egyptian_units_of_measurementhttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Rollhttp://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Rollhttp://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Rollhttp://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Rollhttp://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Rollhttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Rollhttp://en.wikipedia.org/wiki/Kahun_Papyrihttp://en.wikipedia.org/wiki/Kahun_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Berlin_Papyrushttp://en.wikipedia.org/wiki/Berlin_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Berlin_Papyrushttp://en.wikipedia.org/wiki/Second_Intermediate_Periodhttp://en.wikipedia.org/wiki/Second_Intermediate_Periodhttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Second_Intermediate_Periodhttp://en.wikipedia.org/wiki/Second_Intermediate_Periodhttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Berlin_Papyrushttp://en.wikipedia.org/wiki/Kahun_Papyrihttp://en.wikipedia.org/wiki/Kahun_Papyrihttp://en.wikipedia.org/wiki/Kahun_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Rollhttp://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Rollhttp://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Rollhttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Twelfth_dynasty_of_Egypthttp://en.wikipedia.org/wiki/Ancient_Egyptian_units_of_measurementhttp://en.wikipedia.org/wiki/Cubithttp://en.wikipedia.org/wiki/Meidumhttp://en.wikipedia.org/wiki/Mastabahttp://en.wikipedia.org/wiki/Old_Kingdomhttp://en.wikipedia.org/wiki/Narmer_Maceheadhttp://en.wikipedia.org/wiki/Narmer_Maceheadhttp://en.wikipedia.org/wiki/Narmer_Maceheadhttp://en.wikipedia.org/wiki/Abydos,_Egypthttp://en.wikipedia.org/wiki/Ancient_Egypt


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History of Egyptian mathmatics

The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus are so-calledmathematical problem texts. They consist of a collection of problems withsolutions. These texts may have been written by a teacher or a student engaged insolving typical mathematics problems.[

An interesting feature ofAncient Egyptian mathematics is the use of unit fractions.The Egyptians used some special notation for fractions such as and and in some

texts for , but other fractions were all written as unit fractions of the form or sumsof such unit fractions. Scribes used tables to help them work with these fractions.

The Egyptian Mathematical Leather Roll for instance is a table of unit fractionswhich are expressed as sums of other unit fractions. The Rhind MathematicalPapyrus and some of the other texts contain tables. These tables allowed thescribes to rewrite any fraction of the form as a sum of unit fractions.

During the New Kingdom (ca 15501070 BC) mathematical problems are

mentioned in the literary Papyrus Anastasi I, and the Papyrus Wilbour from thetime ofRamesses III records land measurements. In the worker's village ofDeir el-Medina several ostraca have been found that record volumes of dirt removedwhile quarrying the tombs.
http://en.wikipedia.org/wiki/Ancient_Egyptianhttp://en.wikipedia.org/wiki/Egyptian_fractionhttp://en.wikipedia.org/wiki/Egyptian_fractionhttp://en.wikipedia.org/wiki/New_Kingdomhttp://en.wikipedia.org/wiki/Ramesses_IIIhttp://en.wikipedia.org/wiki/Papyrus_Anastasi_Ihttp://en.wikipedia.org/w/index.php?title=Papyrus_Wilbour&action=edit&redlink=1http://en.wikipedia.org/wiki/Deir_el-Medinahttp://en.wikipedia.org/wiki/Deir_el-Medinahttp://en.wikipedia.org/wiki/Ramesses_IIIhttp://en.wikipedia.org/wiki/Ostraconhttp://en.wikipedia.org/wiki/Ostraconhttp://en.wikipedia.org/wiki/Deir_el-Medinahttp://en.wikipedia.org/wiki/Deir_el-Medinahttp://en.wikipedia.org/wiki/Ostraconhttp://en.wikipedia.org/wiki/Ostraconhttp://en.wikipedia.org/wiki/Deir_el-Medinahttp://en.wikipedia.org/wiki/Deir_el-Medinahttp://en.wikipedia.org/wiki/Deir_el-Medinahttp://en.wikipedia.org/wiki/Deir_el-Medinahttp://en.wikipedia.org/wiki/Deir_el-Medinahttp://en.wikipedia.org/wiki/Ramesses_IIIhttp://en.wikipedia.org/wiki/Ramesses_IIIhttp://en.wikipedia.org/wiki/Ramesses_IIIhttp://en.wikipedia.org/w/index.php?title=Papyrus_Wilbour&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Papyrus_Wilbour&action=edit&redlink=1http://en.wikipedia.org/wiki/Papyrus_Anastasi_Ihttp://en.wikipedia.org/wiki/Papyrus_Anastasi_Ihttp://en.wikipedia.org/wiki/Papyrus_Anastasi_Ihttp://en.wikipedia.org/wiki/Papyrus_Anastasi_Ihttp://en.wikipedia.org/wiki/New_Kingdomhttp://en.wikipedia.org/wiki/Egyptian_fractionhttp://en.wikipedia.org/wiki/Ancient_Egyptian


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Numerals in Egyptian mathmatics

Ancient Egyptian texts could be written ineither hieroglyphs or in Hieratic. In either representationthe number system was always given in base 10.

number 1 was depicted by a simple stroke,

number 2 was represented by two strokes, number 10 is a hobble for cattle,

number 100 is represented by a coiled rope,

number 1000 is represented by a lotus flower,

number 10,000 is represented by a finger, number 100,000 is represented by a frog

a million was represented by a god with his hands raised inadoration.
http://en.wikipedia.org/wiki/Ancient_Egyptianhttp://en.wikipedia.org/wiki/Egyptian_hieroglyphshttp://en.wikipedia.org/wiki/Egyptian_hieroglyphshttp://en.wikipedia.org/wiki/Hieratichttp://en.wikipedia.org/wiki/Egyptian_hieroglyphshttp://en.wikipedia.org/wiki/Hieratichttp://en.wikipedia.org/wiki/Hieratichttp://en.wikipedia.org/wiki/Egyptian_hieroglyphshttp://en.wikipedia.org/wiki/Ancient_Egyptianhttp://en.wikipedia.org/wiki/Ancient_Egyptian


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Hieroglyphics for Egyptian numerals


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fractions in Egyptian mathmatics

The Egyptians almost exclusively used fractions ofthe form 1/n. One notable exception is the fraction2/3 which is frequently found in the mathematicaltexts. Very rarely a special glyph was used to denote

3/4.1. The fraction 1/2 was represented by a glyph that

may have depicted a piece of linen folded in two.

2. The fraction 2/3 was represented by the glyph for a

mouth with 2 (different sized) strokes.3. The rest of the fractions were always represented

by a mouth super-imposed over a number.


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Hieroglyphics for some Egyptian

fractions


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Multiplication and division Egyptian multiplication was done by repeated doubling of the number

to be multiplied (the multiplicand), and choosing which of the doublings

to add together (essentially a form ofbinary arithmetic), a method thatlinks to the Old Kingdom. The multiplicand was written next to thefigure 1; the multiplicand was then added to itself, and the resultwritten next to the number 2. The process was continued until thedoublings gave a number greater than half of the multiplier. Then thedoubled numbers (1, 2, etc.) would be repeatedly subtracted from the

multiplier to select which of the results of the existing calculationsshould be added together to create the answer.[2]

As a short cut for larger numbers, the multiplicand can also beimmediately multiplied by 10, 100, etc.

For example, Problem 69 on the Rhind Papyrus (RMP) provides the

following illustration, as if Hieroglyphic symbols were used (rather thanthe RMP's actual hieratic script).[5]
http://en.wikipedia.org/wiki/Binary_numeral_systemhttp://en.wikipedia.org/wiki/Multiplierhttp://en.wikipedia.org/wiki/Multiplierhttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Multiplierhttp://en.wikipedia.org/wiki/Binary_numeral_system


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Algebra in Egyptian mathmatics

Egyptian algebra problems appear in both the Rhind mathematical papyrus andthe Moscow mathematical papyrus as well as several other sources.[5]

Aha in hieroglyphs Aha problems involve finding unknown quantities (referred toas Aha) if the sum of the quantity and part(s) of it are given. The RhindMathematical Papyrus also contains four of these type of problems. Problems 1,19, and 25 of the Moscow Papyrus are Aha problems. For instance problem 19 asksone to calculate a quantity taken 1 and times and added to 4 to make 10.[5] Inother words, in modern mathematical notation we are asked to solve the linearequation:

Solving these Aha problems involves a technique called Method of false position.The technique is also called the method of false assumption. The scribe wouldsubstitute an initial guess of the answer into the problem. The solution using thefalse assumption would be proportional to the actual answer, and the scribe wouldfind the answer by using this ratio.[5]

The mathematical writings show that the scribes used (least) common multiples toturn problems with fractions into problems using integers. The multiplicative

factors were often recorded in red ink and are referred to as Red auxiliarynumbers.[5]

The use of the Horus eye fractions shows some (rudimentary) knowledge ofgeometrical progression. Knowledge of arithmetic progressions is also evidentfrom the mathematical sources.[5
http://en.wikipedia.org/wiki/Moscow_mathematical_papyrushttp://en.wikipedia.org/wiki/Rhind_mathematical_papyrushttp://en.wikipedia.org/wiki/Moscow_mathematical_papyrushttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Egyptian_hieroglyphshttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Linear_equationhttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Linear_equationhttp://en.wikipedia.org/wiki/Linear_equationhttp://en.wikipedia.org/wiki/Linear_equationhttp://en.wikipedia.org/wiki/Linear_equationhttp://en.wikipedia.org/w/index.php?title=Method_of_false_position_(Egyptian)&action=edit&redlink=1http://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Red_auxiliary_numberhttp://en.wikipedia.org/wiki/Red_auxiliary_numberhttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Red_auxiliary_numberhttp://en.wikipedia.org/wiki/Red_auxiliary_numberhttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Red_auxiliary_numberhttp://en.wikipedia.org/wiki/Red_auxiliary_numberhttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/w/index.php?title=Method_of_false_position_(Egyptian)&action=edit&redlink=1http://en.wikipedia.org/wiki/Linear_equationhttp://en.wikipedia.org/wiki/Linear_equationhttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Egyptian_hieroglyphshttp://en.wikipedia.org/wiki/Egyptian_mathematicshttp://en.wikipedia.org/wiki/Moscow_mathematical_papyrushttp://en.wikipedia.org/wiki/Rhind_mathematical_papyrushttp://en.wikipedia.org/wiki/Rhind_mathematical_papyrushttp://en.wikipedia.org/wiki/Rhind_mathematical_papyrus


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Geometry in Egyptian mathmatics

1. We only have a limited number of problems

from ancient Egypt that concern geometry.

Geometric problems appear in both the Moscow

Mathematical Papyrus (MMP) and in the RhindMathematicalPapyrus (RMP).

2. The examples demonstrate that the Ancient

Egyptians knew how to compute areas of severalgeometric shapes and the volumes of cylinders

and pyramids.
http://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Ancient_Egyptianshttp://en.wikipedia.org/wiki/Ancient_Egyptianshttp://en.wikipedia.org/wiki/Ancient_Egyptianshttp://en.wikipedia.org/wiki/Ancient_Egyptianshttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrus


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Area:

Triangles: The scribes record problems computing the areaof a triangle (RMP and MMP).

Rectangles: Problems regarding the area of a rectangular

plot of land appear in the RMP and the MMP. A similarproblem appears in the Lahun Mathematical Papyri inLondon.

Circles: Problem 48 of the RMP compares the area of acircle (approximated by an octagon) and its circumscribing

square. This problem's result is used in problem 50, wherethe scribe finds the area of a round field of diameter 9 khet.

Hemisphere: Problem 10 in the MMP finds the area of ahemisphere.
http://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyri


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Volumes:

Cylindrical granaries: Several problems compute thevolume of cylindrical granaries (RMP 41-43), while problem60 RMP seems to concern a pillar or a cone instead of a

pyramid. It Is rather small and steep, with a seked (slope) offour palms (per cubit). In section IV.3 of the LahunMathematical Papyri the volume of a granary with a circularbase is found is using the same procedure as RMP 43.

Rectangular granaries: Several problems in theMoscow

Mathematical Papyrus (problem 14) and in the RhindMathematical Papyrus (numbers 44, 45, 46) compute thevolume of a rectangular granary.

Truncated pyramid (frustum): The volume of a truncated

pyramid is computed in MMP 14.
http://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Moscow_Mathematical_Papyrushttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyrihttp://en.wikipedia.org/wiki/Lahun_Mathematical_Papyri


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The Seqed

Problem 56 of the RMP indicates an understanding of theidea of geometric similarity. This problem discusses theratio run/rise, also known as the seqed. Such a formulawould be needed for building pyramids. In the nextproblem (Problem 57), the height of a pyramid is calculatedfrom the base length and the seked(Egyptian for slope),while problem 58 gives the length of the base and theheight and uses these measurements to compute theseqed. In Problem 59 part 1 computes the seqed, while the

second part may be a computation to check the answer: Ifyou construct a pyramid with base side 12 [cubits] and witha seqed of 5 palms 1 finger; what is its altitude?


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The similarities between EgyptianMathematics and Babylonian mathematics

1. The similarities between Egyptian Mathematics andBabylonian mathematics are that both cultures created aform of Algebra, and a form of Geometry.

2. Mesopotamian and Egyptian, mathematical systems

were very much alike, and the combinations of bothmath systems have created the modern systems oftoday.

3. Even though the systems were alike, what they lookedlike were very different. The Mesopotamians number

system had more triangles, where the Egyptians numbersystem has more lines and distinct shapes.


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Egyptian Mathematics

1. The Egyptians used a decimal (base 10) system with 1,000,000 as the

biggest unit.

2. They used measurements such as (9 deben or 91 grams or 3.2 ounces).

3. They did arithmetic problems such as addition and subtraction, into

which multiplication and division was made.4. Fractions were usually thought of as 2/3 and 3/4 had 195 as the

numerator: This changed to 7/32 was thought of as 1/8 + 1/16 +

1/32 (using 32 as the lowest common denomonator).

5. Geometry was basically an experiment rather than from a theory. The

theory touched an astonishing correction in its answers to a calculations.The Egyptians could calculate the area of a square, trapezoid, triangle

and a circle by squaring 8/9 of the diameter, and the height and the

angles of a pyramid. Even the volume of a cylinder and a cut off piece of

a pyramid were able to be calculated!


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Babylonian mathematics

1. The Babylonian had an advanced system than our own system. It was asystem with base 60, instead of the base 10 system of today. The basesixty is the origin of the use of time (60 seconds in a minute and sixtyminutes in an hour.

2. The Babylonians were the ones to come up with the twenty-four hourday, and sixty minutes in one hour. This way of counting has been in

existence for over four thousand years now.3. They give squares of the numbers up to fifty nine and cubes of the

numbers up to thirty two. Babylonians counting was complicatedbecause it didn't start like our system one, two, three, four, etc.., Theirsystem went one, sixty-one, three thousand and one, one hundred andfifty thousand and one, seven million five fundred thousands and one,

and three hundred and seventy five million and one etc.4. Babylonians used the formula ab= ((a+b) squared- 4- b squared)

multiplied by 14. Division is harder, and the Babylonians didn't havean algorithm for long division, instead they based their system on factsthat ab-a.


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BABYLONIAN NUMERATIONSYSTEM


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The Babylonian system was a positional,

base 60 system:

Their system had two basic symbols.

A units symbol

A tens symbol


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Here is an example of a number

written in Babylonian:

5x602 + 4x601 + 12x600 = 18000 + 240 + 12 = 18,252


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Problom solving in Egyptian

MathematicsAha Problems in Egyptian Mathematics

Aha is an egyptian word meaning quantity. Aha problems wereproblems to compute an unknown quantity. Today these problemswould be solved with algebra, but then they were solved using'false position', a rhetorical method of solving certain algebra

problems. Several Papyri containing collections of arithemetic and aha

problems, together with their solutions, have been found

Sample problem: A quantity and its fifth, added together, give 23.What is the quantity?

Solution by false position: Suppose the quantity is 5. Then add afifth to get 6. Now do the same thing to 5 and 6 until 6 turns into23. Like multiply 6 by . So the answer is 5= = .

Solution by algebra: Let x be the quantity. So x + x = 23. Solving forx gives x = 23.


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The End

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set b egyptian mathematics

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