ANCIENT MATH

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ANCIENT MATH
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The origins of mathematics

• The Egyptians had a calendar as early as 4800 BC
• In 4200 BC their mathematics and astronomy
  produced a 365 day calendar
• Taking taxes and maintaining a large army
  required some mathematics

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Egyptian numerals

• Egyptian first practice the scientific arts
• Number system was not well suited for
  arithmetical calculations, so the Egyptians had
  extended their use of numbers
• One of the earliest examples of Egyptian writing
  were the hieroglyphs

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Egyptian counting with heiroglyphs
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Egyptian ''Pi''

• The Egyptian value of ''pi'' was 3.16, which is
  much closer to the modern 3.14
• The Egyptian mathematicians probably used
  measurement, experimentation and a theoretical
  analysis of ''squaring a circle'' to obtain such
  an accurate of ''pi''

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Addition Subtraction

• Simple processes using the glyphs

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Multiplication Division
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Egyptian fractions

• Ancient Egyptians used a number system based on
  unit fractions fractions with one in the
  numerator
• E.g. 2/7 ¼ 1/28

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Egyptian mathematics papyri

• Two primary sources
• - Rhind(or Ahmes) Papyrus
• - Moscow Papyrus
• Secondary sources
• - Egyptian Mathematical Leather Roll
• - Berlin Papyrus
• - Reisner Papyrus

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Rhind Papyri

• Contains 112 mathematics problems and solutions
  e.g.
• - Simple equations problem 24
• - Geometric series simultaneous equations
  problem 40
• - Determining p problem 50

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Moscow Papyrus ( 1850 BC)

• Contains mathematics problems (simple equations)
  and solutions
• Line by line translation problem 10

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• Example of calculating the surface area of a
  basket hemisphere.

• You are given a hemisphere with a mouth
  magnitude

• of 4 1/2 in diameter.

• What is its surface?

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• Take 1/9 of of 9 since

• the basket is half an egg hemisphere. You get
  1.

• Calculate the remainder when subtracted from 9
  which is 8.

• Calculate 1/9 of 8.

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• You get 2/3 1/6 1/18.

• Find the remainder of this 8

• After subtracting 2/3 1/6 1/18. You get 7
  1/9.

• Multiply 7 1/9 by 4 1/2.

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• You get 32. Behold this is its surface area!

• You have found it correctly.

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Secondary sources

• The Egyptian Mathematical Leather Roll ( 1850
  BC)
• Table consisting 26 decompositions into unit
  fractions
• The Berlin Papyrus (1800 BC)
• It contains, among other things, two problems in
  simultaneous equations
• The Reisner Papyri (1800 BC)
• It consists of four fragments of rolls containing
  calculations of volume of temples

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Astronomy

• 24 hour division of the day
• Three different system for astronomical reference
• - The zodiac by the Mesopotamians
• - The lunar mansions in India
• - The decans in Egypt
• Decan - 10 intervals led to the division of
  night into 12 equal parts and ultimately the 24
  hour day

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Calendar

• Appearance of the brightest star in the sky after
  the period when it is too close to the sun to be
  seen
• For Sirius this occurs in July and this was
  taken to be star of the year because the Nile
  flooded shortly after this
• The civil year was divided into 12 months of
  thirty days 5 days

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Pyramids

• Famous beccause of precise mathematical
  calculations used in their constructions
• Construction of the Cheops pyramid
• - It was built with the shape of a square
• - In the middle of the construction was circle
  shaped wall

• Two points ( place where Venus showed up above
  the wall and the place where it disappeared
  again) formed an angle of which the bisector
  pointed directly north

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Mathematical interpretation

• Pyramid of Cheops is in a cyrcle divided in 7
  segments
• 3 corners correspond approximately with 3 corners
  of the formed heptagon

• Sarcophagus in the King's room
• - internal volume 1,12 m³
• - external volume 2,24 m³
• - ratio of these numbers is almost the same as
  the ratio of the distance of Mars Sun and Earth
  Sun
• - the angle of the corridor to the Queen's room
  is 27

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Sumerian math

• By about 3000 BC, the Sumerians were drawing
  images of tokens on clay tablets

• They made a metrological numeration system
  (system of weights and measures)

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Babylonian math

• During the third and second millenia BC, along
  the river valleys of the Tigris and Euphrates was
  situated Babylonia

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Babylonian numerals

• There's no zero ? both 60 and 1 look the same
• The most common fractions, ½, 1/3, ¼, 1/5, 1/6
  are represented by a single number (1/2 ltltlt)

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Babylonian mathematical tables

• Reciprocal tables a/b a(1/b)
• Tables of squares ab (ab)²-a²-b²/2
• ab (ab)²-(a-b)²/4
• Square and cube roots two methods of
  approximating square roots
• - Square root of (a²b) approximate is ab/2a
• - aa1 is initial approximation if a1 lt square
  root of 2 then 2/a1 gt square root of 2

• Quadratic equations from ax² bx c using
  substitation y ax they got x² q px
• n³ n² table solving cubic equation ax³ bx²
  c gives the equation y³ y² ca²/b³
• Exponentials logarithms

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• The Yale tablet YBC 7289
• - It consists of a tablet on which a diagram
  appears

• Plimpton 322
• - 4 columns with 15 rows
• - First column is (c/h)²
• - In every row the square of the number c in
  column 2 is a perfect square, c² - b² h²

• The Susa tablet
• -The problem is to find the radius of
• the circle through the three vertices
• The Tell Dhibayi tablet
• - It asks for the sides of a rectangle whose
  area and diagonal are known

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Weights and Measures 60s everywhere!

• Calendar with a year of 360 days, 12 months of 30
  days each
• Circle was divided into 360 degrees

Pythagoras' Theorem a Thousand Year before
Pythagoras

• Some of the clay tablets discovered contain lists
  of triplets of numbers, starting with (3, 4, 5)
  and (5, 12, 13) which are the lengths of sides of
  right-angled triangles, obeying Pythagoras' "sums
  of squares" formula

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Made by

• Bingulac Suzana
• Ivkovic Anita
• Kosanovic Sanja
• Prpic Vedrana
• imic Ivana