Foundations of mathematics

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babylonian and egyptian math

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Foundations of mathematics

  1. 1. Babylonian And Egyptian Mathematics
  2. 2. The Babylonians lived in Mesopotamia, a fertile plainbetween the Tigris and Euphrates rivers.
  3. 3.  The Babylonian civilization has its roots dating to 3500BCE with the Sumerians in Mesopotamia. This was an advanced civilization building cities and supporting the people with irrigation systems, a legal system, administration, and even a postal service The Greeks called this land “Mesopotamia,” meaning “the land between the rivers.” Most of it today is part of the modern state of Iraq, although both the Tigris and the Euphrates rise in Turkey.
  4. 4. Babylonian Cuneiform Script
  5. 5.  Shortly after 3000 B.C., the Babylonians developed a system of writing from “pictographs”—a kind of picture writing much like hieroglyphics. Whereas the Egyptians used pen and ink to keep their records, the Babylonians used first a reed , later a stylus with a triangular end. Because the Latin word for “wedge” is cuneus, the resulting style of writing has become known as “cuneiform.” Triangular Reed end
  6. 6.  Cuneiform script was a natural consequence of the choice of clay as a writing medium. The stylus did not allow for drawing curved lines, so all pictographic symbols had to be composed of wedges oriented in different ways: vertical , horizontal oblique.
  7. 7.  Their mathematical notation was positional but sexagesimal. They used no zero. More general fractions, though not all fractions, were admitted. They could extract square roots. They could solve linear systems. They worked with Pythagorean triples. They solved cubic equations with the help of tables. They studied circular measurement. Their geometry was sometimes incorrect.
  8. 8.  The Babylonian scale of enumeration was not decimal, but sexagesimal (60 as a base), so that every place a “digit” is moved to the left increases its value by a factor of 60.When whole numbers are represented in the sexagesimal system, the last space is reserved for the numbers from 1 to 59, the next-to-last space for the multiples of 60, preceded by multiples of 60², and so on.
  9. 9. For example, the Babylonian 3 25 4 might stand for the number3 · 60² + 25 · 60 + 4 = 12,304and not3 · 10³ + 25 · 10 + 4 = 3254,as in our decimal (base 10) system.But the question is, how did they find out about the base sixty numbers?????
  10. 10.  It was confirmed by twotablets found in 1854 at Senkerah on the Euphrates by the English geologist W. K. Loftus. These tablets, which probably date from the period of Hammurabi (2000 B.C.), give the squares of all integers from 1 to 59 and their cubes as far as that of 32.
  11. 11. The tablet of squaresreads easily up to 7², or 49. Where weshould expect to find 64, the tablet gives 1 4; the only thingthat makes sense is to let 1 stand for 60. Following 8², thevalue of 9² is listed as 1 21, implying again that the left digitmust represent 60. The same scheme is followed throughoutthe table until we come to the last entry, which is 58 1,this cannot but mean: 58 1 = 58 · 60 + 1 = 3481 = 59².
  12. 12. But the question now is, how were they able to identify the translation of the given encryption???
  13. 13.  The simple upright wedge had the value 1 and could be used nine times, while the broad sideways wedge stood for 10 and could be used up to five times. When both symbols were used, those indicating tens appeared to the left of those for ones, as in
  14. 14.  Appropriate spacing between tight groups of symbols corresponded to descending powers of 60, read from left to right. As an illustration, we have which could be interpreted as 1 · 603 + 28 · 602 + 52 · 60 + 20 = 319,940.
  15. 15.  The Babylonians occasionally relieved the awkwardness of their system by using a subtractive sign . It permitted writing such numbers as 19 in the form 20 − 1,instead of using a tens symbol followed by nine units:
  16. 16.  Babylonian positional notation in its earliest development lent itself to conflicting interpretations because there was no symbol for zero. There was no way to distinguish between the numbers 1 · 60 + 24 = 84 and 1 · 602 + 0 · 60 + 24 = 3624, since each was represented in cuneiform by
  17. 17.  Because of this problem in the positional system, in 300 B.C. a new symbol was developed called he placeholder represented by or
  18. 18.  With this, the number 84 was readily distinguishablefrom 3624, the latter being represented by
  19. 19.  The absence of zero signs at the ends of numbers meant that there was no way of telling whether the lowest place was a unit, a multiple of 60 or 60², or even a multiple of 1/60 . The value of the symbol 2 24 in cuneiformcould be 2 · 60 + 24 = 144.but other interpretations are possible, for instance, 2 · 60² + 24 · 60 = 8640,or if intended as a fraction, 2 + 24/60 = 2/25 .
  20. 20. The square root of √2, the length of the diagonal of a unitsquare was approximated by the babylonians of the OldBabylonian Period (1900 B.C.-1650 B.C.) as 24 51 10 305471: 24:51:10 1 2 3 1.414212..... 60 60 60 21600
  21. 21. Thus, the Babylonians of antiquity never achievedan absolute positional system. Their numericalrepresentation expressed the relative order of thedigits, and context alone decided the magnitude of asexagesimally written number; since the base was solarge, it was usually evident what value was intended.
  22. 22. Hieroglyphic Representation of Numbers
  23. 23.  Civilisation reached a high level in Egypt at an early period. The country was well suited for the people, with a fertile land thanks to the river Nile yet with a pleasing climate. It was also a country which was easily defended having few natural neighbours to attack it for the surrounding deserts provided a natural barrier to invading forces. As a consequence Egypt enjoyed long periods of peace when society advanced rapidly.
  24. 24.  By 3000 BC two earlier nations had joined to form a single Egyptian nation under a single ruler. Agriculture had been developed making heavy use of the regular wet and dry periods of the year. The Nile flooded during the rainy season providing fertile land which complex irrigation systems made fertile for growing crops. Knowing when the rainy season was about to arrive was vital and the study of astronomy developed to provide calendar information.
  25. 25.  Hieroglyphs are little pictures representing words. It is easy to see how they would denote the word "bird" by a little picture of a bird but clearly without further development this system of writing cannot represent many word. The Egyptians had a bases 10 system of hieroglyphs for numerals. By this we mean that they has separate symbols for one unit, one ten, one hundred, one thousand, one ten thousand, one hundred thousand, and one million.
  26. 26. 1. The RhindMathematical Papyrus named for A.H.Rhind (1833-1863) who purchased it at Luxor in 1858. Origin: 1650 BCE but it was written very much earlier. It is 18 feet long and13 inches wide. It is also called the Ahmes Papyrus after the scribe that last copied it. The Moscow Mathematical Papyrus purchased by V. S. Golenishchev (d. 1947). Origin: 1700 BC. It is 15 ft long and 3 inches wide. Two sections of this chapter offer highlights from these papyri.
  27. 27.  Multiplication is basically binary. Example Multiply: 47 × 24 47 × 24 47 1 94 2 188 4 376 8* 752 16 *  Selecting 8 and 16 (i.e. 8 + 16 = 24), we have 24 = 16 + 8 47 × 24 = 47 × (16 + 8) = 752 + 376 = 1128
  28. 28.  Although the Egyptians had symbols for numbers, they had no generally uniform notation for arithmetical operations. In the case of the famous Rhind Papyrus (dating about 1650 B.C.),the scribe did represent addition and subtraction by the hieroglyphs and , which resemble the legs of a person coming and going.
  29. 29.  The symbol for unit fractions was a flattened oval above the denominator. In fact, this oval was the sign used by the Egyptians for the mouth . For ordinary fractions, we have the following. 1 1 3 1 7 24
  30. 30.  There were special symbols for the fractions 1/2 , 2/3 , 3/4, of whichone each of the forms is shown below. 1 2 3 2 3 4
  31. 31.  Burton, David (2007) The History of Mathematics: An Introduction, Sixth Edition, Page 12-28 MacTutor, Babylonian and Egyptian Numerals http://en.wikipedia.org/wiki/Babylonian_numerals

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