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Irrational Numbers

Irrational Numbers. When it came to measuring quantities in dissimilar vessels, such a proportion could only be found by finding a unit of measure by which both vessels could be measured as a whole number  Anthyphairesis. Anthyphairesis. GO TO MATH HISTORY LESSON TO SEE PROCESS!!!!.

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Irrational Numbers

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  1. Irrational Numbers

  2. When it came to measuring quantities in dissimilar vessels, such a proportion could only be found by finding a unit of measure by which both vessels could be measured as a whole number  Anthyphairesis

  3. Anthyphairesis • GO TO MATH HISTORY LESSON TO SEE PROCESS!!!!

  4. InComMensurability • Egyptiona and Babylonians calculated square roots • These were approximated • Not appreciated • Hippasus of Metapontum • Credited for discovering Irrationals • Died for revealing the discovery

  5. InComMesurability • First recorded proof that is irrational • Euclid’s Elements • Here is the most popular proof

  6. The History of pi Approximation of Pi • 1650 BC: Rhind Papyrus x = 3.16045 • 950 BC Temple of Solomon: π = 3

  7. The History of pi Approximation of Pi • 250 BC: Archimedes 3.1418 • 150 CE: Ptolemy used a 360 – gon • 3.14166 • 263 CE: Liu Hiu used a 192 regular inscribed polygon • 3.14159 • 480 CE: ZuChongzhi used a 24576-gon • 3.141929265

  8. The History of pi Definition of Pi • Ratio of

  9. The history of This is what Bernoulli was trying to solve when he discovered e • Sometimes known as Euler’s constant. • The first references to “e” were in the appendix of a work by John Napier • The discovery of the constant itself is credited to Jacob Bernoulli

  10. Negative Numbers

  11. Chinese Mathematics • 200 BCE: Chinese Rod System • Commercial calculations • Red rods cancelled black rods • Amount Sold: Positive • Amount Spent: Negative

  12. Negative Numbers in India • Brahmagupta – 7th Century Mathematician • 1st wrote of negative numbers • Zero already had a value • Developed rules for negative numbers • Developed the Integers we know

  13. Arithmetic rules with Integers Brahmagupta’s work Translation to modern day Negative – 0 = negative Positive – 0 = positive 0 – 0 = 0 0 – negative = positive 0 – positive = negative • A debt minus zero is a debt • A fortune minus zero is a fortune • Zero minus zero is zero • A debt subtracted from zero is a fortune • A fortune subtracted from zero is a debt

  14. Arithmetic rules with Integers – cont’d Brahmagupta’s work • A product of zero multiplied by a debt or fortune is zero • The product of zero multiplied by zero is zero • The product or quotient of two fortunes is a fortune • The product or quotient of two debts is a fortune • The product or quotient of a debt and a fortune is a debt • The product or quotient of a fortune and a debt is a debt

  15. Negative numbers in greece Ignored and Neglected by Greeks Why would problems arising from Geometry cause Greeks to ignore negative numbers? • 300 CE: Diophantus wrote Arithmetica • 4 = 4x + 20 • “Absurd result”

  16. Arabian mathematics Also ignored negatives • Al-Khwarizami’s Algebra book – • 780 CE • Acknowledged Brahmagupta • Heaviily influenced by the Greeks • Called Negative Results “meaningless”

  17. Arabian mathematics – cont’d Al-Samaw’al (1130 – 1180 CE) His contribution to math al-Samawal is said to have been developing algebra of polynomials He introduced decimals, well before its appearance in Europe • Shining Book of Calculations • Produced statements regarding algebra • Had no difficulty handling negative expressions

  18. Al-Samawal’s Algebra • If we subtract a positive number from an ‘empty power’, the same negative number remains. • If we subtract the negative number from an ‘empty power’, the same positive number remains. • The product of a negative number by a positive number is negative, and be a negative number is positive.

  19. European mathematics • 15th century • Arabs brought negatives to Europe • Translated ancient Islamic and Byzantine texts • Spurred solutions to quadratics and cubics

  20. European mathematics • Luca Pacioli (1445 – 1517) • Summa de arithmetica, geometria • Double Entry Book-Keeping • He kept the use of negatives alive • John Wallis ( 1616-1703) • English • Invented Number Line

  21. European mathematics • 1758: Francis Maseres • British “ (negative numbers) darken the very whole doctrines of the equations and made dark the things which are in their nature excessively obvious and simple”

  22. European mathematics • 1770: Euler • Swiss “Since negative numbers may be considered as debts ... We say that negative numbers are less than nothing. Thus, when a man has nothing of his own, and owes 50 crowns, it is certain that he has 50 crowns less than nothing; though if any were to make a present of 50 crowns to pay his debt, he would still have nothing, though really richer than before.”

  23. Potential Infinity vsActual Infinity

  24. SOURCES History of Negative Numbers: http://nrich.maths.org/5961 https://brilliant.org/discussions/thread/discovery-of-irrational-numbers/ https://www.math.rutgers.edu/~cherlin/History/Papers2000/wilson.html MacTutor History of Mathematics: http://www-history.mcs.st-and.ac.uk

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