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Delve into the intriguing history of irrational numbers, from the Egyptian and Babylonian approximate calculations to Hippasus of Metapontum's daring revelation. Learn about the first recorded proof of irrationality in Euclid's Elements and the fascinating journey of approximating the value of pi throughout history. Discover the origins of the constant "e" and the development of rules for negative numbers in different ancient civilizations, including China and India. Explore the European mathematical advancements in handling negative numbers, from Luca Pacioli's accounting methods to the conceptualization of negativity by Euler. Uncover the complexities of potential infinity versus actual infinity in the realm of mathematics. Dive into this rich history of numbers, from their mysterious origins to their profound impact on the development of mathematical concepts worldwide.
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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!!!!
InComMensurability • Egyptiona and Babylonians calculated square roots • These were approximated • Not appreciated • Hippasus of Metapontum • Credited for discovering Irrationals • Died for revealing the discovery
InComMesurability • First recorded proof that is irrational • Euclid’s Elements • Here is the most popular proof
The History of pi Approximation of Pi • 1650 BC: Rhind Papyrus x = 3.16045 • 950 BC Temple of Solomon: π = 3
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
The History of pi Definition of Pi • Ratio of
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
Chinese Mathematics • 200 BCE: Chinese Rod System • Commercial calculations • Red rods cancelled black rods • Amount Sold: Positive • Amount Spent: Negative
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
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
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
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”
Arabian mathematics Also ignored negatives • Al-Khwarizami’s Algebra book – • 780 CE • Acknowledged Brahmagupta • Heaviily influenced by the Greeks • Called Negative Results “meaningless”
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
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.
European mathematics • 15th century • Arabs brought negatives to Europe • Translated ancient Islamic and Byzantine texts • Spurred solutions to quadratics and cubics
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
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”
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.”
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
