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!!!!

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: Zu Chongzhi used a 24576- gon 3.141929265

The History of pi Definition of Pi Ratio of

The history of 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 This is what Bernoulli was trying to solve when he discovered e

Negative Numbers

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 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 Negative – 0 = negative Positive – 0 = positive 0 – 0 = 0 0 – negative = positive 0 – positive = negative

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 300 CE: Diophantus wrote Arithmetica 4 = 4x + 20 “Absurd result” Why would problems arising from Geometry cause Greeks to ignore negative numbers?

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

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.”

Potential Infinity vs Actual Infinity

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