Negative Numbers.

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Presentation transcript:

Negative Numbers

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

Negative Numbers 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 YOUR TURN – Write an example that illustrates each rule on your white board

Negative numbers in greece Ignored and Neglected by Greeks On your white board: Mathematics in Greece established through Geometry 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 6 forms of linears and quadratics Acknowledged Brahmagupta Heaviily influenced by the Greeks Called Negative Results “meaningless”

Arabian mathematics – cont’d His contribution to math Al-Samawal (1130 – 1180 CE) Produced statements regarding algebra “If we subtract a positive number from an empty power, the same negative number remains” Also had statements about products and quotients al-Samawal is said to have been developing algebra of polynomials Which one of the statements to the left suggests this? What do you think he was trying to say?

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) Italian Summa Double Entry Book-Keeping 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 that 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 Brahmagupta: http://www.storyofmathematics.com/indian_brahmagupta.html The History of Mathematics: http://www.math.tamu.edu/~dallen/masters/hist_frame.htm Negative Numbers: http://people.cst.cmich.edu/piate1kl/mth_553_f07/negative_numbers_small.pdf MacTutor History of Mathematics: http://www-history.mcs.st-and.ac.uk

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