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

• 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

Translation to modern day

• Negative – 0 = negative

• Positive – 0 = positive

• 0 – 0 = 0

• 0 – negative = positive

• 0 – positive = negative

ARITHMETIC RULES WITH INTEGERS – CONT’DBrahmagupta’s work

1. A product of zero multiplied by a debt or fortune is zero

2. The product of zero multiplied by zero is zero

3. The product or quotient of two fortunes is a fortune

4. The product or quotient of two debts is a fortune

5. The product or quotient of a debt and a fortune is a debt

6. 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 • Mathematics in Greece

established through Geometry

• 300 CE: Diophantus wrote Arithmetica• 4 = 4x + 20 • “Absurd result”

On your white board:

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

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

His contribution to mathal-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.”

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

SOURCES